texto

INTRODUCTION
Unexpected claims from unexpected quarters usually meet
with much resistance and often with resentment if they dis¬
regard what is widely or authoritatively believed to be true.
Attempted past contributions to knowledge are well known to
have been rejected because they conflicted with contemporary
stands, and upon eventual acceptance as of merit of what was
offered intentions have been to forestall similar adverse
responses. Yet the acceptance of new views tends again to
harden opposition to subsequent attempted contributions in
conflict with them. The previous relinquishment of a then
prevailing outlook for what seems a truer one brings about a
reluctance to consider that the later of these, too, might be cor¬
rected. This is especially true if what is offered, however of
value, does not follow current avenues of thought. Men can be
overcome by successes, perpetuating searches in those direc¬
tions to the neglect of what presents itself, in the perhaps most
common fallacy, that of undue generalization.
The opposition to the unacknowledged may intensify if what
is submitted is not merely presumed wrong, but is, even if sup¬
posed right, of determinations regarded as either unlikely or
impossible to make, as are ones propounded in this treatise.
Among determinations regarded long as unlikely or impossi¬
ble to make are the demonstration of axioms or postulates
underlying mathematics and logic, and the resolution of para¬
doxes or other incongruities held to be of the same fields. More
recently certain propositions in these fields came to be believed
demonstrated unprovable though true, and on a wider philo¬
sophical scale statements customarily referred to as metaphysi¬
cal have been thought impossible to demonstrate as true like¬
wise, if not looked upon as meaningless.
The questions in metaphysics, which has been a major phil¬
osophical branch, concern such issues as what sort of funda¬
mental entities, as may be mind contrasted with matter, exist,
and various speculative systems were in the past constructed
to supply the answers. For some time ancient mathematical
accomplishments were of extensive influence. Some thinkers
accordingly essayed corresponding methods in philosophy,
among them notably Spinoza, who built inferentially on as¬
sumed axioms in the manner of the geometry of Euclid. How¬
ever, not only did the inferences lack sufficient rigor, but the
axioms, the initial assumptions supposed true, were far less
self-evident than those of geometry. A point taken up in the
present treatise is whether science indeed must depend on
initial assumptions of fact, or whether all its presumable truths
2
should not alike be ascertained. Aside from that consideration
the progress of natural science, repeatedly overturning the
results of metaphysical speculation, gained a predominant
position for its methods of physical experiment and observation.
And truths in metaphysics came to be held not capable of proof
because they seem not amenable to these methods. It is for not
being subject to similar verification that statements were even
declared meaningless, unless of logic or mathematics. The last
mentioned exceptions have been, further, regarded as merely
tautologies, true by definition.
The tendencies can be viewed as denying the existence in
consciousness of conceptions of things not physical, not to
speak of denial of consciousness, and accordingly pronounce¬
ments like those of ethics or aesthetics, admittedly of meaning,
are considered not to be statements allowing of truth or falsity,
but expressions of other kinds, such as conveyance of feelings.
To deny consciousness itself is, to be sure, not only to forego
finding of any reality at all, since finding it means it is of con¬
sciousness, but because all things, real or otherwise, are of
recognition only be being of consciousness, without it there is
no certainty of any thing.
As regards the meaningless statements contended, the inti¬
mation is that a statement can be inferred meaningless on
determining it not to be verifiable as proposed. But a statement
is understood to be verified in accordance with its meaning,
and its meaningfulness is accordingly not pending its verifi¬
ability. Irrespectively, one can conceive of not only physical
states but what might be termed otherworldly ones, whether
they have actuality or not, and these conceptions can likewise
be made the understood meanings of statements.
Considering statements of mathematics or logic, they can,
regardless of whether of definitions, equally be held to be of
conceptual entities, as exemplified by the ideal figures of
geometry. And that they should be tautologies, supplying no
new information, is refuted by the very dependence on them
as major sources of learning.
AndlrTregard to expressions not looked upon as statements
of truth or falsity, this position does not prevent the reality again
in consciousness of feelings or other ingredients, however
expressions asserting the same be designated.
Distinguishing nevertheless between fleeting elements of
consciousness and realities of a more tangible kind, more
moderate viewpoints than the preceding have for a longer time
been that such realities can only be known by means of sense
experience, not the aid of deduction. A distinction has been
made thus between matters of fact, concerning things of the
world, and matters of reason, concerning logic and mathe-
3
matics, with the latter ones, while highly regarded, not admit¬
ted to furnish any truths about the former. In deliberation of
the usefulness of logic it was suggested that facts are at best
by it substantiated but not discovered. But substantiating a
fact can be taken to mean none other than discovering it.
^Because of the spoken of reliance on scientific method, the
sciences of which it is the province have alone been considered
by philosophers to have the competency to determine what
exists. Hence metaphysics became either frankly conjectural or
was abandoned as a fruitful enterprise. Philosophy in general
is for the same reason no longer concerned with determining
worldly facts. There appears even a prevalent notion that any¬
thing discovered about the world is at once carried to, as it is
said, its logical conclusion, although mathematics is evidence
that conclusions drawn about things, as in findings in that sci¬
ence, can be virtually limitless. In lieu of the other inquiries and
with the prospect of aiding them in other fields, philosophy thus
directs its attention largely toward clarification of concepts, the
nature of many of which, too, is believed to be, if not inde¬
terminable, indefinable.
The physical character of scientific method, however, incor¬
porates a host of presuppositions, of understandings or knowl¬
edge on which the authenticity of the observations depends,
and these understandings, possessed by everyman, are not
verified by that method. They are of metaphysical entities, of
what things have reality, as well as of deductive principles and
last brought up concepts, and their oversight, beside preventing
insights, gives rise to mentioned paradoxes. The last is remind¬
ful that inasmuch as these understandings comprise presup¬
posed knowledge they cannot, in accord with the well known
and in a coming chapter discussed law of contradiction, be
defensibly denied.
Inasmuch as they comprise knowledge they might also be
thought to represent knowledge as an innate attribute of man,
as said to be put forward by Socrates. He is storied to argue for
it by eliciting a mathematical principle through reasoning from
an unschooled boy. The knowledge presently at issue, however,
is not necessarily one that has been ever present, but that
underlies scientific or other particular knowledge about the
natural world. But as knowledge which is presupposed by other
findings, it and anything that may be deduced from it, is by
strength of implication at least equal in certainty to other find¬
ings and susceptible to exploration.
To disclose how such an exploration can be carried out, how
in general fundamental or other truths can be reflectively dem¬
onstrated and thereby an actual rather than speculative edifice
of existence revealed, is together with such demonstrations the
s
4
object of this writing. It is on investigation that, as made known
by the title, includes such subject matter as the existence of
God, which, irrespective of the arguments used, may in its
exalted objective encounter strongest opposition.
For this reason and those given earlier it is urged that in read¬
ing this exposition one set aside preconceptions and rely on
one’s apprehension regarding the determinations herein made.
The existence of God is uppermost in consideration, because
of its momentousness. But as indicated, other realities are in
like manner explored, among them, beside realities in a
customary sense, truths by which, as by logical principles or
even meanings of words, they may be determined, as well as
the reflective methods by which, as by deduction, it is possible
to determine either of these. In the process, the difficulties
associated with the seemingly indemonstrable and insoluble
matters spoken of will be found surmountable.
PROCESS OF INQUIRY
The relied on methods of inference, of reflective determina¬
tions made on the basis of what is known, have traditionally
been those of induction and deduction. Popularly stated, the
first, applying to nature, is inference of principles or laws from
constancies in it although these are conceivable otherwise, and
the second, exemplified in logic and mathematics, is similar
inference of one thing from another but with the other not con¬
ceivable without the first. This distinction may not coincide with
everyone’s. What the difference between the two kinds of infer¬
ence is, or whether there is any, has been debated. It should
suffice at this time that the difference here meant by the terms
is, more exactly than in the preceding, between, in deduction,
inference of what is intrinsic in things, in the sense that it,
although not known before, is part of the things as conceived,
and, in induction, inference of what is extrinsic to things, of
what is not of those parts. The concept of conceivability or not
may not be adequately useful, because in for instance lengthy
deductions in mathematics it is strange to say that the result is
not conceivable otherwise. And it remains later in this book to
find that both connections, the intrinsic and extrinsic, exist, how,
for the matter of that, they become deductive and inductive
knowledge.
The two connections, respective ones also called analytic and
synthetic, can as ones that are negations of each other be seen
exhaustive, in conformance with the likewise known and in a
later chapter discussed law of excluded middle. Further, since
both inductive and deductive inferences are based on unfailing
connections as to things and what is true of them, they provide
5
complete certainty. These two means of inference ore therefore
those utilized herein os well.
As apparent in the forgoing, it is not the intention here to con¬
jecture on possible realities, but to base determinations only on
criteria not wanting in certainty. The criteria used will in point
of fact be seen to exceed in certainty those of the sciences
customarily resorted to. The discovery in natural science, in the
inductive sciences, of particular laws, of invariabilities, encoun¬
ters utmost difficulties, and the scientist relies heavily on theory
for determination of fact. And, as noted, basic principles
in mathematics and logic, in the deductive sciences, have de¬
fied their ascertainment as unquestionable, and, while held
to be so inconsistently with holding them unprovable, they,
too, have been consigned to theory.
The use of theory, beside experiment and observation
another prevailing method in natural science, is, again, for that
reason respected elsewhere. It should be recognized in this
respect that theory unreserved is but a return to speculation.
Still, in sciences of the physical world theory might be of
assistance, because the very nature of an extrinsic connection
allows for unlimited possibilities, and a reasonable but tenta¬
tive judgement could be made as to the area in which to look.
The same is not the case when the connections are intrinsic.
The area is already delimited, and only what is contained in it
can be the object of a search. This does not render the search
easy, but a theory can becloud a survey of what is already in
cognition.
A theory can be a burden in other sciences as well, because
of the conviction it can carry by help of accompanying argu¬
ments. An accepted principle in reasoning has been reductio
ad absurdum, whereby if a hypothesis leads to a false con¬
clusion it is inferred false. It is tempting to think that if it leads
to a true conclusion it is true, which is fallacious as to be found.
Although the fallaciousness is not unknown to science, there is
nonetheless considerable reliance on the hypothetico-deductive method, which follows that reasoning. The point is that a
theory can rest partly on cogent arguments, as by use of mathe¬
matics, instilling certitude, and in consequence an opportunity
for other answers, of which there may be quite many, can be
hampered.
A worse yet effect of implanted beliefs is that they can be
pursued with too much zeal, as are often doctrines and ide¬
ologies and as can be the inexactly or falsely based theses of
such as behavioral sciences, so as to be deleterious to
humankind. Emphasis is given to these pitfalls of theory
since false conclusions may well be responsible for nearly all
of man-made ills.
While to surmise the truth in many situations is unavoidable
in life, it can frequently be avoided in research. In statistical
researches, often of scantest and misleading evidence,
theoretical conclusions need be drawn very far from lightly, and
in the researches engaged in in this disquisition they need, for
the opposite reason, not be drawn at all. As indicated, the
inferences in the present researches are of the certainty of, on
one hand, inductive criteria which apply to experiences more
fundamental than particular physical findings, and of, on the
other hand, deductive criteria which apply to factors that can
be found already contained in the concepts. Unquestionable
knowledge is accordingly possible in both.
In addition to these methods of inference, there is the ques¬
tion of premises, material on which conclusions, the inferences,
are based. This material might be simply referred to, as earlier
above, as that which is known. But to determine what is initially
known is by no means a simple problem, as attested by another
major branch of philosophy, epistemology.
Much of what may be considered known is itself the result of
inference, to include many of the referred to presuppositions,
which will be examined correspondingly. Accordingly, in order
to consider the material for earliest inference, it will be required
to revert to the entities of interest themselves, as represented by
their concepts, before anything, including their existence, is
inferred, whether or not the entities be spoken of as objects of
knowledge. Thus, to establish what truly may be known, the
initial task will be to determine the nature of the entities or con¬
cepts considered, a task usually identified with definition. This
procedure, the determination of the nature of the things con¬
sidered, may, in harmony with the procedures of inference, be
termed eduction. It is differentiated from definition, because
the primary concern will be apprehension of the concept,
rather than its formulation in language.
EDUCTION ^
LANGUAGE
CONCEPTS ^
<-
REALITY
>
FIGURE A
7
Figure A here correspondingly delineates this step alongside
the two of subsequent inference, as taking place in the present
investigations. Inquiry proceeds by educing beyond the
language through which they are in this and other discourse
conveyed the concepts of concern; by inducing from these con¬
cepts realities by which a world is known; and by deducing
from these realities further ones. Since realities determined, as
are other things, are treated of through both, concepts and
language, these two subjects extend beside the third to the end
of the diagram. Their reverse extensions, past that of reality
(solid line), in addition signify imaginary concepts (dashed
line), and language void of meaning (dotted line).
Whereas above the diagram the three procedures are ap¬
plied to the subjects as noted, they may be used extensively in
intermediate searches. Eduction can also be of any concept
later acquired; induction is usually from occurrences within a
reality already determined; and deduction may concern purely
conceptual entities as ground for logical or mathematical prin¬
ciples. It stands to reason that the use herein will be in such a
comprehensive manner.
Similarly, the three subjects below the diagram are desig¬
nated in the titles of the first and second chapters but are
likewise concerned in the further of the four. The third chapter
deals with principles by which one thing can be deduced from
another, the things intended to be of reality, and the chapter is
hence a furtherance of what realities may be determined, with
language and concepts, mentioned as pervasive, accordingly
also involved. And the concluding chapter, similarly proceeding
through language and concepts, is of realities that are derived
via the other determinations and are outside of what is viewed
as worldly experience.
The repleteness and range of this undertaking may appear
discouraging as a reading to some. However, while the material
may be difficult to construct, it should not present the same dif¬
ficulty in comprehension. It is, as known, often the case that the
more extensive an inquiry the simpler its outcome, and the
same proves to be the case at present.
This, too, may be contrary to expectations, for solutions to
philosophical problems in particular, because implicitly encom¬
passing, are looked to as exceptionally complex. So dominant
is this anticipation that a thesis not elaborate to the point of
being unfathomable can be refused outright, unawares that
failure of accomplishment may hide behind a veil of inscrut¬
ability. Authors are often indeed praised for the depth of their
insights, while endless scholarship is expended to estimate
what they may be.
s
8
There may also, contrariwise, be expectations of lesser mul¬
tiplicity, frequently in seeking immoderate uniformity. But in
accordance with the accustomed view of the issues, the find¬
ings mode should be seen to be of unovyoited simplicity.
The writing itself con furthermore, in the light of the volu¬
minous other writings on its subject matter, be seen to be of
substantial brevity. This because of its dependence on conclu¬
sive demonstration. The large number of supporting arguments
through which o speculative work may gain conviction con be
omitted, os con arguments that ore irrelevant, which o work
often includes.
Approochobility of the discussions may be further enhanced
by the common ownership of the ideas discussed. They reside
at the basis of everybody’s conscious existence and require no
specialized knowledge. For the some reason there is no need
for use of esoteric language, and terminology not of wide cur¬
rency is avoided.
This accessibility may likewise not be expected, and the
reminder may therefore be appropriate that one be guided not
by expectations about the nature of the findings or their cir¬
cumstance, but by one’s cognitions concerning them.
It may in addition to the above be mentioned that ports such
os chapters or sections con be read individually by individual
readers. Although premises on which conclusions there ore
based may be substantiated elsewhere, they should appear
true os o matter of course. Pertinent other ports will nonetheless
be referred to, os was done with respect to certain logical lows
in this introduction.
The reader may over and above these be attracted by the pro¬
cess of deductive discovery, os readers hove been for centuries
regarding the geometry demonstrations in Euclid’s Elements, or
in recent times regarding the detective exploits of Sherlock
Holmes and other fictional investigators.
The considerations listed should moke acquaintance with
much of what follows feasible for every contemplative reader.
Chapter I
LANGUAGE AND
CONCEPTS
9
Section 1
THE SIGNIFICATION
OF LANGUAGE
The presence of language in man’s pursuit of knowledge is
critical, since language is almost the only form in which all
known, as well as assumed, is set down and communicated.
Accordingly words are often viewed as inherently standing for
certain entities, and it is endeavored to discover what the entity
represented by a given word is.
What constitutes life, for example, is a matter of appreciable
debate; as an illustration, the Encyclopedia Britannica offers
five alternative and protracted definitions. As another example,
light has been progressively redefined, in the confidence that
it is in the course described more accurately; instead of for
instance defining it as, initially understood, visual sensation, it
would be defined as electromagnetic radiation of wavelengths
beginning and ending with, in contradiction of the first mean¬
ing, invisible ultraviolet and infrared.
Language as used by man does not presume undiscovered
meanings, hawever, which man is to investigate. Language is
by definition, as would be said, and in conformance with the
beginning of this chapter a form of conveyance of things in the
awareness of man, who thereby assigns the meaning to the
words.
The preceding statement, as can any other, may afford the
first occasion for disputation in the body of this treatise. What
linguistic expressions are about or mean is a matter of exten¬
sive theory, known as semantics. Notwithstanding a frequent
acknowledgement that the meaning of words is decided by
consent of its users, it is maintained that it has a more remote
inception. It is likened to government, which likewise may have
s
10
a more obscure beginning than a sometimes supposed social
contract. However, the origin of a meaning is not relevant, as
need not be the origin of a government. Men do agree or not
to use the meaning, or to abide by the form of government.
As was indicated in the introduction, of account with regard
to language herein is that the inquiry centers on entities how¬
ever worded, not, in a linguistic study, on the words. That is to
say, what is herein of account is not the initial meaning of the
language used, nor what other meaning it might have, but
what is wished to be expressed by it. It is here thus dependent
on agreement or decision what the words mean. The same can
be taken to be the case in regard to the customary philosoph¬
ical or comparable search for the meaning of expressions. The
search is not for the meaning of words as used by someone
else, in which case the issue might be considered a linguistic
inquiry into a foreign or obsolete language. The search is for
the meaning of words as used by the inquirers themselves, as
happens with ‘life” as used by physiologists, or “light” as used
by physicists. In that event, however, the inquirers know the
meaning, which is the one intended by them.
The truth of this inference, as of any, would still be chal¬
lenged, and a frequent query concerns the understanding of
language as conveyor of things in man’s awareness, in man’s
consciousness. The contention is that, at one extreme, abstract
words like “and” and, at another extreme, concrete words like
“chair” do not designate things residing in consciousness, aside
from the mentioned view that there is no consciousness. It was
also mentioned that all things are of recognition by being of
consciousness, which is to say more generally that anything
that might be considered, whether something merely con¬
templated or of a determination like a decision or discovery, is
so by manifestation in consciousness. Thus an abstract word like
“and”, although not designating an independent entity, is
represented conceptually by the presence of more than one
thing in some circumstance, and a concrete word like “chair”,
although designating an object viewed as separate from the
perception of it, is represented conceptually by that perception.
The words are used to designate those entities in accordance
with those conceptual representations. The representations in
awareness need not even be what are held to be direct percep¬
tions of the objects. The objects one speaks of may only be
thought of, and in the case of hypothetical ones they further¬
more need not exist although the words used for them, not
referring to the thought of them, would refer only to existing
ones. What matters is that one has a conception of the thing
referred to, perhaps indirectly, rather than using words empty
of such content.
11
This issue is not the present one, however, which is the
awareness that certain words stand for certain things, rather
than the awareness of those things. It is similarly beside the
point to question whether that awareness of what the words
mean can be said to be knowledge. That the users of words
have knowledge of what they mean was asserted in the above
inference, and inasmuch as previous to it only such as consent
to the meaning by the users was spoken of, it might be argued
that its knowledge is not the consequence. The argument may
be advanced because of the alluded to belief that knowledge
can only be acquired through sense experience. It concerns the
synthetic connections spoken of as contrasted with analytical
ones and held alone to comprise facts, compared to the sup¬
posed definitional truth of the other. Someone may therefore
propose that definitions do not constitute knowledge, and that
as a result the above determination does not hold. But as sug¬
gested regarding statements supposed meaningless or neither
true nor false, a decision over language, as presently whether
something may be termed knowledge, does not dispose of
underlying matters. The users of a language were said above
to know their meaning because the of concern understood
awareness of it is normally referred to as knowledge. And it
may be specified here correspondingly that by knowledge of
a meaning is meant awareness of it, instead of awareness of its
assumption, however the awareness be obtained, as by one’s
own definition. It may be remarked that by awareness and
hence knowledge is not meant present consciousness alone of
a thing, but its mentioned being sometime considered, in the
sense that it may be stored in memory. And the question is
whether the users of a language do in this sense not know their
meaning, accordingly seeking it, or whether they do.
The affirmative reply should by the foregoing be obvious, the
meant knowledge consisting in the very having decided the
use, and it should be observed that the inference is made about
other persons by induction. The elucidation of how the induc¬
tion is made can be postponed, however, because the com¬
prehension altogether of this writing, as of the writing or other
forms of communication by others, presupposes, is proof, that
the induction has been performed. It has been learned that
others convey meanings through language in the same man¬
ner as oneself does. As a matter of fact, since a form of lan¬
guage is generally not devised by oneself, one uses it in that
manner because one learns that that is the usage by others.
This writer thus has likewise induced that language is used
in this manner by others, and the induction of understandings
held in common extends farther. It concerns also any observa¬
tions herein made through the language, observations which
were noted to be of basic matters in man’s life. The observations
can be of the mentioned educed entities themselves which
man has cognition of, as well as of the inferred facts about
them as brought out in this treatise. In all of this is implicit of
course that the induction as pertains to language is not only
that it is used by men alike for conveyance of their meaning,
but that particular meanings are likewise held in common.
All particular meanings, it is clear, need not be understood
by every speaker of the language. Language is an inconstant
thing, differing not only with communities, but with individ¬
uals. Of pertinence at present is that while the hearer of a
language need in instances be informed of its meaning, the
user need not, but knows it.
The recognition of this knowledge can be of estimable value
to thinkers, since corresponding calling forth of what entity is
named provides a basis for further knowledge about it. The like
took place in inferring that since consciousness is meant to
extend to all objects of man’s consideration, all meanings man
considers for words are of consciousness. The recognition that
a meaning is known can also dissuade from supposing a dif¬
ferent one, to thereby wrongly infer a fact about, ironically, the
originally meant. Such inference takes place in the frequent
argument that if the self is held to include the body then the
destruction of one’s body is the destruction of oneself, although
the self first meant with respect to immortality is one dissoci¬
ated from the body.
The presence of knowledge of the meaning of the words
man uses does not imply that a verbal definition, or even a
suitable eduction in thought alone, is an easy task if not a pro¬
hibitive one. One often thinks of only familiar instances of what
is named, and as with other knowledge a taxing of memory
may be required. The knowledge does imply, however, that the
search for knowledge of various fundamental entities, which
has been a concern of philosophy, is misconceived. It has been
continually sought to know what entities like virtue are, with
volumes sometimes devoted to the questions, without cogni¬
zance that, insofar as the concern is not with an unfamiliar
language, what the entities named are is known to the seekers
in accordance with their sometime deciding.
These mistaken searches for knowledge already possessed
are perpetuated among leading schools by phenomenology
and analytic philosophy, in their main enterprises of investigat¬
ing essences or concepts. It is the investigation mentioned in
the introduction as having replaced former ones in philosophy,
so as to furnish information for others elsewhere. But in concord
with observations in the same place and by the above, pos¬
session of that information belongs to the understandings pre-
13
supposed in science as in all human activity making use of
language.
Accordingly, as regards linguistic meaning the formulation of
theory is not only not required, similarly to the situation in
logic, but it is misplaced. In deduction, though conclusive an¬
swers may be reflectively available, they may be difficult to
attain and theory the only substitute. But what the meaning of
words is has been determined, and theory but infringes on it.
Be determinations dependent thus on definition or on other
factors, once any have been appropriately made they cannot,
lest engaging in contradiction again, be controverted by argu¬
ment. It has as remarked of Socrates (page 3, fourth para¬
graph), assuredly, been put forward that man possesses what
is sometimes called a priori knowledge, knowledge that
precedes any determination. It is contrasted with a posteriori
knowledge, knowledge derived from experience, and it should
be explained that it is meant here in the sense in which it
excludes any finding, physical or reflective. Thus certain logical
principles are held to be known without confirmation. But
something proposed to be true, as would be a theory in any
field, becomes by definition admittedly known to be true only
if it is found to be so.
The preceding speaks again of knowledge, but with respect
to more than meaning, and its implied definition as including
a certainty would be questioned. It would be maintained that
the knowledge one has before findings need not have the cer¬
tainty attached to things afterward. But the issue is exactly the
certainty that can be attached to things. By knowledge here is
meant, as can be held to be its common meaning, the com¬
plete certainty connected with having determined that some¬
thing is the case. Otherwise a fact would not be held known,
but merely perhaps quite certain. In other words, in following
through from the definition of knowledge of a meaning (page
11, first paragraph), by knowledge in an inclusive sense is meant
awareness of something actual, in whatever the form by which
something is meant to be so, instead of awareness of something
assumed, however probable the thing. Among the assumptions
are meant to be beliefs, which accordingly, howsoever infused
with conviction, fall short of knowledge. The definition of
knowledge as justified true belief, frequent in philosophy, is
accordingly inadequate. Objections to it are of the contention
that it is not apparent what makes a belief justified. But the justi¬
fication can be characterized independently, by again the par¬
ticular way by which something is meant to be fact. Justification
for a belief only, however, is insufficient because of its very
limitation-by its meaning, even though the belief may be true.
In the light of the foregoing it should upon appropriate dem-
s
14
onstration of something not be requisite to address opposing
arguments. If therefore such arguments are in this exposition
intermittently surveyed it is because they may be well known
or their discussion instructive. In the one case the inaccuracy
may not be discerned, with men laboring under misapprehen¬
sion, and in the other case additional observations can be
made, supplementing primary ones.
With reference to linguistic meaning, theories dealing with
it as a whole tend correspondingly not to investigate the mean¬
ing of individual expressions, but what sort of things are meant
by them in general. Proposals on individual meanings will in
this treatise in the main be considered when they come into
question. As to what sort of things are meant by words in gen¬
eral, any attempt to particularize those things, since they are up
to the speakers, is as misapplied as are theories about indi¬
vidual meanings. The issue has been moreover confounded
with the question of what is sometimes phrased as the mean¬
ing of meaning. It is a question concerning an individual word
again, rather than words in general.
The phrase ‘”meaning of meaning” is of course begging the
question, since, contrary to the intimation, if what “meaning”
stands for is not known at the end of the phrase it is not known
at its start, the purport of the whole not understood. But the
question being, in using here a presumably understood
synonym, what “meaning” stands for, the situation is such that,
in asserting to try to know what words in general mean, one
cannot make up one’s mind about what one wishes to say by
“mean” or other words used in the process.
Furthermore in the questioning of particular expressions
others, less accustomed ones, are often substituted for them,
forgetting that since unusual, they are more difficult yet to com¬
prehend. In place of thus asking “What does this word mean?”
it is asked “What is it to say this word?”, and the second
phraseology is evidently the one unclear to the usual hearer.
Rather than responding with a definition as customarily
desired, one might try to assist in pronunciation. How by replac¬
ing in these attempts other words as well beside “meaning”
language is, contrary to intention, not clarified but obfuscated,
is illustrated with respect to speaking of a person as “one”, as
done at the end of the preceding paragraph. The notion being
that the second and third use of the word in a like sentence can
be taken to speak of someone other than the first does, writers
would replace them by “his” and “he”. But by usage the inter¬
pretation is the opposite. The substitutions signify thus a reading
into words meanings at odds with those agreed to.
That the word “meaning” itself be in this way held ambigu¬
ous is interestingly attributable to a supposed twofold significa-
15
tion of words in general, regardless of whether the signification
be called meaning or not. The duality does not concern an
expression’s two what are generally called senses, but two
kinds of things an expression is in, inconsistently, a single sense
thought to designate. In further inconsistency as observed,
since what expressions designate is arbitrary, they cannot all
designate the same particular kind of thing, let alone two.
That they are believed to can be viewed as due to the cogni¬
tion that while language can designate worldly objects, it is
made to do so by reference to conceptual representations of
them. Approximately corresponding to these conceptual and
worldly interpretations, the respective two signfications sup¬
posed of words or composites of them have been variously
named intension and extension, connotation and denotation,
or sense and reference. The last mentioned and most recent
division is argued by adducing the difference in expressions by
which a given thing can be referred to. Accordingly it is con¬
tended that whereas “the morning star” and “the evening star”
both refer to the planet Venus, they cannot have the same
meaning. If they did, it is maintained, then by knowing the
meaning of both, one would know that they refer to the same
thing, when in fact the sameness is an astronomic discovery.
The specific argument can again be seen a matter of defini¬
tion. “Refer to” and “mean” can be considered synonyms, and
by “morning star” and “evening star” is in fact respectively said
to be meant a certain star seen at sunrise and one seen at
sunset. That is to say, the star itself, not the circumstance of its
observation, is said to be meant, and despite thus meaning in
both cases the same thing, one may not knovy that the same
star is at issue.
To hold that meaning only concerns a certain character of the
thing referred to, not the thing itself, also contradicts the pur¬
ported dual aspect of meaning spoken of in the same regard,
by which one of the aspects concerns the thing itself. This dual¬
ity further, whether termed one of meaning or not, does not
take place in the above designations of the planet. While they
concern it as wholes, their parts, those of the composite desig¬
nations specifically, concern other things, ones connected with
it. One and the same unit does not have two kinds of meaning.
The different meanings regarding the two phrases do not con¬
cern their wholes but their parts, simply because different
things are named by them. The duality is for the same reasons
not true of complete sentences, to which it was also proposed
to apply.
What a sentence, a declarative one in particular, as a unit
refers to is, naturally, also subject to debate. The referent is
usually said to be a proposition, and it is not always clear what
s
16
that word refers to either. The issue is again one of consent, and
o declarative sentence, herein in short called a statement, is,
alike with other linguistic units used in more than one sense.
In its widest use a statement can be held to be of a straightfor¬
ward assertion of fact.
By an assertion, to be clear, is not again meant a statement,
but its content, the discussed object of consciousness. That con¬
tent of a statement is, worded otherwise, the internal ascertain¬
ment, the affirmation, of the state described in it. As an object
of consciousness the affirmation can be characterized as a tak¬
ing into cognizance the actuality of that state. The affirmation
may have been made at any time past though conveyed by a
present statement. A statement, it should be noted, does not,
contrary to some contention, stand for the state itself, which is
done by a phrase. The state in ‘”Birds fly” is referred to by “the
flight of birds” but whereas the first expression, the statement,
signifies the affirmation of the state, the second expression,
the phrase, signifies the state, as if its affirmation had been
granted.
Statements can, certainly, also be falsehoods, in which case
they may signify intent to delude that the stated is true, or un¬
warranted certitude of it. Statements are, further, frequently
used to signify suppositions regarding which something is to be
demonstrated, furnishing ground for speaking of them as about
propositions. A discourse may set out to demonstrate the truth
of a proposition, as done in mathematics, or to assume it for the
sake of argument, as done in logic. When its substantiation is
conclusive, the proposition can in truth be regarded as an
above assertion of fact, and when it is not, or the proposition
merely serves as a postulate, then the sentence stating it in¬
forms in effect tacitly that the supposed fact is a supposed one.
Of interest is that sentences and other word combinations
have, as do words, in single senses, redundantly, single mean¬
ings. And the word “meaning” can as others also be used in
more than one sense, by a sense understood, as suggested by
in the last sentence mentioned redundancy, a meaning as in
the preceding explicated. What is of pertinence is that when
two meanings, e.g. connotation and denotation, are said to be
associated with expressions, said is that associated with them
are two things, that may happen to be called by the same
name. But of interest in the linguistic inquiry is a particular
sense of meaning, which word it will be seen can be used in
more than two senses, as can more than two things be
associated with a word. One can associate with a word its pro¬
nunciation, as remarked, its etymology, and so forth. And one
can associate with it in connection with the thing named
various things known about the thing. The point is that in the
17
case of for instance the planet Venus, one can associate with
it things about it such as being the morning or evening star,
without those things being what is meant by its designation.
It is in fact because different utterances speak of the same
thing that something what is called informative is said about it.
This thought was in the arguments for two meanings expressed
by saying that whereas “A equals A” is uninformative, “A
equals 6″ is not. Hence, it was argued, A cannot mean the
same thing 6 does. And by the present sense of “”mean” it is
owing to the very identity that the same thing is meant. What
is not the same is the name, the description, or perhaps the
unspoken particular attribute considered. Only in the last case,
the other cases assumed to name the attributes, can an addi¬
tional content different from the present sense of the meant be
spoken of. To explain, when making a statement about some¬
thing named by a word, one considers under it, unless the word
itself is the subject, some particular attribute known of the
thing, of which more is then stated. “”Earth is the planet man
lives on”” may merely be a definition of a word, but in “”Earth is
the third planet from the sun”” some particular attribute other
than stated may be considered under the subject. In other
words, both the subject and what is said of it speak of the same
thing, but under the first less of particular attributes is
understood. All this is natural to all learning, in knowing about
things talked about only so much and then determining more.
When the particulars known are unspoken, they can be con¬
sidered understood premises, with demonstrations including
them familiar in logic as enthymemes, in which conclusions
may be reached from unstated facts. By the preceding it should
be clear that the silent understandings under words, inasmuch
as things talked about are seldom described, abound, and they
can generally be held to coincide with the particular attributes
by which the entity first becomes known or defined, whereafter
other attributes are determined. They can also be, however, of
particular attributes known besides, and in either case one
might speak of these understandings as meanings, in another
sense of the word than described. But because it is another
sense, one would be dealing with something else associated
with words, not with their signification as expounded. This other
sense of meaning can further unlike presumed differ from oc¬
casion to occasion, although the same name is used. When
making a statement about the planet Venus, one may think of
it as either the morning star or the evening star. In speaking of
particular attributes as a meaning, furthermore, meaning can
be split into additional two senses. One is of attributes in con¬
sciousness when a thing is considered, and the other of
attributes by which a thing is fundamentally understood and
often defined. Both meanings, it will be seen, con be drown
upon for inference.
When the limited attributes by which a thing is considered,
moreover, are intended to be what is meant when using its
name, the distinction between this meant and the whole entity
referred to is not the assumed one between the conceptual and
the worldly. The goddess Venus may be considered by the
limited attributes of being either the Roman goddess of beauty
or of love, but the possessor of the whole of the attributes need
herself not exist; the opposite takes place concerning the planet
of that name, the attributes of its morning and evening appear¬
ance existing with it.
In contrast to intending meaning to apply to those aspects of
what is named that are conceptual, meaning is sometimes
regarded as applying only to things of the physical world, as
suggested in the introduction in relation to statements held
meaningless. Those statements, it was noted, were considered
ones whose truth or falsity could not be verified through natural
science. It was noted also that some pronouncements were in
conjunction acknowledged not to be so verifiable though
meaningful. They were accordingly viewed as nondeclarative
sentences, neither true nor false. Irrespectively, however, of the
often declarative form of those sentences, stating such as that
something is beautiful, all sentences can be seen as declarative
in content, asserting something that is true or false. As lan¬
guage they, in keeping with what was said, are meant to trans¬
mit information about something in awareness, the informing
of it a declaring, an asserting, true or false. A question informs
of the wish to know something; a request of the wish to receive
something.
This sameness of assertion in sentences notwithstanding, it
has been maintained that the meaning of the sentences in
question has to do with such as the attitude of the speaker or
listener, as would be their expression of approval or disap¬
proval. The speaker’s wishes referred to in the preceding could
indeed be held attitudes meant, explicit or implicit. But to hold
the attitude of the listener as the meaning adds but another,
nonlinguistic, sense of it, that of the purpose, the use, of the
sentence. The use language is put to is another of the things
associated with it and not meant to be its content. That content
as its meaning is a precondition for its use, as when informing
someone about something in order to gain a certain response.
It has been also submitted that the meaning of words is the
accompanying outward behavior of their users. Overlooking
how farfetched, this opinion has only to do with a further non¬
linguistic sense of meaning, that of what goes with a thing,
what is implied by it. It is another association, now of the user’s
19
behavior with the words, to wit with their meaning os content.
What that meaning is by consent of its users, by which it is
therefore known to them, was expounded in this section. And
because of the pervasiveness of the correspondingly mistaken
search for the meaning of linguistic expressions, in discussing
various entities herein it will recurrently be stressed that the
meaning of the terms used for them will not be determined by
on estimation of it, in accordance with on enigmatic origin.
Their meaning will instead be determined by choice, in accor¬
dance with the entities of concern. If there should be estima¬
tion it will be os to the suitability of the terms in the light of
usage.
Since the entities dealt with ore of common acquaintance,
there should for the most port be no trouble correlating on
entity with o sense of o word or term in use. The attention is
nevertheless on the entity considered, and that it should be that
particular entity which is considered will be either expressly or
tacitly mode apparent. That is to soy, it may for instance be at
times specified in this writing that on expression used in it has
o certain meaning, with the understanding that that is the
meaning of concern to the reader as well as to the writer. Not
to impoverish language, moreover, the use of words will not be
needlessly restricted to a single sense, their particular sense no
doubt recognized from their context, and more than one ex¬
pression may be used for the same entity as well.
The search may then proceed toward what may be known
about the entities, the objects in awareness, that are of concern,
however named. And since objects in man’s awareness take for
man the form in which they are presented to it, the knowledge
will be sought with respect to those forms, which may inclu¬
sively be termed concepts.
CONCEPTS AND
THEIR CONSTITUENTS
Because it does not seem out of the ordinary that entities that
are the contents of man’s awareness be called concepts, it
should not be important to give reason for, though doing so, the
use of the word. As in instances herein where the intended
meaning of a term is only given, there is no problem in affixing
the word to the entity. The preponderance of the terminology
used will evidently not be defined at all. The defining terms
may be asked to be defined also, and so on endlessly. Most of
the meaning must therefore perforce be grasped from the
context, as it is in all discourse and is of course practicable
accordingly, as explicated. It may be added that far from giving
a reason for use of a word for an entity of a discussion, giving
definitions of key terms is in many expositions refused on
grounds that it is unworkable, in resemblance to the believed
uncertainty of meaning described in the first section. These
presumed difficulties were in the introduction mentioned with
regard to concepts, by which term, it may be remarked,
discussants, though disputing the same, refer to the very things
represented by words, in accordance with the present
definition.
Despite these considerations, because ‘^concept” in the
present sense refers to all things of human apprehension,
including those of language itself, it should be favorable to
supply ground for the word’s use, in order that its signification
be comprehended with ease.
Unlike “consciousness”, “concept” is less frequently used for
all apprehended, this time for anything belonging to that all,
rather than for the entirety alone. Accordingly it would be
objected that the definition of the word is too sweeping, that
there are important differences among objects of human
awareness. The domain of an expression, however, can be as
wide as desired, without taking away from either inside or
outside its boundaries. Any intricacies associated with the
subject should therefore not be layed at the door of definition,
but investigated separately. The more particular things in
consciousness that sometimes would be meant by concepts are
usually the ones distinguished from things in the world, and
concepts are at times even within that limited range distin¬
guished from else.
As concerns the exclusion of worldly things, one can again
be misled into wondering if one knows in any way the nature
of for example chairs, contrary to the observed knowledge of
what entities one names by words. One identifies a thing,
including a worldly one, by the nature by which considered,
and should one suppose a nature different from that, it would
contradictorily not be of the thing considered.
As pertains to concepts as still more particular things of the
part of consciousness not of the world, they are regarded in
opposing ways. Concepts are sometimes viewed as ideas of
classes of things, while classes of things, in the traditional
debate about universals, are sometimes considered to exist in
name or in reality rather than as concepts.
The problem of universals offers a prominent example of the
pursuit of linguistic meaning already possessed, and of the
21
potential deceptiveness of language side by side with its utility
in succinctly denoting multitudinous things.
A universal is, or is what is represented by, a term such as
“^man”, which does not refer to only one individual but to a
class, in this instance to all men. But since a universal term does
not refer to a class as a unit, man e.g. being two-footed rather
than of all the feet of mankind, the question has been outstand¬
ing as to what sort of being a universal term designates.
As can be seen all words except proper names can be held
to be universal, applying to many instances of something. And
if it is taken into account that some dictionaries do not list
proper names, all words in a language may be held to be
universals. The problem can accordingly be a wide-ranging
one. In conformance with the aforesaid, however, the problem
is illusory. What the words are about is decided by their users.
The users can nevertheless be deceived by the words of their
choice, especially since anyone’s language is largely received
from someone else.
If the intended meaning is obscured, then “man”, as an
example, could, being a singular noun, be expected to refer to
a single entity, as was suggested. What that entity is is accord¬
ingly wondered. However, the word is not meant to refer to such
an entity, but to all humans individually, despite its concise
singularity.
Varied arguments have, regardless, attempted to demon¬
strate the presence of behind universal terms universal single
entities as well, of, as sometimes worded, one over many. The
arguments are irrelevant, however, since it is each of the many
to which the universal term is intended to refer, whether or not
there is a single other one.
A most frequent argument for a single entity is that it is
required as a standard by which members of the class can be
recognized as alike. It has been sought of late to circumvent the
question by asserting a frequent absence of common attributes
in those members. Things named by a universal term are thus
said to often have only family resemblances, some of the things
being similar to some of the others, without any attribute be¬
ing shared by all. That observation, however, deals with the
universal term in differing senses, when the things denoted are
not, as otherwise, defined by the same attributes. As diction¬
aries already inform, the same term often designates through
differing senses things differing in the attributes concerned, not
the case with single senses. And as noted elsewhere, the
interest in what is named by a term has to do with one of its
senses at a time. The question as to a standard by which things
named by a universal term are found to be alike can accord¬
ingly remain. But as implied in the last paragraph, such a
standard, thought of as the universal behind the term, is usually
not a thing named by it.
Alluded to are named things of the world, and by the time
names are used for classes of some found in some respect
alike, an attribute by which one of them is determined to be
like another is usually thought of apart from actual observation.
That is to say, it is in accordance with a thought of the things
named that an actual thing is determined to be one of them.
But the thought, the standard, is not one of those things. In con¬
formance with an earlier remark, the universal ”chair” refers to
actual chairs, not to the thought of them by which their being
chairs is determined.
It may be noted that the oft debated question of what form
a thought of a universal entity takes is likewise irrelevant. Since
consciousness, a part of which other than apprehension of the
world is meant by thought, is seen as the ultimate arbiter of all
that takes place, there is no need to support by argument what
takes place in consciousness. The taking place in it is its own
confirmation.
With the foregoing in view, it may be added that the ancient
argument of “the third man”, disputing in fact single universal
entities, is erroneous. By the argument if two things are alike
through their likeness to a third, then the third is like the others
through their combined likeness to a fourth, and so on indefi¬
nitely. Not so. The presence of a standard in thought by which
real things are judged to be the same in some respect does not
mean that they cannot be so judged without the standard.
Its use, as suggested, is due to the passing quickly into
memory of things perceived, and it is much apter to compare
things to that memory than to seek out past seen instances,
which action makes use of memory regardless. Memory indeed
serves as indicated (page 11, first paragraph) the function of
storing knowledge, on which to draw in the future for compar¬
ison, and a sense like hearing can compare alike sounds only
through memory.
But a likeness between two things at once perceived is, as the
mere fact of consciousness of the resemblance again confirms,
observed without a third thing, provided the pertinent attribute
is, as in weight, not observed somewhere else only. A worldly
thing can respecting some attributes, as in the case of size,
indeed not be compared to a memory but only directly to
another worldly thing.
With universal terms not referring to at least the standards
only, or to other single entities which somehow represent
classes, but to the members of the classes themselves, they bear
a likeness in meaning to concepts as here considered. These,
too, concern in such as classes of worldly things the actual
23
entities as presented in awareness. They can, however, also
concern the single as well as objects of thought. In brief,
“‘concept” is here used in the sense in which one would ordi¬
narily query what concept any linguistic expression refers to.
One may speak of the concept of the single and real universe,
or of the concept of the multiple and ideal geometric figure.
Sometimes objects in human awareness in general are in¬
stead spoken of as ideas, as happens in British empiricism or is
reflected in references to some philosophical doctrines as
instances of idealism. But nowadays “idea” appears to connote
mainly a proposition, with the broad application accorded to
concept. That application, it should be clear, includes propo¬
sitions or any assertion as described, in concord with the
inclusion of all that is in awareness, as represented by any
linguistic expression.
Although all that is considered may accordingly be named
concept, it may be added that, in keeping with the varied
usefulness of language, there is no abstention herein from
speaking of, rather than concepts, entities or things, especially
when concerning realities. Similarly, there should be no reluc¬
tance to speak of the conceptual in the sense contrasting it with
the worldly.
The inclusive meaning of concept being thus specified, the
material of which a concept is constituted may, further, be
termed perception. Specifically, by perception is meant all that
is a distinct ingredient of consciousness, all that, unlike for
instance quantity, is conceivable not only in connection with
something else. Among perceptions are for example colors,
sounds and other qualities associated with the senses. But such
as emotions or even intellectual effort likewise belong to them.
It should also be noted that no differentiation is made here
between perception as the object confronted and as the act of
confronting, because the act of perception can be considered
to consist in the characteristic that the object is a perception, a
certain occurrence to someone.
The term “perception” is in philosophy today mainly confined
to sense perception, to conscious occurrences found to be the
result of the processes of the senses. In a previous time John
Locke for one instead defined perception as the first operation
of all intellectual faculties. Such a definition was rejected by
others as failing to distinguish between perception and other
cognition. The rejection can again be seen itself to fail, by not
recognizing that meanings of words are assigned.
Through the debating of what words should mean, as was
suggested, it has been attempted to resolve factual problems.
Views in opposition to each other have thus been tried to be
vindicated by adjustments in language. Behaviorism, eschew-
s
24
ing consciousness, would define names for man’s perceptions
with language for man’s overt behavior, thereby thinking to
eliminate them. Phenomenalism, uncertain of physical objects,
would refer to them with descriptions of their perceived attri¬
butes, although the single words for the objects refer to the
same phenomena.
As pertains to perception itself, the tying of its meaning to
physical occurrences may give rise to the notion that as a result
they exist. This was to counter an extreme skepticism about
worldly reality, but to have a word concerning it does not
imply it.
there may be objection to speaking of perceptions even with
respect to worldly reality, insofar as they identify it with some¬
thing in awareness. It is often argued that worldly things cannot
be identified with a content of private consciousness, because
of their public reality. But as observed (page 20, fourth para¬
graph), the objects require no more for their identity than their
nature in awareness. If they can be considered public, it is
because they coincide in appropriate ways among numbers of
persons, in contrast to parts of consciousness such as dreams,
which can accordingly be regarded as private.
The word “perception” as used here should rather be an
accustomed one, inasmuch as things of which consciousness
is composed are in general said to be perceived, while
things that like quantity are not distinct entities as described
are not said to be perceived independently. The use of the
word need as before yet not be restricted to this sense, it being
sometimes convenient to call anything in consciousness, distinct
or not, perceived.
However, the consideration of the distinct elements of con¬
sciousness presently named perceptions is important, because
they consist in the qualities by which concepts, perhaps fanci¬
ful, are apprehended, and by which accordingly anything
factual about them comes to be known.
These perceptual elements may, as in the preceding, also be
referred to as qualities, as is the practice in some of science,
with the attention less on entities as objects of consciousness.
And it may be remarked that these elements are themselves
concepts as spoken of, which are meant to, again, include all
kinds of entities, from the simplest to the most complex, from
perceptions of a moment to objects that, as in the case of
material ones, are conceptually constructed out of myriads of
perceptions repeated.
And the just mentioned matter of apprehending concepts by
those qualities, by perceptual elements, and of accordingly
acquiring about the concepts further knowledge can be im¬
peded by their intimate connection with language.
25
Considering that in interdependence among men a large
part of information one obtains is received from others,
language plays by its basic communicating function already an
important part in acquiring personal knowledge. In additionally
serving to preserve knowledge in forms like writing, language
functions similarly as a medium. For these purposes languages
developed into systems of most economical signs, taking the
place of the likes of pictorial representation. For that reason
language serves well even in thought, where rather than trans¬
mitting information, it codifies the concepts contemplated. But
language became simultaneously an obstacle, for by formu¬
lating thoughts in language, the concepts behind it can be
obscured, and findings about them correspondingly prevented.
In order to observe concepts and what is resultingly true
about them, therefore, it is necessary to take special care to
separate them from their names. Frequent pronouncements to
the contrary, it is required to so to speak think by means of con¬
cepts instead of by means of language.
To that end it is requisite to, in the parlance of the intro¬
duction, educe the concepts in accordance with what may be
termed their perceptual attributes, in accordance with
described perceptions by which they may be apprehended. To
elucidate, since what is sought in concepts are objects of con¬
sciousness apart from the language signifying them, if a con¬
cept, such as quantity, is not a distinct object of consciousness,
not a perception, it must be educed by some other concept
aside from the language. Quantity for example can be educed
in relation to things as it applies to them.
In connection with this observation, a still further sense of
linguistic meaning, next to those of the last section, can be
spoken of. It concerns the question of whether or not an expres¬
sion has meaning, and it was alluded to before (page 10, last
paragraph). In its primary sense the “meant”, as noted and
used in dictionaries, concerns the actual entity considered, not
something regarding it. In that sense by “man” are meant
actual men, not the thought in some form of them. But by that
sense of “meant” it could be said that nothing is meant by an
expression for an assumed entity that happens not to exist. And
upon misunderstanding, the expression could be dismissed, as
thinkers have tended to. Assumed, looked for, realities are,
however, an integral part of man’s everyday as well as scientific
pursuits. They are put into words not only in parts of but whole
sentences, in statements and questions with responses of true
or false and yes or no. The positive response signifies the reality
of the state in question, the negative response the unreality. But
in the last case, too, the sentences, or their concerned parts, will
be held of meaning.
s
26
This meaning can be said to in lieu of the entity or state be
about the thought of them. It can even be about something
logically impossible, though expressed in words. What matters
is that language for assumed things expresses thoughts of
something as connected with something else as the assumed,
possibly existence, even if the connection between the two
cannot occur conceptually either, each of them conceived only
separately. For example a sometimes cited round square can be
considered though contradictory, as can any deductive
proposition though proving to be false. More generally, to
conceive behind expressions certain things, to specifically have
perceptions regarding them as explained, is the requirement
for meaning in the present sense. Quantity need not have a
distinct existence, in reality or in thought, but the term has
meaning because of perceptions it represents.
This sense of meaning appears similar to one in which the
things named are considered only by particular attributes. Even
when so considered, however, the attributes can be ones
assumed real, and hence if they are not, there is no thing that
is meant. Instead what is meant in the present sense of mean¬
ing behind words is merely the described perceptions, so as not
to render the words vacuous.
Knowledge of what in particular cases the perceptions, the
perceptual attributes of the things named, are is needed for, as
indicated, findings to be made about them, findings specifi¬
cally that are not linguistic only, having to only do with names
of the things. That a thing of a name is also of another name
could be learned without knowing what the thing is, the saying
so by one who uses the words the criterion. But knowledge that
something else is true about a thing, the issue being one’s own
confirmation, requires more. Any truth about something, any
fact, becomes one for man, it will be recalled, in the form in
which presented to consciousness (e.g. page 24, third para¬
graph). Put differently, for something to be true about some¬
thing for man, it must be presented to consciousness in the form
concerned. That form was observed to consist of concepts,
identified by perceptual attributes. Specifically now, in order to
know whether something is true about something, the things
are meant apprehended by their perceptual attributes, be they
the contents of words or words themselves. Regarding the last
this took place in the above example, and the same takes partly
place in usual definitions. They speak partly of words and partly
of concepts named by them, in which case the concepts must
be apprehended by their perceptual attributes, as are the words
by theirs. And if what is to be determined, nonlinguistically, is
something about something when these things are taken as of
attributes behind their names, then it is those attributes which
27
must be of perceptual awareness.
With regard to attributes of things treated of in this writing,
the determining, the ascertaining, is, as made clear, by what
may be held reflective inference, rather than by, as concerns
things of the world, physical observation. Therefore the percep¬
tions relied on, if things of the world are at issue, are of their
conceptual representations, whereafter something may be
induced or deduced of them. As was remarked (page 22, fifth
paragraph), memory is relied upon for knowledge. In the case
of induction laws of nature are inferred from memories of
constancies already perceived, and in the case of deduction
inferences about nature from its conceptual awareness are
made in what may be held conformance with logical prin¬
ciples. The conceptual awareness is adequate for application
of constancies of this kind, because logical principles hold, as
it is said, in all possible worlds. And the awareness is required,
in order to know that the particulars conform to the conditions
of the logical principles, or in order to, as more common,
perceive the inference directly from the particular concepts.
That logical, or mathematical, principles themselves hold must
of course be perceived likewise, in consonance with the
requirement that something is true only if in awareness in the
form that makes it true.
Some of the findings may only be made step by step, when
a connection is not perceived at once but after a train of reason¬
ing, it being of account presently that for these and other
purposes concepts behind words be sufficiently educed by
perceptual attributes.
The accurate identification of concepts as a result of their
sufficient eduction will as suggested be seen to prevent often
found erroneous inferences, as well as to promote revelatory
new ones.
This can in symptoms of linguistic deceptiveness be true
because, as seen in the case of ”meaning” itself, the meaning
of one word can be variant, and the meaning of more than one
word the same, these characteristics frequently called respec¬
tively homonymity and, referred to, synonymity. The fallacies
connected with homonyms are well known as equivocations;
they will be found nevertheless committed professionally. The
unwarranted differentiation to which synonyms are liable is
recognized less; the contradictory results will be observable in
paradoxes. More contributively, synonyms can through their
identification lead to additional knowledge, by enabling recog¬
nition that things of attributes known under one of the syn¬
onyms have attributes known under another. Likewise the
uncovering of ambiguity in homonyms, which virtually all
words because of their multiple senses can be said to be, aids
s
28
in separating the meanings, whereupon what is not true of one
may of another be found to hold.
It is worthwhile to add that the harm that can be wrought by
language without due attention to the represented concepts
exceeds failures in inference. At issue are particularly homo¬
nyms or synonyms, in uses like those of metaphor and
euphemisms. Euphemisms are known to make the bad sound
good, leading to the acceptance of numerous evils, and
metaphor makes as often the good sound bad, with words of
disparaging senses that bring equal injustice.
Language can conceal other factors beside those from which
synonyms and homonyms mislead, factors the eduction of
which enables further findings, and therefore if in discourse
something about an entity is to be demonstrated, it may be
requisite that the term used for it be appropriately defined.
DEFINITION
It is said that some entities, called simple or unanalyzable,
are undefinable. Among them are held to be the elementary
perceptions or qualities spoken of, because they cannot be
reduced to other entities. But considered among them are also
concepts such as the good, for the very reason that they
are thought not to be identifiable with such perceptions or
qualities.
That to define an entity should require that it be analyzable
can be understood from the usual definition of one term by
several others. An explanation of the entity, not furnished by a
single term, is often desirable. However, the corresponding
reference to things by which an entity is identified is, in what
can be held an equivocation of “‘analyzable” confounded with
the reference to things of which the entity is composed.
Not only the many concepts that like “and” are indistinct as
described are identified by circumstance. Material objects are
likewise often defined by function, as are localities by place,
and so forth. Circumstance can similarly identify mentioned
qualities, as blue is in dictionaries defined by the color of the
clear sky.
The issue as said is that the meaning of the words be so
educed as to make known what is talked about, of especial
importance for demonstrating what is true about the things
named. For that purpose any definition bringing the concept to
consciousness will do, even a synonym, whereupon one may
29
assure oneself whether or not something about the concept
holds. All sorts of defining attributes made explicit are in fact
commonly part of deductive proof, illustrated by the traditional
syllogism. In likely the best known it is from definitional
”Socrates is a man” and knowledge of “All men are mortal”
deduced “Socrates is mortal”. It is immaterial to the deduction
whether a defining, or other, attribute of the thing in question
is about some of its components or about something else
connected with it. What matters is that the attribute figure in the
inference, and that it be known through identifying its possessor
by whatever definition.
As an early proponent of indefinability of simple entities,
Locke defended it by noting that a blind man could not compre¬
hend the concept of light however described. But a subsequent
reference to the rainbow inadvertently reveals that the problem
need not apply to simple entities alone. The rainbow, a com¬
plex thing, can, also, not be described to a blind man. The
question indeed is not whether something, simple or complex,
can be defined for someone who is not acquainted with it or
with, at least, things by which it is conceived.
As indicated in this chapter, men in communicating use lan¬
guage to speak about things in possible mutual awareness. If
the participants are unable to share the meaning, then the
language does not have its communicating function. In the
example of the blind man the inability to acquaint him with a
concept is not due to the weakness of language, but to the lack
of visual meaning for him.
In the present treatise, as in many others, definition of
unknown concepts need, as suggested, not even arise. The
matters dealt with are predominantly inseparable from every¬
body’s conscious life, and the definitions accordingly are in
general of concepts well versed in by the average reader.
As pertains to definability of these or other concepts, the
point is, as explained, to bring them to consciousness by
perceptual attributes.
Correspondingly, if definition of perceptual elements, the
described qualities, itself is the issue, it is doubtless enough to
use a synonym, having understood that the term defined is not
comprehended, perhaps not in the intended sense. More likely
it is comprehended, and no definition is needed. For inasmuch
as these qualities are the elemental things perceived, they
have an abundance of well known names, the use of which
will make the concepts specific.
As to definitions of other than the perceptual elements, defi¬
nitions of the nondistinct concepts not containing them or of
those containing more, they can be performed by way of per¬
ceptual elerrients likewise. The nondistinct concepts, too, must
30
be, as noted (page 25, third paragraph), identified by some
perceptual attribute, by way of which circumstance, as noted
also (page 28, last paragraph), they would be defined. By cir¬
cumstance definable were also indicated (page 28, next to last
paragraph) to be mony that are composites of perceptual
elements. In defining either concepts, furthermore, defining
words can refer to nondistinct concepts themselves, to ones not
perceived alone. While the name for the concept defined is
again taken not to be appropriately comprehended, the same
is not necessarily true of the words defining it. As in the case of
perceptual elements, there is a richness of understood lan¬
guage for all objects of knowledge held by men in common, in
particular for basic concepts by which further things are char¬
acterized, and including ones not perceived alone, such as
motion. As remarked with respect to words in general, these
concepts can at the least be grasped from their context, without
continued definition.
It is the through shared cognitions avaliable choice of all
kinds of words in all kinds of arrangements that makes defi¬
nition of all concepts that can be shared by men possible. That
fundamental concepts and correspondingly the meaning of any
language for them are shared was discussed as determined by
induction, which as observed will be amplified later, since the
sharing is presupposed by comprehension of this writing.
It should be added that there are qualities not of acquaint¬
ance to everybody who can perceive them. Whereas one may
have no trouble calling forth an image of a color by defining
it, perhaps by a synonym, the same may not be true regarding
a taste. In those cases it is the difference between actual and
possible conception, one said of account, which is in effect.
Though the thing defined may not be conceptualized there¬
upon, it can be conceptualized on finding the defining circum¬
stance. These definitions are of little significance here, however,
such as tastes not even bearing names, the concepts of concern
enjoying as indicated universal familiarity.
Because there should in either event be by the preceding no
concept that, as sometimes said, defies definition, it is justified
in discourse to ask for a definition if a term is not understood.
It is justified, specifically, to ask for identity by perceptual
attribute as expounded, be the identity unspoken in the defi¬
nition, so long as apprehension does not remain linguistic.
Much of deductive demonstration includes, as observed, such
definition, insofar sometimes as recourse is not had to men¬
tioned enthymemes, arguments in which some pertinent factor
is not stated but taken to be in consciousness. The omission to
state the factor is acceptable in obvious situations, but evidently
not when consciousness of it is in doubt. Such can be the case
31
when the to be omitted is a definition, and it is, as was indi¬
cated, in fact maintained that all things deductively derived
follow by virtue of definition.
Definitions, in spite of their avoidance, are thus accorded a
central place in logic. They are regarded as referred to (page 4,
last paragraph) analytic statements, as accordingly necessarily
true, that status often explained by using as example the state¬
ment ‘”No bachelor is married”, it is noted that if “bachelor” is
replaced by its definitional equivalent “unmarried man” then
the resulting statement, “No unmarried man is married” is true
logically and accordingly analytically, or necessarily. Since the
definitional statement can thus be turned into a logical truth,
it is inferred that it is likewise analytic or necessary.
This explanation can be viewed as prompted by the desire to
demonstrate that definitions, while not appearing as logical
truths, possess the same internal coherence discussed in the
introduction as intrinsic or deductive. The interchange of
definitional equivalents in getting logical truths has been
supported by what is known as Leibniz’s law, by which if two
expressions refer to the same thing then on replacing in a true
statement one of them by the other the statement remains true.
By thus complying with that law, however, any statement what¬
ever, even when of an accidental fact, can by that interchange
be made logical. In recognition it is attempted to in various
ways qualify the statements in which the interchange can be
carried out. But the interchanges are irrelevant.
Allowing that on replacing “bachelor” by “unmarried man”
in “No bachelor is married” the resulting statement remains
true, the statement is not the same one as the first. The process
is an instance of deduction by to be treated transitivity, as when
from “A equals 6” and “6 equals C” inferring “A equals C”,
without the inferred stating the same that either of the inferred
from did. “A equals 6″ can stand for ” ‘not unmarried man’
equals ‘not bachelor’ “, “6 equals C” for ” ‘not bachelor’ equals
‘married man’ “, and “A equals C” for ” ‘not unmarried man’
equals ‘married man’ “. Indeed whereas the two premises, or
the reversely ordered previous “No bachelor is married” and
equivalence of “(no) unmarried man” to “(no) bachelor”, may
be held to state definitions, the conclusion, or “No unmarried
man is married”, states something else, a logical truism as it
were. That is to say the statement revealed logically true
through the interchange is the derived one, not the initial one.
It may be wondered though whether in consequence the
initial one is not logically true also. It may in fact also be won¬
dered whether basic logical principles, notoriously resisting
proof, cannot in some such manner be proven. But the logical
truths derived in the preceding are the consequence of their
s
32
being already employed in the statements from which derived.
For instance ^not unmarried man’ equals ‘married man’ ” is
logically true by later examined law of double negation, used
in the second of the premises from which the statement is con¬
cluded. The first premise, to explain, follows from the definition
” ‘Bachelor’ equals ‘unmarried man’ ” by also later trans¬
position, and the second premise is the equivalent of the first
but for the replacement by “married man”. But this replacement
is itself licensed by the law expressed in the conclusion, which
follows as a result. The conclusion expresses the law because
it is applied to the definition of bachelor, which merely serves
as a premise with the others, none of which results likewise in
a logical truth, in an intrinsic one, as to be seen.
A similar situation occurs when oppositely a statement like
that premise is through the interchange obtained from a state¬
ment like that conclusion. In the dispute as to necessity in a
definitional statement like “No bachelor is married” it is
contended that that interchange cannot work for facts that lack
necessity. A statement like “Necessarily no unmarried man is
married” is construed as expressing the one previously inferred,
and a statement reversely inferred from it is maintained to like¬
wise have necessity, since the interchangeability means that
what is true under one name is true under another. It is thought
in other words that if the interchangeability were always to hold
then from for instance “Necessarily no toothless feathered
creature has teeth” would follow “Necessarily no bird has
teeth”, taking the interchanged as equivalent. And the inferred
statement would be held to falsely assert necessity.
But what can be inferred is here, too, mistaken for something
initially posited. The conclusion that no bird has teeth is truly
necessary, provided the premise that all birds are toothless is
true. In a general explanation, if, or when, “A equals B” is true,
and “6 equals ‘not not 6’ ” being by law of double negation
necessary, then “A equals ‘not not 6’ ” is necessary as well,
without the same required of the first equality. To be more
specific, the last necessity, namely logicality, is here one not
only because resulting by mentioned transitivity law, but
because it, due to the first premise, is an instance of the second,
logical, one. Were the second premise a causal law, the con¬
clusion would similarly be an instance of it.
The preceding would suggest that Leibniz’s law, by which
the equality admits the interchangeability, holds after all. But
the law relies too much on language, whose delusiveness was
noted in the foregoing. Expressions overtly naming certain
things refer covertly often to only particular attributes of them,
possibly the names themselves. Thus it was observed in the past
that if something true is said about a word, perhaps that it has
33
so many letters while using the word as if the thing meant by
it were spoken of, the same may not be truly said about a sub¬
stitute word of the same meaning. Likewise with other attri¬
butes. “Being the city of the Acropolis makes Athens a favorite
tourist spot” may not be true if the part describing Athens is
replaced by “the capital of Greece”.
If the sentence began with the addition “The attribute of” it
would be explicit that something is said about the particular
attribute, not the city. Things differ if something is said about
the city, even if still identified by a particular attribute. Since
Athens is both the city of the Acropolis and the capital of
Greece, the city of the Acropolis is the capital of Greece. The
issue is that if one thing is identical with another then what is
true of one is by transitivity in fact true of the other, but that
what the things talked about can be obscured by language.
The effort to prove definitions, definitional identities, to be in
an indicated sense necessary is thus ill-conceived if only
because the reasoning offered as ground for proof, e.g. some¬
thing like Leibniz’s law, applies to all identities, including
accidental ones. It can moreover on a much simpler basis
be determined that identity by definition is not an intrinsic
necessity. Since names are arbitrary, what is named by a word
can be identical with a certain thing at one moment and not at
another.
A reason that the necessity is held plausible can be found in
a confounding of definition with properties that a thing by its
nature always has, in likeness to mentioned analyzability. But
as likewise indicated, things are not always defined by such
properties. The moon is defined as the heavenly body revolving
around the earth, but that it always revolve is not required in
order to be the same object.
Argument is also advanced that a statement like “All bach¬
elors are unmarried” is somehow necessary because an ana¬
logous statement can be found in most any language. But aside
from no logical basis, other languages are apt to have a word
for unmarried men, as they have words for the many other
things of common human acquaintance.
More pertinent than these, a definition has to do with eluci¬
dating what is meant by an expression, and that meaning can
be changed at will. At the base of instead arguing that that
meaning possesses some sort of necessity can be held to lie the
proposal that deductive truths are somehow grounded on defin¬
itions rather than unchanging facts. Since those truths are none¬
theless considered to possess that necessity, so are definitions.
In defense of this point of view inconsistency is engaged in.
In presenting truths of logic as ones founded on meaning,
their traditional understanding as ones contained in the
concepts concerned is discarded. It is customary in logic to not
surprisingly view its problems in terms of sentences or state¬
ments, in place of states of things. Hence logical containment
has been expressed as containment of predicate in the subject.
This is objected to as limited to subject-predicate forms and as
making containment unclear, as evident from the linguistic
separation. However, there is no such limit or unclarity as
regards containment in concepts, explained to cover all appre¬
hension, with the containment factual. It is more the notion
offered that the logically true is so by virtue of meaning that
lacks clarity, and interpreted as following only from definitions
it is these truths that became limited, precluding deduction
from other conditions. But really of account is that what com¬
prises deductive truth depends as with other entities on what
is chosen to be meant by it. Meant by it traditionally and in its
present role was noted to be inference of something part of the
concept of the inferred from. The inconsistency in the argu¬
ments disowning this meaning resides in their attempt to show
that the definitional replacement possesses the disowned
internal necessity nonetheless.
Thus definitional statements were as discussed endeavored
to be reduced to logically necessary ones, or to be explained
as describing the necessary nature of the things defined. The
last necessity is sometimes called essence, and because it and
other conceptions of necessity, together with those of possibility
and related ones, all usually named modal, appear, due to
their relation to certainty, important to definition as well as to
subsequently explored actual and logical truths, an examin¬
ation of them should be timely.
The question is again what is meant by them, and the defi¬
nitions here will again be in conformity with what the writer
understands to be their generally held meaning and, more
pertinently, the concern. With all definitions herein, if a
reader’s concepts of concern are not the same, it is hoped that
the concepts discussed will at least be found of interest.
As regards “necessary”, it might as frequent be defined by
“indispensable”, or “what cannot be otherwise”, or similarly.
These, however, contain forms of “impossible”, itself a modal
concept and hence wanting in the desired definition.
The necessary can in a broad sense rather be simply defined
as that which is completely certain, in a factual sense rather
than the previous one regarding knowledge (p. 13, fourth par.).
It is the sense in which what has happened would be called
necessary in that it will no longer, for good or for bad, be pre¬
vented, whereas the same may not hold for the future. That
is to say that if all is not determined beforehand, if there is
chance or free will, then some of the future is not of a neces-
35
sary outcome. Otherwise, should all be predetermined, as by
unbroken casual chains or by divine decree, all things would be
necessary. In either event, those things are broadly necessary
which have been the case or are predetermined to be the case
at some future time.
By what reason something may be held predetermined may
further be asked. And as indicated (e.g. p. 4, last par.), correct
inference, in which ascertainments about the future by having
to by what was said be based on previous perceptions consist,
is meant based on unfailing finding. Things are certain to at
some future time be the case, accordingly, if they always are
the case under any condition or a particular condition an
instance of which has occurred. To the first of these belong the,
inductive or deductive, principles themselves, by which other
things to be the case in the future are inferred.
Whichever the situation, in the light of the preceding the
necessary in a wide sense may be newly described as that
which—as little as by chance—has been the case, or that
which—as might be the effectuality of a deity—is always the
case, whether independently or in connection with something
else.
The necessary so comprehended can be seen as identical
with fact in a likewise broad sense of the word, when by fact,
also, are meant those things that have been true together with
those always true, independently or in connection with some¬
thing else.
It may be objected that the necessary is meant to be some¬
how more substantial than fact in this sense, that for something
to be necessary it should have a reason other than something
with which it is connected however lastingly, not to speak of its
merely being the case at any time. The reason for anything is
seen to be none other, however, supposing that such other be
wished to be called necessary. For let something be the reason,
in whatever sense, for the first thing. But then the last thing
is none other than something with which the first thing is
connected.
That fact as well as the necessary be about what has been
true along with principles and what consequently will be true,
can be attributed to the regarding as fact or necessary those
things that need be taken into account in man’s presumed free
actions. It is for that reason that the necessary as the completely
certain was above described as what will not be prevented. If
mentioned chance be possible too, then though it by its present
meaning cannot be caused, it still would be pertinent as not
necessary if it can be prevented. Of bearing is that the neces¬
sary is about no more than the, by man dependent on, certainty
of fact.
s
36
Although the necessary is accordingly synonymous with fact,
its terminology can be useful, as exemplified by ‘”need” in the
preceding paragraph. It is simpler to say ‘This is necessary for
that” than to state as a factual principle “Whenever that occurs
this had occurred”. Consequently forms of “necessary” or
acknowledged synonyms, or other terms proving not to stand for
supposed additional concepts, are again not avoided in this
writing if they facilitate it.
In the last paragraph the mention of forms of “necessary”
with regard to principles suggests that the words may in a
narrower sense be used for that purpose only. It is indeed the
practice to say of a past event that it was not necessary,
meaning that there was no compulsion by such as a causal law.
It may be remarked that the very speaking of necessity
customarily with respect to laws of nature, which are referred
to synthetic truths contrasted with the analytic, makes the term
as suitable for them as for laws of logic, often argued to be the
only applicable ones. What the term is used for is, to repeat, a
matter of choice, and it is in logic that in fact, in contrast to
above, principles are stated, making the designation of
necessity superfluous. It would there be stated that if A then 6,
not as in causation that an event, 6, is necessary due to another,
A; this likely since in logic the interest is more abstractly
principles, and in nature concrete particulars. A stricter sense
of necessity can thus herein concern principles in general. The
narrow sense of fact instead, it may be added, usually concerns
only what has occurred.
Having defined the necessary, by “impossible” are meant
indirectly, by denying the possible, facts which are contrary to
those meant by the necessary. That is to say, those things are in
the wide sense impossible which in fact have not been or never
are the case, independently or in connection with something
else. The narrow sense is as before confined to what in fact
never is the case. In either sense, accordingly, whereas “neces¬
sary” asserts affirmative facts, “impossible” asserts negative
ones. For the matter of that, the necessity associated under
determinism with all things refers to the impossible as well, to
absences as to presences.
By “possible”, next, is meant the denial of the impossible, of
negative fact. It may be wondered if this meaning, taken as
common, does not without warrant, by saying that the possible
is the not not possible, presume common knowledge of the
mentioned law of double negation. Ordinary life is continually,
however, guided by, as cases of the discussed presupposed,
knowledge conforming to these and other principles, without
formal acquaintance with them.
Specifically, then, in denying the broadly impossible the pos-
37
sible refers, by later law of complements, to o thing os having
been the cose, or to what is sometime, maybe always, the case
in itself or with the specified, or would sometime be the cose
in itself or with the specified if certain things were done or be¬
cause, os may be on act of will or event of chance, it is merely
not precluded by o principle. In denying the narrowly im¬
possible, that something never is the cose, the possible refers
to the some os in the preceding, but for o thing os having been
the cose. Viz., that something in the post was possible does, os
normal, not mean that it was the cose, but that it was not
impossible by some principle.
the possible thus includes the necessary, inasmuch os it in
the brood sense refers to what has been true and what beside
sometime is always true os explicated, and in the narrow sense
to what is always true likewise.
It might be proposed that the necessary should not be in¬
cluded in the possible. It con be heard stated that something is
not possible but necessary. This con, however, be regarded os
on emphasizing figure of speech, there being likely no mis¬
understanding if after ^”not” in the preceding sentence is
inserted ‘”only”. Not to be possible and to be necessary, both
asserted in that sentence’s figure of speech, ore ordinarily
viewed as contraries, and the present definitions will be fitting
for the purposes.
It may also be noted that as it can be useful to speak of the
necessary in place of stating entire principles, so can it be use¬
ful to speak of the possible in place of denying them. It is
simpler to say “This is possible” than “It is not true that this is
never the case”.
Corresponding to the possible, by “not necessary” is of
course meant the denial of the necessary, of affirmative fact,
the things specifically included being the negative counterparts
of those included in the possible.
At this point it may be observed that these modal expres¬
sions, the necessary, the impossible, and their negations, are
often without qualifying used with respect to human knowl¬
edge rather than its objects. When it is said that something is
not necessarily true it can be meant that it will not necessarily
turn out to be so. A future event may thus be held not necessary
in accordance with experience, while the event may in fact be
necessary as a result of others. For the same reason the neces¬
sary would in that usage refer to only facts which are known in
accordance with experience, not also to those true independ¬
ently of it. The possible and impossible are spoken of in the
same dual manner, leading to later observed equivocation.
The significance of the above defined concepts is at this time
their referred to correspondence to what is considered the
essence and any related characteristic of entities.
Designating herein anything that is true about an entity its
attribute, by ‘”essential” can regarding an entity be held to be
in general meant an attribute without which an entity is not the
one considered. But since without any attribute a thing has had
it is not the thing considered, any attribute a thing has in fact
had is in that sense essential. There appears thus the same factuality that was encountered with regard to the necessary, and
the sameness is seen to extend throughout. When considering
broadly as necessary what has been true, then since truth
concerns a fact regarding something, something has been true
of something, which by the preceding is the same as saying
that something has had a certain attribute. The necessary,
further, concerned principles, constancies, likewise happening
to essential attributes. Things necessary by being always true
independently are, again, about something always true of
something, namely about constant attributes. And things
always true in connection with something else are about
constant attributes under some condition, e.g. about an effect
on a thing under a cause. Only the undetermined future is of
things not counted as true, as attribute.
The only distinction between essence and necessity that
might be made is that in a narrow sense by essence the referred
to as true of a thing, as its attribute, can be that by which the
thing is first considered, not what is learned about it later,
though perhaps a principle. This strict sense of essence has to
do with the interest presently, which is concepts as mainly first
considered, of which more is then determined.
Both, the attributes referred to by the broad and the narrow
sense of essence, can be seen of importance in the light of
earlier observations. The first because a thing can be defined
by any attribute applying that distinguishes it, and the second
because a thing cannot be redefined by denying a first meant
attribute without speaking of something else.
The contradiction in similar denials is also pertinent to the
longstanding contention that existence is not an attribute. As
regards that contention it should suffice to remind that as
elsewhere what is named an attribute, the present meaning of
it as anything true of a thing another common one, is arbitrary.
More to the point, if a thing considered is an existing one then
something not existing is not the thing considered. That is to
say, the existence is, along with other things, essential, whether
or not called an attribute. To repeat a former example, of a yet
more stringent requirement, for something to be a chair,
although there may not be any, it has to, beside else, exist. A
thought of a chair is not a chair.
It may again be remonstrated that the essence of a thing
39
should be something deeper than on attribute that in fact
applies, even something other than an attribute, as has long
been argued in reference to substance.
A distinction made between substance and attribute can be
understood as counterpart to the grammatical distinction
between subject and predicate. However, in line with the dis¬
cussed disparity between language and its content, the utility
of making a statement in terms of subject and predicate does
not imply a like separation in actuality. To suppose the separa¬
tion amounts to needless differentiation mentioned as to syno¬
nyms, which in fact are knowingly used in definitions, which
are instances of statements with subject and predicate. Simi¬
larly, subject as part of a sentence is as a homonym equivo¬
cated with subject as object of consideration. With these mis¬
conceptions the impression that attributes differ from subjects
has led to varied interpretations of the two, akin to the dis¬
cussed twofold meaning assumed of words, although both
words and attributes can in their respective designating and
characterizing way be of any content.
Most often the subject, in relation to the notion of substance,
is equated with the concrete individual said to be denoted, and
the attribute with the abstract universal said to be connoted. But
when the former is indeed a concrete, an existing, entity, then
so can be regarded the latter. In ‘^Socrates is a man”, the
mentioned and oft cited paradigm of a singular proposition, a
living Socrates is also a living man. Even in “All men are
mortal”, the companion universal proposition, living men are
living mortals. And the difference between the individual and
universal is merely that the second concerns more than one.
Moreover, it can be the subject which is abstract, a con¬
ception, without the attribute the same, in reversal of the
contention. This happens every time a thing is stated to exist.
Since the information is new, the thing is known without its
existence before declared. In cases where the thing is a
conjectured reality, it may be wondered how it can be referred
to without its being. Thus by a quark, as by a chair, is not meant
a thought, but a physical object, if not a confirmed one. Here
again the flexibility of language can deceive. In “Quarks exist”
by quarks are provisionally not meant real things, but the
conception of them, in concord with the last discussed sense of
meaning (p. 25, last par., through p. 26, second par.). As long
as hypothetical, with all entities the referred to, often as subject,
is thus their conception.
When it comes instead to substance, it, as could existence,
can also, perhaps as material, be called an attribute by fiat, it
has been argued besides that there must, based on the
assumption of a world independent of the perception of it, be
40
an unknown substratum in objects, one that may be called their
substance and the truly essential. But as observed (p. 20, fourth
par.), things are identified by the nature by which appre¬
hended, not requiring a further nature to be the things meant.
Furthermore whatever unknown things may be true about
something, such as the composition of matter, they still would
be none other than what in fact is sometime or always con¬
nected with it, as explicated of the necessary.
Many entities do in addition, as indicated above, not concern
substance, mostly conceived as concrete matter. Apart from
sometimes being abstract in the conceptual sense, entities can
be so by not having a distinct, an independently perceived,
nature as previously described. Size is an example. And in
many material things attributes like function, as in tools, are, in
place of the substance of which they consist, considered the
essential ones.
Designating as the essential in likeness to the necessary thus
factual attributes, the same should not be required in likeness
to the impossible. The essential should be adequate to desig¬
nate factual absence of given attributes as well. As to the nonessential, it designates the denial of the essential, in continued
likeness. Since the interest now is identification, definition, this
means that nonessential attributes, such as those a thing has
not had, are those by which a thing would not be described as
desired.
In seeking accordingly definitions, of account is that in
concord with the aforesaid an entity is fully characterized by its
attributes, simply because ”attribute” can, and does, designate
anything that an entity is, e.g. the entity itself.
This state of the matter notwithstanding, counterargument is
advanced. Thus it is maintained that if things that basically
identify an entity be, like other things about it, termed attri¬
butes, a contradiction arises. The first accordingly termed attri¬
butes, identifying the entity, then include the perhaps later
discovered others. Namely, it is argued, the number of attri¬
butes is both that of the first ones and its combination with the
number of the others.
Needless to repeat, anything that identifies a thing can at
will be called an attribute, and dependence on language,
therefore, does not provide a solution. It is dependence on
language, rather, which brings about confusion. A combination
of attributes is commonly spoken of as a single one, in seeming
contradiction. This is due to the compactness of words, wherein
combinations of often a multitude of attributes are given a
single name. The generality of this is as a matter of fact
manifested in most of definition, where one thing, as repre¬
sented by a word, is defined in terms of a number of others. It
41
is a process involving classification, when for example man is
defined as animal and certain further attributes, animal is
defined as living thing and certain yet further attributes, and so
forth. It would be awkward then to say that being a man is
attributes instead of an attribute, in keeping with the singularity
of “man”. The same holds for animal and so on, either of which
can be said to have numbers of attributes, though spoken of in
these contexts as single ones.
Taking into account the foregoing, certain theorems or prin¬
ciples of definition may be formulated. These can be thought
of as eductive principles in similarity to deductive principles or
laws and inductive laws of nature, because they comprise rules
by which the process of eduction or associated definition may
be guided. The truth of these principles, as of others, may
appear self-evident, but what may be regarded as formal logi¬
cal proof is for the sake of better assurance furnished.
Instead of, then, speaking of what defines a particular thing
as something necessary or essential to it, it can be spoken of as
the factual attributes of the following.
THEOREM 1.1. A thing is defined if and only if referred to by
attributes ail and only those things have.
As diagramed below in Figure I, a chair (circle) could accord¬
ingly be defined as a single seat with a back, which in fact all
and only chairs are. It would not be defined as only a single
seat (left oval) or a seat with a back (right oval), which in fact
not only chairs are, and it would not be defined as also a seat
with arms (bottom oval), which in fact not all chairs are.
Proof. (Formal proofs of theorems are in this treatise con¬
tained in single paragraphs.) The theorem is an instance of a
principle treated in Chapter III, and it can—in at least its “if”
half, the “only if” half often not given—be held equivalent to
what is known as the axiom of extensionality. In accordance
with the principle, if and only if all and only things of one kind
SEATS
SINGLE CHAIRS WITH
ARMS
FIGURE I
s
42
(those named chairs) are of another kind (those of the attributes
listed) then the first and second things are identical.
The principle is disclosed in the diagram, by which all of and
only the circle (the first thing) contains the combination of (the
second thing) descending and ascending diagonals, while not
all of the circle contains also verticals. As the singularity of the
circle as a thing may suggest, by all can be meant a single
thing alone, the principle applying to also a definition of an
individual entity, when the question of not all having an attri¬
bute does not enter.
For the above definition of chair the figure does not assign
to being a seat a separate attribute. Since the attributes are
viewed as characterizing things, those a chair is, the listed
attributes other than of merely a seat, which are not suited as
nouns, would separately be referred to by phrases like “‘things
with a back”. Seeing that being a thing can likewise be held an
attribute, the attributes listed by separately giving that of being
a seat would needlessly multiply. That fewer are listed will be
remindful of the practice of regarding as a single attribute a
combination of them, often by a single name. The one attribute
chair is defined by a few others. An entity, as was indicated, can
be defined by sparsest attributes, so long as the above theorem
on identity is conformed with.
This is accomplished even in definition by a single term, a
synonym, although it may not be as illuminating as required.
For that reason, because a concept can require eduction by
some other attribute, a further theorem is furnished, for which
the preceding concepts of term and synonym will first them¬
selves be defined.
By “term” is meant a word or phrase which, as approximately
defined in dictionaries, has a definite meaning, but which,
more precisely, refers, through its whole, to only one kind of
entity, not, through its parts, to more than one, which combined
would identify it. When a term is a phrase it is sometimes called
idiom, and “bring about” serves as an example. Its meaning is
to be “cause” as a verb, without here an independent meaning
of the words within the phrase, whereby it would be explicated.
A single word can by this understanding, as a matter of fact,
consist of more than one term. “Knowable” refers to both the
concept of knowledge and its possible object.
By “synonym” is then meant a single term referring to the
same entity a given other term does. It may be remarked that
it has been proposed, in particular in connection with the
spoken of interchangeability, that no two expressions are
exactly synonymous. But as with other meanings, it is merely
a matter of choice that one expression refer to the same thing
as another.
43
THEOREM 1.2. An expression, perhaps in part, does not
define a term by a synonym if and only if it defines it solely by
more than one term.
Proof. By the above definition of synonym os one term, if on
expression, perhaps in port, defines o term by o synonym, it
does not define it solely by more than one term. Therefore by
later principle of transposition, by which if from A follows 6
then from the negation of 6 follows the negation of A, the “if”
half of the theorem is true. Further, since part of a term has by
the above no independent meaning, if an expression does not
define a term solely by more than one term, it, perhaps in part,
defines it by a synonym. By transposition therefore the ^”only if”
half, equivalent to placing “if” in front, is true.
When accordingly a definition by a synonym is not suitable,
as in defining discussed nondistinct things by distinct ones, then
by the “only if” half of the theorem more than one term must
be used, and by the “if” half none of them can be a synonym
of the term defined.
Having set forth these principles of definition or eduction, of
identifying concepts of concern, inquiry, in keeping with the
process outlined in the introduction, may begin as to what
realities can be discovered through concepts so identified.
s
44
Chapter II
REALITY
Because the object of this treatise is to determine realities, it
appears especially important that reality be comprehensibly
defined.
The determination of fundamental realities has been, as
noted, the primary interest of metaphysics, which had been
called the queen of sciences, and its general omission of a
definition of reality, its identification, reveals how a major field
of inquiry can fail to decide what the inquiry is about. In con¬
formance with the expounded, when without an established
concept behind the word, the query about reality amounts to
a linguistic one, about what sort of things would be referred to
as real. In contrast once the nature of the concept intended is
settled, it is possible to, sometimes immediately, learn with
certainty what particular things have reality, obviating much of
the speculation and doubt of metaphysical doctrines.
Since by ‘”reality”, to accordingly proceed, is not meant a
mere quality like color, the concept—in order to indicate with
some specificity what qualities, what perceptual attributes,
apply to it—must by what was said be defined by language
other than for its whole, and consequently a definition by a
synonym, such as “existent” or “actuality” falls short. The con¬
cept must for that purpose therefore, by above Theorem 1.2, be
defined by more than one term. But in accordance with
Theorem 1.1 that definition, too, can be inadequate. Were
reality defined as, for instance, the object of sense perception,
then sense perception, if itself thought a reality but not its own
object, would be excluded.
In consonance with what went before, the question of the
meaning of reality should be better answered by asking what
the meaning of concern is.
It can then be said that the reality sought in this treatise and,
it can doubtless be safely presumed, in most every quest for the
real, are those objects of man’s possible awareness, possible
perception, that are the case and on which, good or bad, the
outcome of man’s pursuits can depend, which are accordingly
consequential in man’s purposes. While a mirage of an oasis as
45
nearby may be the case, the oasis would not be considered real
without further possible perception such as through touch and,
ultimately, fulfillment of a desire such as quenching a thirst.
The real thus bears a resemblance to the necessary and fact
again, these having been noted (p.35,.last par.) as of things
that need be taken into account in man’s actions. The now con¬
sidered stricter sense of reality, of being, does not concern,
however, anything that has been or is the case, nor merely
principles, constancies. As indicated, something that is the case,
e.g. the oasis of a mirage, may not be considered real, whereas
something not a principle, e.g. an actual oasis, may. More
specifically, when describing reality as something the case on
which the outcome of man’s pursuits can depend, by this speci¬
fic “can”, a synonym of a form of “possible”, is meant, by what
was said about the possible, that this reality, existence,
concerns facts on which the outcome of man’s pursuits either
depends always, or sometime, or if certain not by principle
precluded things are done, not to mention dependence simply
not precluded. On reaching an oasis and drinking its water,
these not precluded in principle, one quenches one’s thirst.
There is also a broad sense of existence or synonyms, cor¬
responding to that of fact or the necessary. Any occurrence,
even an object of thought, would be said to have being. Forms
of the last word are frequently used to connect subject and
predicate, when the word is likewise a means for expressing
asserted facts, existences of states, in conformance with earlier
observation (p.l6, first through third pars.). And this use is
suggestive of why words like “existence”, similarly to “fact” and
the like, would be applied in the wide as well as a narrow
sense. All that has been the case is in the vast contents of man’s
consciousness, with many of them found crucial to man’s pur¬
poses, ascribed a tentative existence like that in propositions,
which may be confirmed or disconfirmed.
Reality as described has an affinity with fact and the neces¬
sary also by supposing a free will (p.35, last par. again),
through which upon knowing realities, things in the above way
consequential in one’s purposes, one may strive to realize those
purposes, rather than be at the mercy of events. The freedom
is especially supposed by the things most often looked upon as
real, those that occur in the material world. Only a fraction of
facts in that world is depended on in one’s purposes. It is of
indifference whether a larger portion of particular facts in it,
e.g. the location of some object, occurs or not, the more so the
more removed from one’s life. That is to say these occurrences
are in the required sense consequential only inasmuch as they
would be depe.nded on if through a free will one acted
accordingly.
Whereas one may at no time depend on all things in the
world, one may at times, as in an act of concentration, virtually
depend on all of one’s consciousness. This observation further
highlights the discussed importance of consciousness in man’s
endeavors. By these endeavors are meant, in point of fact,
efforts made through the employment of consciousness. They
do not pertain to the human organism’s activities in which one’s
consciousness is not made use of. Of consciousness to be sure
are as said earlier all things under human consideration. What
differs regarding the body’s activities and all things material is
that they are considered to be present also when not perceived.
But like things stored in memory they like all else acquire actu¬
ality in their form in man’s consciousness (p.26, last par.), and
for that reason presently discussed reality was described as of
man’s possible awareness, possible perception. That material
realities, or contents of memory, be considered present when
not perceived thus signifies that, as in other possibilities, under
certain in principle not precluded conditions they would be
perceived.
The real was spoken of as perceivable by man, and it cannot
be argued that, for instance, animals depend on the same
things, things accordingly consequential to them, and that
these things are by the aforesaid therefore realities even if man
did not exist. The reality of animals might conceivably be only
theirs. In other words, in order that something be man’s reality,
it must be consequential to and by the preceding therefore
perceivable by man.
That it be perceivable accords somewhat with the pro¬
nouncement of the 18th-century philosopher George Berkeley
that to be is to be perceived. A difference is that he did not
specify man as perceiver, positing instead that all reality is
perceived by God. He correspondingly also speaks of the
perceived instead of the perceivable. And what is perceivable
is in the present writing, further, not identified with what is real.
I.e. unlike the converse, not all perceivable is real, as opposed
to how Berkeley’s view can be interpreted.
What is real can, however, be identified with what is knowable, knowable by man. In agreement with the definition in the
first chapter (p.l3, fourth par.), knowledge can be understood
as awareness of something actual, compared to awareness of
something assumed, and “actual” can, as indicated (p.44, third
par.), be replaced by “real”. By the real is for the meantime
meant, if not all that is, the explained consequential broadly
enough to include, as suggested in the introduction (p.4, third
par.), things like logical principles or even definitions, things on
which one may depend in seeking more proximately conse¬
quential facts. They include negative facts as well and represent
47
truths that may be known, in contrast to hypotheses assumed.
Transforming then the definition of knowledge, so as to speak
of the known as the real meant to be the object of the aware¬
ness, something known is something real of awareness. This is
the same as saying that something knowable, as what can be
an object of the awareness, is something real of possible
awareness. This sameness here of the known and the knowable, or of any occurrence and its possibility, is due to the com¬
prehensive meaning of a word like ‘”known”. Used as in defini¬
tions, the word covers not only the actually known, but anything
known possibly. And since the possible was seen to also cover
the actual, the known and the knowable are in this use the
same. By saying in the preceding thus that something known
is something real of awareness, all possible times of the knowl¬
edge and awareness are meant. TKence something knowable
is something real of possible awareness. But in accordance with
the meaning of real (p.44, last par.), all of it is of possible
awareness, which qualification of it is therefore superfluous.
I.e. what is knowable is what is real.
A few more remarks about reality as perceivable and
knowable.
The perceivability may be questioned in the light of accepted
realities like the atom, seen as unviewable. The issue may be
likened, however, to that of nondistinct entities like quantity.
Particles of matter are determined in accordance with what are
regarded as their effects, and while invisible, their description
accords with attributes of visible things with similar effects. The
descriptions are thus conceptual aids, the things referred to
being truly the connections between things observed, as
between certain experiments and their results. What matters
with respect to the realities is that things of possible
dependency are in fact perceived. Without the perception the
dependency would not be held ascertained.
With regard to knowability it may be added that for some¬
thing to be known does not imply strictly that it has been per¬
ceived. The entire process of correct inference about worldly
reality rests on an absense of perception of the inferred. Never¬
theless all that is known presupposes its perceivability, its
possible presence in consciousness in the form making it
actual, instead of an image. For even when something is known
by inference, meant is by the very inference that what it con¬
ceives in the form of an image is actual, namely perceivable
accordingly. It is by the same token that it is the circumstances
of such as atoms that are actual, real. In whichever case it is not
the image but the real which is the described consequential,
correspondingly perceivable.
Doctrines inconsistent with the finding that what is real is
s
48
humanly perceivable are certain realisms, sometimes including
an idealism. The concerned things in the latter can be exem¬
plified by Plato’s forms, or universals, associated with attributes,
and those in more exclusively the former by Aristotle’s assumed
underlying substance. Both, abstract archetypes of properties,
and concrete substrata as their bearers, have been thought real
but unperceivable by man. That what is real is what is knowable controverts, in addition, the skeptical position that no
reality, some being provided, is knowable.
On the positive side, the furnished understanding of reality,
as was indicated, makes it possible to establish whether and
what realities exist, what accordingly, because of their identity
with the knowable, is or can be known. In consequence spec¬
ulation in epistemology, dealing with theory of knowledge and
comprising the other major sphere of philosophy, is made like¬
wise unnecessary. What is or can be known is what has been
found to be or is real, i.e. consequential as explicated.
In the search for the real, for knowledge, there is between
the real, the known or knowable, and knowledge also the issue
of means by which knowledge is acquired. What these means
are, a question in epistemology, is equally determined by what
in that regard is consequential, whereby the means, too, can be
held real, as suggested near the beginning of this chapter with
respect to the senses (p.44, third par.), and elsewhere with
respect to reflective methods and other matters. The reflective
methods, distinguished before as eduction, induction, and
deduction, and by means of which other realities that include
broadly comprehended ones are herein sought, are examined
in the course of seeking those realities as treated in various
chapters. And the chapters immediately preceding and follow¬
ing the present one treat of further things that are realities by
being means to knowledge. They concern reflective general¬
ities, the first as to concepts of awareness, whereof subsequent
realities may be discovered, and the second as to findings in
logic, bases for discovery of additional realities. It is for the pre¬
sent chapter to for the first time explore realities other than of
general reflection, although the reflective means of eduction
and deduction used in the other chapters mentioned, and used
here abundantly as well, are yet prominently supplemented
here with the one of induction. The presently explored realities,
represented by nature, are largely no longer means to knowl¬
edge, but more exclusively its sought after objects, as often,
more directly, means toward human fulfillment which is no
longer a means, but is desired for its own sake.
What fundamental entities belong to these realities, includ¬
ing any that might be held to extend beyond nature, has, as
previously indicated, been a matter of much metaphysical
49
debate. And the general requirement that the real be of those
facts of one’s possible awareness on which one can be depen¬
dent in one’s pursuits was observed in the foregoing. Particular
laws of nature as things of that dependence are for example
accordingly realities.
It may be further observed that since the concern in realities
is in what can be of knowledge, if something is presupposed by
that knowledge, something one may find to be dependent on
for it, then the presupposed can be held a prior reality. The
dependency of one thing on another is a form of implication,
and by later law of transitivity, hence, if something is depen¬
dent on the first thing, it is on the second. Accordingly since the
knowledge at issue is of dependency in man’s pursuits, and the
presupposed spoken of is of dependency in that knowledge,
the presupposed is of dependency in man’s pursuits and is
therefore by the discussed a prior reality mentioned. It is so
insofar as it at least may be found a means to that knowledge,
as are the senses found in relation to the outside world. And
evidently if such a presupposed reality presupposes something
else then that is by repeat transitivity a prior yet reality.
The priorities are still more emphatic if representing, as
presuppositions by knowledge of another reality, not only
realities but their knowledge. It can correspondingly be found
that the common acceptance of particular realities presup¬
poses, as noted in the introduction (p.3, third par.), not only cer¬
tain realities questioned, but knowledge of them.
It is in accordance with the preceding thus possible to deter¬
mine an order of priorities of reality, supplying a framework for
more extended searches regarding it.
1
MIND
As observed, all things are known by man through the me¬
dium of one’s consciousness, past, present, or future. In agree¬
ment with the above, hence, since knowledge of all reality
accordingly presupposes consciousness, consciousness is a
prior reality. This is of interest in view of the traditional opposi¬
tion between espousals of only a material world and of only a
conceptual one, more on which subject later. Since all appre¬
hended is, to proceed, an ingredient of consciousness, however,
it remains to be asked which ingredients have reality prior to
which others.
Because the justification for things to be considered real was
50
seen their consequentiality in man’s pursuits, in man’s purposes,
these purposes, hence presupposed by knowledge of other
realities, have prior reality. Many of the purposes, to continue,
have as interest more basic ones. One has certain ends as
objective so as to lead to ones of more concern. There are then
as was suggested (p.48, third par.) eventual ends aimed at not
for the sake of others but for their own sake, fulfilling what can
be regarded as a person’s fundamental purpose of being
satisfied, of attaining one’s what is spoken of as happiness. This
what can be held a fundamental quality of purpose or concern,
whose knowledge as ground in one’s conscious pursuits is by
the foregoing presupposed by all other knowledge, can be
accordingly seen as primary as a reality. Since what is, further,
principally meant to identify a person, a self, is, as in hope for
surviving the decease of one’s body, that quality of purpose, the
concern with being satisfied, that quality can be considered to
be the self.
This definition of the self may seem unusual. The self is fre¬
quently identified instead with the whole of consciousness, or
what is called mind, although the soul as the self and subject
of consciousness may be heard of. And materialism identifies
the self with the body, alone or with consciousness as a com¬
ponent. Whether or not consciousness is accordingly insepar¬
able from the body, it can be separately considered in its man¬
ifestations, and these manifestations are often held to represent
the self. It is the self conceived of again as possibly disem¬
bodied. Consciousness encompasses, however, all of what is
called the external world, and the self is not meant to consist
of also that world. Nor is it meant to consist of all the internal
world, of objects of such as dreams or the imagination. The self
is rather thought of as different from these objects, sometimes
as somehow unperceivable except for a manner of functioning.
Viewed from the outside it might thus in fact be considered
vaporous and not discernible, as are all of another person’s per¬
ceptions. What is accordingly meant to distinguish continued
life after the death of the body is indeed the functioning that
is meant to distinguish all life, namely the functioning, the
presence, of certain purpose. The writer will not at this point
address the alluded to arguments (p.9, second par.) on what
constitutes life, but for reminding that it is a matter of definition.
And by the preceding whereas the purpose found to distinguish
living organisms is preservation of them and others, the pur¬
pose found to distinguish sentient, conscious, beings is satisfac¬
tion in general. This purpose distinguishing the self is while
perceived from outside by effects only nevertheless like other
purposes an internal perception or quality, which may be
named individual sensibility, called at times ego.
51
The question may be raised again as to how a perception,
said to occur to someone, can occur without a subject different
from it. The topic is sometimes presented by proposing the self
to be a substance like matter, and the argument is irrelevant,
the point as in other cases being what is chosen to be meant by
the self. In addition the linguistic nature of the issue may by
now be evident. Though perception might in subject-predicate
fashion be said to occur to someone, the language does not
require that a perception, a quality, may not exist as an autono¬
mous subject of other perceptions, and as perception conscious
of itself in what is called apperception. It may be asked how
that can, in view of the seemingly more palbable processes of
such as the body, occur. But it does and, as in the case of e.g.
resemblance (p.22, sixth par.), requires only its own confirma¬
tion. As a matter of fact a person of a body, viewed as the
subject, is, as all else known, itself made up of perceptions.
These perceptions of the person accordingly could themselves
be thought to require a subject of them. But since all is percep¬
tion, there is no subject outside it to which it occurs. Indeed one
would be loath to declare that something is perceived by one’s
body, and not by oneself as the inner sensibility of concern.
The maintenance that the self be an entity distinct from the
body is by some thinkers a most vehemently assailed one. The
attitude can be understood as recognition of only material
reality, and it is irrelevant once again, the meaning of the self
being a matter of choice. As presently disclosed, furthermore,
the described self, far from subordinate as reality to the body
or other material things, is a prerequisite for them.
Other arguments in behalf of a bodily self have been ad¬
vanced, likewise containing irrelevancies. Upon finding no
entity thought to qualify as a mental substance representing the
self, former philosophers, adhering to a conceptual one, ques¬
tioned the presence of a criterion by which many perceptions
can be held to belong to one person. Since attempts at an
answer were by later thinkers thought to be useless, the body
was again proffered as criterion of an individual. But the issue
is not relevant to the, once more arbitrary, definition, instead
concerning what makes the defined a single entity.
This question can be asked about a sought after substance as
well, spiritual or corporeal. And what makes an entity a single
one, whether at a moment or through time, is characteristically
left out of definitions. The comprehension is normally implicit,
and there should be no wonder, since there seems no common
criterion, with designation of unit another arbitrary matter. The
simplest unit can be the smallest discernible part of a percep¬
tion, and a single perception, one that may be held to be of a
moment, can be one that undergoes no change. Units can
s
52
become more complex depending on the interest, familiar
ones among these being material objects, of a single identity
though of, as earlier remarked (p.24, sixth par.), manifold
simultaneous perception and subsequent changing ones.
With many of these objects it may be of comparatively small
importance whether they remain one and the same. The same¬
ness of the self is of paramount importance, however, because
it figures in the very seeking of fulfillment of one’s purposes, by
which other things become important. The sameness figures as
stated inasmuch as one wants to know it is the same oneself
whose purposes may be fulfilled. In the long run the interest,
as explicated, is that oneself as the sensibility, the fundamental
quality of purpose or concern, is satisfied. It is in the form of this
sensibility, perhaps lastingly satisfied, that one wishes to sur¬
vive. This sameness of the self is also the interest of persons
outside it, in what is known as the question of responsibility.
Others wish to know that an action is performed by the same
person that, depending on the action, is treated correspond¬
ingly, so that the person’s purpose in the actions be in harmony
with the purposes of others. The sameness of the self accord¬
ingly hinges on self-awareness as of the same purpose, con¬
cern, since the interest both of the self and others is whether
perceptions, pleasing or displeasing, of different times are
regarded by the self as its own, and its unity is related to that
sarheness. A self is chosen to be called a single one because
it identifies itself with the selfsame purpose concerned. In
similarity to the above singleness of a least perception of a
moment, it identifies itself with a purpose by its whole, not
separately by any part, and it continues to do so through time,
not undergoing change, any different part as the purpose, or
any through change different purpose, not being by that
difference, as self-determined, the selfsame. How the identi¬
fication is accomplished is again immaterial, it mattering to self
and others that it is.
Thinkers who identified the self with an internal entity de¬
scribed its unity through time by, while seeing only conscious¬
ness in general, its memory of awarenesses of things. Con¬
sciousness as a whole—made up of varied things, including
phenomena of the material world—cannot of course be held
the subject of past awarenesses, nor is it clear what makes any
a unit at a time. But as factor in defining unity of an individual,
memory was irrelevantly criticized as leaning on something
unreliable. An entity, including unity through time, can, again,
be defined by any attribute of interest.
The objection to the memory was as before intended to
support the concept of a bodily self. It is contended that one’s
memory of having perceived things cannot, in view of failures
53
of memory, be a criterion for one’s post existence, that it is
therefore the sameness of the body which identifies the indi¬
vidual. It can be noted that even if the body were better suited
for that purpose, it would not thereby have to be a part or the
whole of the self, not to repeat dependence on definition.
Things can be signs of something, e.g. of the same individual,
without being its constituents. What is more, memory is relied
upon for all that is past, including acquaintance with someone’s
body. Still more, the body can change to a point of losing any
individual resemblance to its previous state.
Regarding failure of a memory in general, it has to do mainly
with the very particulars of the external world, including
individuals. An amnesiac will even fail within a short span to
recognize his immediate family. But he will remember general
things such causal connections. Similarly men at large
remember to considerable detail fundamental things required
in their lives. Since they remember these as of account in their
purposes, they remember themselves as of those purposes. Or
simply, since their remembrances are of perceptions by them¬
selves, they remember themselves as of the perceptions.
As last of the arguments put forward, the contention may be
mentioned that since by person is often meant the outer
appearance of someone, that aspect be held to be constituent
of oneself. But in this case merely a choice of definition takes
place. Reference to persons was in what went before here made
sparingly, because the self as discussed can apply to animals
as well. Apart from this, bodies are known to be accoutrements
of sentient beings, and it makes sense to give a name to the
combined attributes. But the question is what entity is wished
to be talked about, and presently it is the discussed individual
sensibility, referred to customarily as soul or the self.
By the foregoing the knowledge or reality of this entity, as the
concern in seeking other realities, is presupposed by all other
knowledge while not presupposing any of it since not requisite,
and it can therefore be stated that
THEOREM II.1. The self is the primary reality.
Proof of theorems in this chapter is contained in the discus¬
sions preceding them, instead of being formally given after
them. Since the chapter functions as a digressive whole in
establishing these theorems, an informal style is preferable.
It will next be suitable to return to the concept of con¬
sciousness, noted at the start of this section to be presupposed
by knowledge of all other realities. Among these is the self,
and the reason it can be considered the primary reality none¬
theless is that- its knowledge presupposes consciousness by
being an instance of it, and thus does therein not presuppose
knowledge of another thing. The self knows all further realities
by consciousness of them, however, and hence consciousness
con be placed after the self as a presupposed reality, as can its
knowledge, by reason of one’s very knowledge of things as
phenomena of consciousness.
The consciousness under discussion might also, and in a way
less confusingly, be called mind. Consciousness often connotes
apprehension of the physical world, in contrast to, for example,
dreams; the consciousness of this writing, as should be evident,
is of all human, or animal, apprehension. Consciousness also
connotes apprehension in the present, not in the past, or per¬
haps the future; in the now written about, consciousness is
meant to include, as in the case of knowledge, apprehensions
committed to memory, from which they might be recalled. The
consciousness meant here is in brief the sum of all perceptions
that have been those of an individual and to a large degree
preserved in memory. This totality may in accordance with
usage more suitably be termed mind.
This naming is reminiscent of mind as spoken of in psychol¬
ogy and including the unconscious. With the unconscious to
mean things that were never in consciousness, however, it is not
included in the present concept of mind. Things that were never
in one’s consciousness though part of the individual are simply
by definition part of other than mind in the organism. This as
dictionaries attest is manifested as the common understanding
of mind, and efforts to prove it false can again not accomplish
more than give the word another meaning. That meaning is
also unable to delimit what part of the organism is the uncon¬
scious, which since all is known through consciousness nothing
can be. In response to frequent further objection it may also
again be noted that, as observed in relation to the self, mind
need not be described in terms of substance, perhaps as recep¬
tacle for perceptions, but can be comprehended by the attribute
of perception alone.
With mind accordingly naming consciousness as the known
reality determined above, it may be stated that
THEOREM 11.2. Mind, including the self, is a reality prior to
every other.
It should be added that by the mental, as by the conceptual
or the like, can as common be herein also meant only the in¬
ner, nonworldly, objects of awareness, as normally understood
from the context, or occasionally referred to as purely mental.
In compliance with what was said this narrower sense of mind
was seen likewise not to be identical with the self, and taken
strictly it does not include it, but is of things in its cognition that
are not of the external world, examined in the next section.
The conclusions stated by the preceding two theorems are in
certain agreement with the one in the famous “I think, there-
55
fore I am” of Descartes of the 17th century, asserted in his search
for indubitable reality. His conclusion that he exists was not well
substantiated, however. He did not furnish, primarily, a concept
of being, of reality, and, to a lesser extent, of the I, the self,
from which the inference within the statement could be drawn.
He made the assumption that the world is as unreal as are
dreams, and since even this form of doubt consists in his
thought, he inferred that at least he and his thought are real.
The contents of even dreams and the world assumed imagined,
however, could in their occurrence be regarded as having as
much being as he and his thought does, and explanation is
wanting as to why the latter entities should have more being
than the former. Further, the use of “I” in “I think” does not
imply the existence of an I distinct from the doubted physical
matter. The phrase can be a succinct way of saying “Inside this
bodily shape occurs thought”. Or, the “I” can be merely the
discussed subject of a sentence, not implying an entity distinct
from the predicated thought. Descartes nevertheless recognized
that a thing can be identified by its attributes, in this case mind
by thinking.
The much debated dualism of mind and body, or mind and
matter, propelled by his views is further of interest. Man, reality,
and other things can in truth be arbitrarily and correctly seen
as twofold, threefold, or more by repeated dichotomy. By con¬
sidering any particular feature as distinguished from the rest of
a thing, a twofold division is made; by continuing the same in
regard to either of these parts, a threefold division is made; and
the procedure can go on. Thus in accordance with above, man
as known might be divided into the self and other attributes,
which attributes might similarly be divided into mind, in the
restricted sense, and body. But the insistent questions regarding
the dualism have been whether either mind, in a wide sense
and as often equated with the self, or matter is not part of the
other, or whether one or the other exists at all.
In the case of whether matter is the inclusive entity, as pro¬
posed by materialism, the questions can be understood by con¬
sidering that matter and its events can seem to, compared to for
instance the imaginary, be the only objects of cognition that are
effectual with regard to man’s interests. This explanation illus¬
trates as basis for a reality the possible dependence on it in
man’s pursuits. By holding then matter and its behavior to be
the only reality, mental occurrences are accounted for by
regarding them as part of it or by denying them altogether, in
what can be viewed as needless equivocation of worldly reality
with any form of being.
In the case of whether only mind is real, a proposition of
idealism, the questions can be understood in the light of the
s
56
same cognition, the manifestation in mind of all, the world of
matter as well as other things, as aided by finding all percep¬
tions to converge in the brain. Matter can be considered to
nevertheless exist in that manifestation, and not as part of a self
not synonymous with mind.
These remarks notwithstanding, since presently mind and the
self are at issue, the bearing of these questions on their status,
namely whether they are forms of matter, if not nonexistents,
may be considered.
That mind or the self be part of matter, specifically the brain,
can be seen to be impossible. Whatever the brain portion
assumed to be indentical with mind, with consciousness, might
be, inasmuch as it would be manifested in particular percep¬
tions characterizing matter, it cannot be identical with other or
all perceptions, i.e. mind. And inasmuch as the self was ob¬
served to be of a quality distinguished by no more than the
sensibility spoken of, it cannot be identical with certain matter,
distinguished otherwise.
At the basis of this impossibility lie as well the mentioned
diverse ways in which things are identified as selfsame. Beside
the character distinguishing individual perceptions of material
objects from the other perceptions, those objects were indicated
to admit of different perceptions of them through time while
held to retain their identity and even be unchanging. The self
and states of consciousness, however, do, as noted alongside
(p.51, last par., through p.52, second par.), not allow that variety
of perception, whereby a version of them could be found
identical with some perceived matter. A state of consciousness,
a perception of a given time, is not the same as another
perception, e.g. of some matter. And a perception that differs
from one of a sensibility which is a self is not of the same
sensibility, if any.
As to the existence at all of mind or a self, it, their reality
before others in fact, was determined precedingly herein.
Doubts about it have extended from the denial of discern¬
ment of oneself in consciousness to the mentioned denial of
consciousness itself. The first of the denials is associated with
the 18th-century philosopher David Hume, who on introspec¬
tion averred to find nothing but a bundle of perceptions. His
views differed from materialism, by giving no more credence
to matter than to a self. But he fell short of recognizing that ei¬
ther of them need be manifested by nothing more than certain
of the perceptions he spoke of, ranging from touch, signifying
the material, to concern, signifying the self. The second of the
above disavowals is as indicated associated with behaviorism,
which seeks to understand men through observations of their
overt actions. Its refusal to recognize consciousness, as readily
57
apparent and suggested before, belies the nature of its own ob¬
servations, the form in which the behavior of others is known.
The lost doctrine is on example of the influence of observa¬
tional science, with o related scientific determinism, the belief
that there is no freedom of action, that all events are deter¬
mined by physical or chemical laws. If they are then men’s
behavior might indeed be studied from their physical response
to stimulus, without paying to their conscious at least attention.
That men’s behavior be studied would then not seem of interest,
however, with the actions of those who study it predetermined
likewise, whatever they may learn of others.
Whether or not man’s mind is thus subject to physical or
chemical laws may be seen presently, in its further exploration.
While the self and mind were found to be prior as realities
to any other, it has not yet been determined what further
entities, entities apprehended through the mind, may be
regarded as realities, as objects of knowledge. Traditionally the
major dispute in epistemology, in theory of knowledge, has
been between rationalism and empiricism. The two doctrines
contend that the source of knowledge is, respectively, reason
and experience. The distinction between the two, though, is not
entirely clear. There is little question that the rational is meant
to be of mind contrasted with a physical world, and the empir¬
ical had been referred to in terms of sense perception. But the
classical empiricists tend to discount a reality of senses and
other physical entities, relying only on mental impressions. It
may be asked in that case, which of the mental occurrences,
before acknowledgement of a physical world, consist in exper¬
ience, and which consist in reason.
An answer suggesting itself is that the first are the involuntary
and the second the voluntary. The division should reveal itself
to be a useful one, but it can now be asked whether such a divi¬
sion exists if there is no free will, if volition is not free. For by
voluntary things are meant here, as appears customary, things
resulting from volition independent of external compulsion. If
they were compelled by something else, they should be con¬
sidered involuntary.
It might be suggested that by voluntary mental occurrences
those be meant that emanate from the mind distinguished from
the physical world, without being free from compulsion. The
distinction of a physical world is assumed not to have yet been
made, however, preventing knowledge of whether something
emanates from it or otherwise. And should such a world, of the
all-pervading physical and chemical laws argued, be assumed,
then.no mental events could be free of them.
It may thus be of benefit to see whether a valuable dif¬
ference cannot be established in accordance with the initial
s
58
meaning of the voluntary and involuntary, whether some occur¬
rences, purely mental or other, arise from o free will, and
whether some do not.
It should be understood that the freedom at issue is not one
completely unrestrained. Supposing on initiating free will, even
the imagination may be hampered by conditions like fatigue,
and bodily action is always constrained to function in accord
with material lows of nature. At issue is whether one’s will,
one’s effort directed at on intellectual or physical end, is wholly
determined by factors external to it, such os divine or natural
lows, or even logical ones in the sense that one could not
deduce o false conclusion from given premises. The question
of complete jurisdiction of mon-mode lows is understandably
not often raised. The question of their futility in absence of free
will, however, is raised frequently, since in that absence monmode lows ore seen os not influencing men’s actions, and it
may be noted that in absence of free will, again, legislation
would be likewise not o matter of choice.
In order then to find if free will exists, it will be well to recall,
from the discussion of necessity, that o universal low, on invar¬
iable fact about something, depends for its truth on no more
than that invariability. The mentioned lows of nature ore of
course accepted os preponderous realities. Questioned is that
man’s will is to any degree exempt from them or perhaps other¬
wise not compelled to take o certain course. But in confor¬
mance with the preceding, if some other fact about something
is determined equally invariable, then that connection is
equally o principle. It is o principle, o corresponding reality,
even more emphatically if the connection con always and fully
be observed, and if, in keeping with the order of priorities
spoken of, knowledge of the other realities depends on it.
And the finding is not merely that circumstances do some¬
times but perhaps not always not impose one’s will. If that were
the cose, it could be presumed that when on act of will seems
to hove no cause, it is because none has yet been found. Such
presumptions ore mode about physical states, because oil of
them ore token to hove o cause, due to the constancy regarding
them in observable coses. The constancy, however, applies to
will in the opposite way, on absence of connection. One’s will
is always found unimposed by any circumstance. One is free to
choose or not any object of volition, even if the object, os may
be o recollecting or o physical action, cannot be realized. That
one con do so con in addition be found at every try and without
o lapse in particulars, in contrast to such os physical causation,
not always observable of on event, while inferred os indicated
above, and intractable in particulars, os remarked in the
introduction (p.5, second par.). The knowledge by principle that
59
one is free to will any object, that one can in fact thereby
realize many of them and their desired consequences, is
furthermore presupposed by regarding, as suggested (p.45, last
par.), as realities any of those results and their world, including
the corresponding laws. One is cognizant that one is free to
make choices regarding one’s fate, accordingly considering the
entire world that enables the same consequential, real.
It has been argued that volition does have a cause in its pur¬
pose or reason, in the objects of one’s will. As one is not com¬
pelled by any other circumstance to will something, however,
one is not compelled by the objects of one’s will, by given pros¬
pects, but can will any other thing or none. The belief that the
reason for a volition may be its cause can be attributed to
equivocation regarding senses of reason and cause. The cause
of an event is sometimes called its reason, and a reason for an
action is sometimes called a cause for it. The first case can be
taken as reflecting mental explanation of the event, and the
second as of reasons as preceding volitions, as causes precede
effects. But the issue concerning free will being that a volition
is not required by something else, cause and reason in this con¬
text are of opposites. In concord with former observations, one
thing requires another by being always accompanied by it. This
can apply to cause and effect as well as to logical implication.
Accordingly the cause requires the effect. But a volition instead
requires, by its meaning, a purpose, the reason, not the reason
the volition, contrary to the supposed causation.
The purpose contained by meaning in volition has in some
arguments for will as predetermined been omitted. Presumedly
thinking a purpose a cause, it is maintained that if a will is un¬
determined it is tantamount to chance, having then no reason,
by which persons are expected to act. It is in this vein put for¬
ward by some that widely accepted free will exists though
predetermined, its freedom residing in an absence of impedi¬
ment to carrying it out. But such an absence was observed miss¬
ing. One can indeed be prevented from carrying out one’s will,
whether by physical factors or lack of knowledge. Furthermore,
the contended freedom from impediment is not the one meant,
which is just that one’s will is not predetermined, that one is
free to seek required knowledge and to accordingly choose
one’s actions, so that one may correspondingly pursue one’s
ends, not otherwise ensured.
The will can as a matter of fact be regarded as free by, in con¬
sequence of its, definition, if understood as the power of com¬
mencing purposive activity. This meaning is the one herein
adopted in a precise sense and can be viewed as the general
one considered above, when volition is regarded as a starting
force directed toward some end, as in the frequent definition of
s
60
will as the power of self-direction, contrasted with direction by
something else. And, as noted to also apply to organisms at
large (p.50, second par.), by purposive activities or events, by
ones directed toward some end, are more specifically meant
those that by appropriate adjustment accomodate whatever cir¬
cumstance toward what does, or appears to, attain, or is likely
to attain, the end, at issue.
The qualifications of appearance and likelihood should be
elucidated. The appearance has to do with man’s supposition
that the object of the will can be attained, and the likelihood
concerns actual probability of the attainment. The difference is
analogous to that between possibility of something insofar as
man knows and insofar as not precluded by a law, in accord
with observations in the last chapter. But purposiveness also has
a more solid base. After the initial force now identified with the
act of will, assuming it to have at least the most immediately
aimed at result, comes what is accordingly already a purpose
fulfilled although most often only a possible means to a further
end. Ends that are purposive outcomes, toward which cir¬
cumstances are adapted and which may be called purposive
events in a restricted sense, are consequently much more fre¬
quent than the above qualifications make it seem. For instance
man’s acts of will result more often than not in the intended
bodily motions, toward which intermediate processes also func¬
tion, and which normally succeed in, in what can be held pur¬
posive.events in a nonrestricted sense, some further material
purposes, like taking care of mundane needs.
With purposive events so understood, and the will the power
giving rise to them, it may be asked if some other condition
does not compel the unqualified will to do so. But if it did then
since the will’s behavior would be decided by it, that behavior
would not adjust to whatever circumstance to fulfill the purpose,
but would be the same irrespectively. Hence the will would not
be the force initiating purposive activities, contrary to its mean¬
ing, and therefore it is not compelled by something else, but is
free.
By this determination the previous finding that one has a free
will is equivalent to one that men, and animals, possess the
power of will, the otherwise redundant ascription of freedom
usable in making it explicit. The finding takes the form of a
principle in that one can if not at all times then when of a
minimal alertness exercise one’s will, exert, purposive, effort.
The by the foregoing established free will should be of much
pertinence to behavioral or social sciences, when on the order
of natural science endeavoring systematic studies of men’s
actions. The tendency has been to look for hidden causes, to
the dismissal of men’s conscious grounds. Men, not to mention
61
animals, are often thus degraded to the rank of objects, if
somewhat complex ones. For the solution of problems profes¬
sionals may try to influence or gain information from men
regarding their actions not by addressing their reason, but by
eliciting unreasoned replies, as in psychoanalysis, though even
animals have reasons for their actions. Similarly, history is
thought to be moved by inexorable laws, and in thinking to
know what they are, governments impose them on populations.
To boot, in the well known notion that responses to stimuli can
be conditioned is implicit a presumption that the experimenters
can create natural laws of their own, in conformity to which
mankind is expected to eventually acquire attributes according
to plan. The conditioning is based on reward or punishment of
its subjects according to their behavior, in the familiar method
of coercion, made worse by the detachment of assumed sci¬
ence. The assumption is additionally that men can be stupefied
into holding the connections between their actions and the
requital to be laws of nature, in denial of an intelligence even
animals, fleeing a cruel master, possess. The denial is in tan¬
dem with the belief that one’s will is but subject to forces
imposed on it. The general error of behavioral sciences is
indeed their thinking to be ones, neglecting that one’s behavior
is guided by one’s free will.
That one has a free will can be regarded as, in view of the
constancy by which it is determined a principle, the first induc¬
tion about now discussed reality. The finding of the continued
existence of something, as would be the case regarding the
world of matter, could likewise be called an induction, and
since the reality of that world is not known without the reality
of selves, their continued existence might be considered an
induction also. Whereas there is little argument about the
continuance of the physical universe, however, there is much
argument about the continuance of individual selves. The
induction of a free will concerns instead, similarly to customary
causal inductions, a power, a capability, namely that of exerting
effort, perhaps a minute one of refraining. The induction, it is
again notable, is a prescientific one, found to be present in the
very holding of knowledge in science as elsewhere worthwhile,
by understanding that one has corresponding choices of action.
Speaking of knowledge and action, there are other pre¬
scientific inductions having to do with the will and requisite in
one’s pursuits. The action is brought about by the will, as is for
that purpose the recalling of remembered, known, things. The
inductions are that, in conjuction with other enabling con¬
ditions, willing of the recollections and actions will produce
them.
It has been questioned what the willing consists in, and
things suggested in that regard, such as a thought of the aimed
for, were determined not to have the desired effect. This ques¬
tion, too, is, however, irrelevant. Men and animals have knowl¬
edge of the nature of a volition, be it only referred to by such
a general word or none, of account being that whatever the
willing consists in, one learns of it as a cause of certain events
regarding oneself.
Insofar as this finding of causation is about bodily action, it
has been called interactionism, which also refers to causation
of mental events by bodily ones. The existence of the inter¬
actions has been beside free volition denied, by holding it
incomprehensible that an immaterial mind be causally con¬
nected with a material body. The connection is of course com¬
prehensible and fact in the very finding of it, and causal
interaction among material things rests on scarcely as firm a
ground. Its legitimacy is secured by being through physical
causation, as through light rays and the sense of sight, con¬
veyed to consciousness, where it is as a mental effect mani¬
fested. Should this causality between matter and mind ter¬
minate, the cessation would be physical at all events, since if
mental then the physical by being dependent on its perception
for its reality would likewise lose it.
Whereas the perception of physical reality is involuntary, not
the direct result of volition, the before spoken of actions, at least
their beginnings, and the rememberings were seen as volun¬
tary. These, too, are in a sense involuntary, by being results of
volition in accordance with spoken of causal laws. What is truly
voluntary are the volitions themselves, rather than their results.
The existence of these voluntary mental contents, of those of
free will, was determined above. And it is possible to deter¬
mine with no lesser ease the existence of involuntary ones. As
it is discernible that one can produce voluntary thoughts, so it
is discernible that one has involuntary experiences, as
suggested at the beginning of this paragraph. Among these
experiences are those of physical realities, which since depend¬
ent on the involuntary perceptions render them also realities.
The common awareness of both the voluntary and involuntary
is sometimes expressed by the distinction between active and
passive mental contents, referring to the use or not of volition,
both contents otherwise involving mental activity, change,
unlike connoted.
There should be thus no trouble to examining the contents of
mind with respect to the distinction between the voluntary and
involuntary, another division exhaustive by dichotomy. Speci¬
fically, the division can serve as a stepping stone for exploration
as to what ingredients of any of the two parts may be further
realities.
63
Having determined the real to be what is of possible
dependency in man’s pursuits, it follows that all reality must be
subject to universal law, to explicated constancy. As indicated
before (e.g. p.4, last par.), the reason that something is found
of possible dependency in some regard is that it, perhaps as
existence itself, always accompanies a certain other thing of
account. This lawfulness is exemplified by the implication by
which the self or mind are real because other realities depend
on them.
As a result those mental ingredients, those concepts, that are
solely voluntary, that are not confirmable by involuntary per¬
ception, are not realities. Since voluntary concepts are free from
compulsion, they are free from constancies by which they
would be of dependency. Concerning either intrinsic, logical,
truths, or extrinsic, natural, ones, one can if relying on volition
make any, e.g. a false, determination, even if contradicting the
concept in view.
This conclusion invalidates the contentions of rationalism if
understood as upholding that, intrinsic or extrinsic, knowledge
is acquired through the mere use of thought or reason as a
voluntary exercise of mind. The understanding is in accordance
with (p.57, fourth and fifth pars.) distinguishing between the
rational and empirical as between knowledge through the
voluntary and involuntary, and it was adumbrated when dis¬
tinguishing between the a priori and a posteriori (p.l3, third
par.). As remarked there, some of logic is thought to be known
true without proof, and the same was proposed of such as
causality. The latter finds less support today, but the former is
likewise excluded by the foregoing. This negation of rational¬
ism does not apply to knowledge by way of deduction. That
method will be seen securely grounded in involuntary percep¬
tion, and as purely mental might be called a rational means to
knowledge.
The preceding does not of course deprive voluntary mental
activities of value. They were already a few times referred to as
source of man’s actions, in the interest of which knowledge,
dependent on the involuntary, is required. And they equally
assume an extensive role as pertains to that knowledge. Rela¬
tive to their likewise referred to use for recollecting, the will is
used for calling forth the concepts, the premises, from which
above mentioned deduction can proceed, as well as for recall¬
ing experiences leading to induction. As observed, further,
anything learned is equally summoned from memory, to apply
it to subsequent purposes. In addition to these uses of volition
for actions and for evoking, for their sake, such as past deter¬
minations, volition is also used for making, to that end, the
determinations, as when affirming what is perceived, the act
N
64
distinguished before from perceived states alone (p.l6, second
par.), or when deciding a definition. These determinations
serve to fix situations in memory, and they are for like reasons
made into linguistic statements, to represent forms of
knowledge.
The voluntary mind thus also has a direct part in acquiring
knowledge, along with the involuntary as conveyor of that
which is to be known. It should be noted that although aware¬
ness of a definition was earlier viewed as loosely a form of
knowledge, definitions naturally are not of the dependability
accorded other objects of knowledge, those comprising stricter
realities and observed to in their law-based dependability
require involuntary perception. There is a degree of man-made
lawfulness in definitions too, insofar as it is for utility agreed
upon to abide by them, and when that consent by others is
learned, it is so in an involuntary way. But when a definition is
specified by oneself, it can be quite arbitrary, and though still
useful to adhere to, it rests the less on established and to that
degree dependable meaning. The upshot is that above volun¬
tary determinations need not be assertions of, dependable,
realities, involuntarily perceived, but may be, changeable,
choices made where desirable. Voluntary mind nevertheless
partakes in the acquisition of knowledge by ascertaining what
is supplied by the involuntary.
Volition participates in that acquisition also more indirectly
than observed before. Recollecting for and acting is done also
to gain knowledge, not only to put it to use for other purposes.
Whichever the purposes, both the voluntary and involuntary
mind are by the preceding indispensable to knowledge of any
of the mind’s content as reality.
While both are thus indispensable to knowledge and hence
are realities, they understandably do as before not assume
therein priority over the self, which after all is the possessor of
these capacities. More in keeping with the aforesaid the prior
reality of the self may in the following again be explained by
the presupposition by other realities of not only the self, but the
knowledge of it.
The introduction above of affirmation as, in acquiring knowl¬
edge, another voluntary act brings more subtlety into the dis¬
cussion of mind, as befits its all-encompassing nature. But the
discussion should be distinguished from those which propose
theories of mind as comprising what is labelled man’s psychol¬
ogy, somehow distinct from man’s conscious and freely pur¬
poseful life. It was already noted that by the usual meaning of
mind any such subconscious life would be consigned to other
parts of the organism, not to mention the impossibility of an
unconscious as a real entity. More critical is that this psycho-
65
logical structure, rather than chosen conscious purposes, is pur¬
ported to supply the drive behind man’s actions. As indicated,
such conceptions of mind find their basis in holding the
behavior of matter as foundation by which to know all reality,
the grounds for human actions sought accordingly in physical
causation. But to seek the grounds one need only resort to
introspection, it being origin in purposeful consciousness which
belongs to the meaning of action, as compared to other bod¬
ily activities. And what truly matters is that it is the mind of con¬
sciousness, presently the free volition responsible for action and
omitted by psychology, that is now explored.
It might be as well to dispense with the discussion of finer
points in mind’s volitional part. The discourse could proceed by
simply observing that the self is part of conscious reality as soon
as anything else is, and then seeing what subsequent priorities
can be found. But volition as mere affirmation or the like is of
interest both in making determinations as wholes and in con¬
sidering subjects of which determinations are to be made.
These become issues in logic. Something is often said to follow
from propositions, whereas it actually follows from the things
proposed. It is not from acts of proposing, but from the con¬
ditions, that Socrates is a man and that all men are mortal that
it follows that Socrates is mortal. Instead the propositions func¬
tion in mentally setting down the conditions, from which to
deduce results. Similarly one may mentally regard a subject,
e.g. a triangle, from which to deduce an attribute, e.g. three
sides. The like referred to before as a calling forth, these
preparatory mental acts of bringing conditions or subjects to
awareness then are volitions, designed here to lead to what
may be deduced. Notable is that the things deduced, as earlier
explained, are contained in the initial conceptions, though not
likewise brought to awareness.
The significance is that these things can hence be viewed as
dormant ingredients of consciousness, of perceptions, and the
bringing them to awareness in deduction a volitional taking
into cognizance, an affirmation, of them. These and other
ingredients of consciousness become in this manner objects of
awareness or knowledge, which may thereafter be committed
to memory. Counting now is that there are many things in per¬
ception that, though often implicit as in the preceding, are not
brought to awareness, which bringing may be done in varying
degrees, as one exerts varying amounts of effort regarding
things. This lack of awareness is sometimes voiced by saying
that someone is looking at something but does not see it. One
may see it as a result of slight attention alone, as in ascertain¬
ing particulars of a natural scene, or through lengthy inference,
as in mathematical chains of equations. Volition thus serves
many functions, from slight attention to thorough determina¬
tions—these figuring mainly in gaining knowledge—to more
active employment.
Returning to the self and other parts of consciousness at
issue, they were seen to be, as deductive facts, implicit in the
knowledge of all other reality. They were in fact indicated to be
of awareness, known, and can be so regarded even with the
addition of the act of affirmation to the gaining of knowledge
by man or animal. Unlike when something is contained in a
conception but is not in awareness and must be deduced, and
as when bringing to awareness conditions or subjects to find
what can be inferred, the present entities belong to those of
awareness, those known, with respect to which other things are
or become known. One again has in awareness, is knowing of,
oneself as the subject of concern in one’s purposes, one’s pur¬
suits that render other things realities, and one is knowing of
one’s voluntary and involuntary mind as means required for
those pursuits. That the knowledge of the self should as said
have priority over the other two is decided on the basis that
knowledge of these mental powers is only required for those
pursuits, while knowledge of the self as the concerned is
required for knowledge of those powers.
In detailed scrutiny common to philosophy, it may be asked
how if knowledge requires a volitional act of affirmation can
the self be an object of knowledge prior to volition being one.
However, as can other things be part of perception without
being affirmed, so can volition. One can by help of volition
affirm oneself, without having so affirmed the volition. As a
matter of fact each volitional act cannot be affirmed, since the
attempt leads to a so-called infinite regress. An affirmation
requires volition, whose affirmation requires another volition,
and so forth. What is more, for affirmation of oneself to be as
a volition likewise affirmed, the volition must have been
performed, namely the self affirmed first.
There is little need in this matter, however, for further reason¬
ing so intricate. That one possesses or not knowledge of things
is made certain without the preceding consideration of volition
in gaining it, and it will be apparent from what went before that
knowledge, or the reality, of one’s voluntary and involuntary
mind presumes the knowledge or reality of oneself as the
described sensibility possessing those powers, that the knowl¬
edge or reality of all else presumes the knowledge or reality of
those powers, and that therefore
THEOREM 11.3. Voluntary and involuntary mind, including the
self, are realities prior to every other.
The certainty of the reality of objects of voluntary and invol¬
untary mind seen supplied by the second of the two, the
67
succeeding task is to find which of its objects may be consid¬
ered realities beyond the already determined.
Section 2
EXTERNAL REALITY
The further realities presently sought concern a world and its
inherent laws, through which that world is consequential in
man’s active pursuits, by linking one’s purposes with their ful¬
fillment. In conformance with what was said, this world as
compared to other objects of consciousness, such as dreams, is
a reality because it is thus consequential, and since it functions
as the mentioned link, extents of knowledge of it by man are
requisite.
This knowledge, as observed, is obtained in involuntary per¬
ception, and it may then be asked what, apart from that consequentiality, distinguishes that world from other objects of
involuntary perception, to which such as dreams also belong.
The question is rightfully asked, because the interest is in what
things the consequentiality applies to, so that they may be
heeded for that reason.
EXTENSION
The things in question are normally designated as matter,
and its identification cannot be made in accordance with the
most recent findings on submicroscopic particles. As mentioned
previously (p.47, third par.), such particles are known by their
effects, and by the preceding the question remains what the
things of those effects are. More pertinently, these and other
realities detected in some regard through instruments concern
in those instruments objects of the macroscopic world, the
familiar world found to be perceived through the unaided
senses. That is to say it is the objects of this larger world, e.g.
the instruments, which are found in this and other ways
consequential, and it is these objects that are initially desig¬
nated as material. If the meaning of matter is accordingly to
remain the same then, although it might also be defined by
later findings, it must be definable by attributes of the above
objects first considered.
More important, since what is meant by a word is a matter of
choice, of concern are the above worldly objects in general as
consequential as found. And the question is what characterizes
these suitably called material objects, whereby they may be
distinguished from others.
It could be proposed that material objects are those that
occupy space, since space as a three-dimensional expanse is
considered the container of physical objects. Objects of dreams
and the like, however, seem to occupy space also. Space can in
point of fact be seen not to be the presumed entity distinct from
the objects in it. A distinct concept referred to by a term, it will
be remembered, must be independently perceivable. Any
worldly entity independently perceivable, however, is regarded
at most as occupant of space, space itself meaning to designate
a void. The word is applied to quantitative attributes of these
entities or objects, to their volume and location, chiefly to any
absence of them in some part of reality, the part being deter¬
mined by some dimensional relation of further of the objects.
In either case space is distinguished in accordance with ma¬
terial objects, rather than material objects being distinguished
in accordance with space.
Physical reality need therefore be distinguished by some
other characteristic, and the three dimensions mentioned can
furnish the answer. It is those objects encountered in perception
that are regarded as of three dimensions, of extension, which
are found subject to the laws at issue and are accordingly real.
Other objects encountered, those that are purely mental, are
only accorded an existence that is two-dimensional. The defin¬
ition of the objects of the physical world as those of extension
is fitting because in conformance with Theorem 1.1 of the first
chapter all and only objects of that world are of extension.
An understandable reply to the definition may be that the
contents of a dream or hallucination can have as much a sem¬
blance of extension as they can of spatiality. They can, and a
distinction between a real and an illusory world is not neces¬
sarily easy to make. One frequently believes to be awake while
dreaming, and an argument from the objective, or public,
experience versus the subjective, or private, will not do. The
assurance by someone else that one is not dreaming is, as all
experience, perceived by oneself and can be part of a dream.
It is left to one’s own judgement to make the discernment.
The discernment, if accomplished, is through the constancy
justifying other inferences of reality. Real objects manifest
themselves as extended things of continuing existence, perhaps
changing in form, compared to the evanescence of the illusory.
This constancy of the material world of extension is implicit in
finding it a reality, expounded to be of dependency through its
constancy.
Whether that world is truly extended or, more particularly,
external to mind has been debated. In both cases the question
69
revolves around the meaning of the terms. By the extension can
be understood the above explicated continued perception itself
of volume, and it is accordingly and by the expounded an
object of knowledge, viz a reality. As to externality, if mind is
understood in the broad sense delineated, then physical reality,
determined in perception, is not external. But if mind is under¬
stood in the also viewed narrow sense, as what is purely
conceptual, then physical reality is external to it by meaning
alone. It is external at all events to the self, which was seen as
a sensibility separate from other objects, those of its perception
and including extended physical reality.
This reality of physical objects is one because, as explicated,
corresponding knowledge of them and of what happens to
them can be of dependency to man. The knowledge involves
identifying these objects, distinguishing one from another, at
a single moment and through time, as considered before in
connection with the self (p.51, last par.). What the identities
consist in has, as indicated there, been a subject of philosoph¬
ical inquiry, and the determination of them, as likewise indi¬
cated, is one of choice, though decided by suitability in
accordance with experience. The issue is also confused with
the one of identity of a thing when contemplated by different
attributes, perhaps names, as treated in the first chapter. The
present issue is any one attribute that would define a thing as
it may last through possible changes.
Changes of one kind or another are indeed profuse as re¬
gards a physical object viewed as remaining the same one.
They are so not merely in a living thing, drastically changing
during development, and most often the subject of discourse
about them. Inanimate objects as well as the animate are
continually perceived from different angles, under different
lighting, or by other phenomena not held to signify changes in
the objects though of changing perceptions, as before sug¬
gested. Particular material things are thus identified through a
changing complexity of characteristics.
The description of the things, as in a definition, may be sim¬
ple, because attributes named concern already understood
material entities. But even the most general character of indi¬
vidual material things consists of accumulations of many
perceptions all of which are of the same objects. Their threedimensionality, their extension, may sometimes be perceived
through the combination of perspective, light and shade, and
touch. Perception through both eyes, and hence from different
mentioned angles, is a further means. The angles together with
lighting were also indicated to change in time, while still
signifying the same unchanging objects. The lighting can also
change their degree of lightness or darkness or their color, and
s
70
distance or foreshortening con change their size, the objects
regarded os oil the while not changing. They remain the some
ones by equally elaborate associations, by which an object of
certain like perceptions is seen to have a, perhaps moving,
continuity, although the continuity is not always observed. The
continuity applies to other, changing, material things that
remain the same things, as well as to ones that do not remain
the things. Among the first of these, the now discussed ones,
lifeless objects are considered to remain the same ones if
retaining some principal feature though changing otherwise.
Living organisms, further, seem often, as indicated, not to retain
any individual physical feature at all, their identity viewed as
remaining by their function to preserve a form of themselves,
if not by, unlike in the case of plants, being of a sentient self
described.
Despite this multifariousness of perceptions by which
material entities may be recognized, men do know what they
mean by words for them, as they do by other expressions. Not
only men, but animals, recognize material things as of re¬
peated daily experience, knowing that they behave according
to certain laws, constancies, by which the perceivers regulate
their own lives.
Knowledge, the reality, of the material, extended, world is
accordingly presupposed by the knowledge or reality of those
laws by which men live, and therefore
THEOREM 11.4. The world of extended matter is a reality prior
to laws applying to it.
It can hereupon be inquired what kind of laws, laws that
make it consequential and therein real, this world is subject to.
CAUSALITY
In consonance with what has been said, the answer immedi¬
ately suggesting itself is that it is causal laws. It is causation of
events, leading to the satisfaction or dissatisfaction of man’s
concerns, which makes the world the reality of interest.
Similarly to the case of that world, it may then be asked what
distinguishes causal laws from any others.
It could be proposed that causal laws are of occurrences in
time, wherein the cause is followed by the effect, since by the
foregoing the issue is what event that has not occurred would
if some other did. Sometimes cause and effect appear
simultaneous, however, as when one object is pushing another.
Time can as a matter of fact like space be seen not to be the
presumed entity distinct from the occurrences in it. It is not
meant to refer to something independently perceivable. The
71
word is applied to quantitative attributes of occurrences or
events, to their duration and occasion or absence, each
determined in relation to further events as characterized by
change. In analogy with space, time is distinguished in accord¬
ance with occurrences, rather than occurrences being distin¬
guished in accordance with time.
Causal laws, the constantly connected events called cause
and effect, need therefore be distinguished otherwise, and the
answer can be supplied by the preceding characterization of
events as changes. A causal law can be characterized as a
constant connection of certain changes.
What distinguishes, what defines, causal laws is of course
again a matter of deciding what laws are to be called causal.
The interest herein is in, as said above, what event that has not
occurred would if some other did. In other words, the causation
of the effect means its coming about as a change from one
state into another, a change that by knowledge of causal laws
may be brought about if desired, or prevented if undesired; and
since the cause implies by its meaning the effect, and the effect
is a change from its absence, that absence implies by trans¬
position the like former absence of the cause, namely that it
also consists in change. The common comprehension of this
meaning of causality is exhibited in regarding occurrences not
preceded or succeeded by a change as having respectively no
cause or effect.
The just mentioned concepts of preceding and succeeding
were made use of somewhat freely. The causally connected
changes, as noted, appear sometimes simultaneous. But many
do not, and, in accord with their usual sequence, in a simul¬
taneity the event considered the cause is customarily the one
known to be independently of the other caused by an event
occurring earlier. The reference to this sequence as customary
is made because the ends of purposive events are, reversely,
sometimes called final causes. But even then it is the aim rather
than the outcome which is of account, and the ordinarily meant
sequence of cause and effect can be adopted here without
confusion.
The difference between cause and effect clarified accord¬
ingly, it can be stated with exactness that a causal law desig¬
nates the constant accompaniment of one event as change by
another, the first of which would be the cause and the second
the effect. The cause and effect, it should be understood, can
be combinations of things, each of which things can be
regarded as a cause or effect, though not the only one.
It should be worth repeating here that the accompaniment
of one thing by another is the only criterion by which a con¬
nection can be established, as brought out in the last chapter’s
s
72
discussion of necessity.
That no causal connection but only a constant conjunction
can be found between events is known as the observation of
Hume. But that only a constant conjunction is possible does not
repudiate a causal connection. Whatever is meant by it, taken
as constant, the meaning cannot be of other than a constant
conjunction, and therefore the connection is not denied by it.
Hume thus did not refute causal connections, but he saw cor¬
rectly that there are none other than conjunctions. However, the
same applies to logic, with which he contrasted causation, or
to any other relation between things.
And on examination it can be seen that the oft heard dis¬
tinction noted in the introduction, that inability to otherwise
conceive something inferred applies to logical, deductive, laws
but not to natural, inductive, ones cannot on more serious
grounds than before be maintained.
It was before observed that in difficult deductions it is odd to
say that the result is not conceivable otherwise. But in view of
the involuntary perception found in the preceding section
requisite for determining a principle, it is indeed not possible
to so perceive a false deductive result, although the true one
may also not be perceived. The conceivability really means this
perceivability, extending in deduction to so-called all possible
worlds, including the actual one. The reason for that inclusive¬
ness is that the conditions to which deductive laws apply are
extremely general. The laws nonetheless apply to certain con¬
ditions, such as combinations of numbers. Laws of nature, how¬
ever, likewise apply to conditions, though much more particular
ones. They are conditions regarding nature itself, not con¬
ceptual ones, and under them results opposed to its laws can
equally not be perceived, while their confirmation may not
either. That is to say, what above is thought to distinguish
deduction is true of induction as well.
It has also been thought that compared to deductive laws
inductive laws concern only the possible, in an equivocation
alluded to earlier (p.37, sixth par.). Because laws of nature are
difficult to observe with precision, many of its manifestations
are inferred only probable, and hence the belief that the laws
are exceptionable. The possible has in this case to do with
man’s knowledge, however. A law, a principle, is by definition
true of all instances, and should a connection turn out not to be
invariable, it is for that reason not a principle. It may in this
regard also be recalled that deductive principles, sometimes
called demonstrative, have in their fundamental cases been
equally unsubstantiated.
Despite these similarities between the deductive and induc¬
tive, there remains the difference that in one the inferred is
73
found to be port of the inferred from as conceived, it is intrinsic,
and in the other it is not, it is extrinsic.
It may yet be wondered whether the observation of an extrin¬
sic constant connection between two events is truly sufficient for
the connection usually called causal, whether perhaps a causal
connection is not an intrinsic one. Night always follows day, but
it will not be alleged that the occurrence of day causes the
occurrence of night. Like acquaintance with an external world
of complex persisting objects despite their ever changing
appearance, however, acquaintance, in man and animal, with
persisting causal connections is much more profound than
isolated observations admit. The occurrence of day and night
is comprehended by a child as depending on illumination by
the sun, and even an animal will discriminate in its attributions
of causation. That causal connections are not intrinsic ones can
furthermore be simply confirmed, by seeing that perceptions of
familiar causes do not contain as part of them the familiar
effects, which can moreover be largely found to occur
subsequently.
Regardless of the type of connection in causal laws, the
material world is found subject to them. This is the case even
if, as noted earlier, the connections are not always observed,
and the exact nature of the laws is not known. The physical
world has as indicated the peculiarity that, unlike the contents
of consciousness, things in it are taken to occur even when not
perceived. Correspondingly laws in it will be considered to hold
even if not precisely recognized, provided there is good reason
for the inference. It could be mentioned that even the above
probability can sometimes be regarded as a law, as appre¬
hended in statistics. If the probability is based on a constancy
within a range, then that constancy can be viewed as a law. If
for every hundred occurrences of A there are from forty to sixty
occurrences of 6, then this proportion of the connection, though
not a constant connection between A and B, can be held to be
a law. But the presence of causal laws in the external world can
be inferred on more substantial ground.
The experience of particular causations is more overwhelm¬
ing than the preceding proportion. A blowing wind is expected
to bend the grass; a stone released in air is expected to drop;
a pot of water placed over the fire is expected to boil. One’s
entire life is regulated by such particular reliabilities. One’s
understandings go deeper. One learns of more assured general
connections behind particular ones. Not only does the wind
move the grass, but objects of many sorts move many others,
and they do so more efficiently in some ways than in others. If
an animal does not secure a fruit from a tree by a certain
motion, it will try other motions, till hopefully succeeding. Its
s
74
lessons were that while there is some reliability that certain
kinds of actions bring certain results, there is an accuracy of the
actions yet more fruitful. One finds that underneath the
likelihood of a certain connection there is a more dependable
one, on which the first one rests. This experience generalized
means that one induces from progressive findings of fairly well
working first connections, then better working ones on which
based, etc., that all deficient connections are based on more
exact ones, with an unvarying one, in which exceptions are
reduced to zero, at the basis. It is so to speak induction of
induction, of inductive laws of nature. The experience that all
that occurs in this world is thus causally connected, whereby
that world is as expounded consequential and hence real, is so
thoroughgoing that the occurrence of things in that world is
considered the criterion of their reality, if only in a general,
material, sense and not a particular one exemplified by the
artificial, which may not be a real specific thing, but is real by
being of the world. And since the reality of things in that world
is established by its conformity to causal laws,
THEOREM 11.5. That the material world is subject to causality
is a reality prior to other realities in that world.
This theorem and what went before can be seen to entail
certain familiar principles of nature. They are that every,
extrinsic, material state, every condition of matter aside from
its existence, has a cause and at least a possible effect, and that
elemental constituents of the world persist or are conserved.
In agreement with the previously established nature of
reality, the material world is found real or consequential
because its states can be caused by and of effect on man. In
consequence, since the states of the world of matter, future
ones that are not determined, can be caused by man, they
cannot be uncaused. If they could, if a future state could occur
by chance, then man could not effect it, could not be respon¬
sible for it. In other words, with the future requirement learned
as a principle from the past, every material state has a cause.
And since the states of this world can be of effect on man, every
material state can have an effect.
As regards the permanence or conservation of elemental
entities, it has to do with what is known as conservation of
matter and energy, or of mass-energy. By their conservation is
meant that the total quantity of the entities is constant, and the
constancy of the world of matter, which does not appear and
disappear, was already observed as distinguishing it from
illusory ones, i.e. as true by definition, and as in that depend¬
ability presupposed by its reality. As to energy, it is defined as
the capacity for doing work, and it is hence a causal property.
It can at once be noted that energy cannot therefore be an
75
independent entity, let alone the constituent of material reality,
as proposed in physics. A causal property is meant to hold of
something other than itself, in this case the very material
objects. Regarding energy’s conservation, it is often explained
through concepts of potential and kinetic energy. A \A/eight
suspended above ground is considered to have potential
energy, in being able to do work on being released and falling
to the ground. After release the energy is considered to in fall
become kinetic, energy of motion, for the same reasons. The
magnitude of the two energies, the capacity for doing work
they signify, is determined unchanging, conserved. This deter¬
mination is, however, not a matter of ascertaining conservation,
but a redundancy. Both names for the energy stand for the
same capacity of doing the work due to the combination of
factors, e.g. the height of suspension and the amount of weight,
the different names merely used to speak of that energy with
regard to stages before and after release respectively. The
energy is in fact an expression of forces like gravity, and energy
is thus a reified fictitious entity, unduly added to those forces.
The forces, having to do with what occurs under what condi¬
tions, comprise nature’s causal laws, and it is they that are con¬
served. This conservation, this constancy, is implicit in the laws,
proclaiming constancy. Like that of matter, it is required by their
consequentiality, also presupposed by their reality.
As in other matters here observed to be presupposed by
acknowledged realities, the principles of the last two para¬
graphs have in some of their form been disputed.
In particle physics, the exact sequence of events has been
found indeterminable, and thinkers have inferred that causality
must be abandoned as a scientific concept. This is formulated
in the dictum that science should not explain, only describe.
But the two procedures can be seen to be here synonymous.
With an individual occurrence, a description differs from an
explanation, which amounts to giving a cause. But in now con¬
cerned observation of generalities, the observation is its own
explanation. Reasons in the present sense, not in the earlier
sense of motives, are generalities, accounting for instances and
not required for generalities themselves, unless they can be
found to be based on broader ones. The interest is in principles,
and, as noted several times, there cannot be any other than
constant conjunctions. Moreover, the observations that in
particle physics are thought to dismiss causality themselves con¬
tain causal conjunctions. The physicist makes the observations
about particles under certain conditions, which constitute
causes, resulting in the observations as effects. Particles were
mentioned before as known by effects, and there exist series of
causal ties producing them. The researcher knows that his will
V
76
causes his actions, his actions the managing of implements,
and so forth till the observations.
In relativity, the famous equation of Albert Einstein,
E = mc^,
enunciating the equivalence of mass and energy, led to the
denial of certain laws of conservation, in particular of matter.
Matter is regarded as synonymous with mass, and since mass
and energy are found to be mutually convertible, conservation
of mass-energy is instead upheld. Matter, as mentioned (p.74,
last par.), cannot be formed of energy, and therefore if energy
is convertible into it, not remaining energy, its conservation
should likewise have to be rejected. The notions find their basis
in a conceptual confusion, however, traceable to the inception
of classical mechanics. Isaac Newton spoke of the principle of
inertia, the property of bodies to as regards their motion resist
forces impressed upon them. Inertia has thus been treated as
a body’s property separate from those forces. It is, however,
merely a manifestation of them. To hold that a body has a
certain resistance to a force is to hold that the force has a certain
magnitude, because the magnitude of the force is measured by
the effect on the body. Formulated either way, inertia is, by
concerning effect, a causal property. But conceived as separate
from force, it is today the property by which mass, namely
matter, is measured, rather than by such as bulk and density,
which concerns matter as here considered but may be difficult
to gauge. The concept of matter was accordingly misconstrued
to be a causal one, and its maintained equivalence with like¬
wise causal energy does not assert a lack of matter’s conserva¬
tion, but a causal relation.
In point of fact, the energy of the above equation was origin¬
ally a kinetic one and can correspondingly be found identical
with the mass, viewed as above resistance. The energy can be
held to be the force of a moving body on another, and that
force can in accord with the above be translated into the mov¬
ing body’s resistance, to wit mass.
Apart from the mistaken differentiation of the force on a
body from its mass as resistance, there is a similarly false dif¬
ferentiation of mass as either inertial or gravitational, but not
in conformance with a like tenet of relativity. By “inertial mass”
is meant the amount of the said resistance of a body to a force,
as when pushing the body, and by “gravitational mass” the
amount of attraction of a body by gravity, as when weighing the
body. The two measures seem opposed, because in one case
the motion of the more massive body under the respective force
is smaller, and in the other case larger. That the two measures
are proportional is thought to be evidenced as follows. It has
long been observed that all bodies falling side by side toward
77
earth, supposing no obstruction such os air, do so at the same
speed. The reason is believed to have been supplied by
Newton, who argued that since resistance to a force, e.g. to
gravity, differs according to the body’s mass, the pull of gravity,
in order that all bodies fall at the same speed, must be pro¬
portional to that resistance. As explained, however, the amount
of resistance is the very manifestation of the force, the pull of
gravity, making the resistance of all bodies and hence pre¬
sumably their mass during fall in fact the same. That the motion
of the more massive body be larger during its weighing and
smaller in the other cases is accordingly not due to opposite
measures of mass. It is due instead to the same resistance,
inertia, by which these objects, resisting motion in other cases,
resist hindrance of their motion during weighing. Of present
account is that since the inertia here has to do with the pull of
gravity, inertial and gravitational mass are indentical by
definition, to which the uniform fall is irrelevant.
The uniform fall of objects can be explained by way of
Newton’s law of gravitation itself. In accordance with it equal
particles of matter are attracted with equal force. This is to say
that a more massive object does not, so as to actually fall
slower, resist gravity more, which reaches every part of the
object to result in equal fall. In contrast in normal pushing or
pulling of one object by another, the force is acting on only
those particles of the first that are held to be in contact with the
second. Hence in order to affect through these parts the others,
the force must surmount the combined resistance, mass, of all,
which increases accordingly with more massive objects.
An interpretation of the uniform fall and with it, os indicated,
a contention of the identity of inertial and gravitational mass
was put forward in relativity. This identity is instead, as ob¬
served, unrelated to that fall, but holds by the understanding of
mass in terms of force. Relativity’s explanation of the uniform
fall of objects, further, commits a rudimental fallacy, known as
affirmation of the consequent and alluded to in the introduction
(p.5, fifth par.). By it from true premises “If something is a bird
it is an animal” and “This is an animal” it would falsely be
concluded “This is a bird”. In the present case in place of the
second premise is relied on a recognized relativity of motion,
specifically motion as change in position. In a vehicle travelling
on earth, a reposing passenger appears at rest in relation to the
vehicle, but in motion in relation to the earth. This relativity is
a logical matter, to be substantiated in the next chapter, and it
does not mean that in actuality a further attribute may not
distinguish what may be viewed as a moving object from one
at rest. The other premise is, further, that if bodies fall equally
toward earth then, as in all motion, the motion is relative, the
I
\
78
earth for instance viewable as moving toward those bodies. To
simplify, by this premise if bodies fall equally then their motion
is relative, and by the preceding premise their motion is rela¬
tive. It is concluded that they fall equally, i.e. that this reasoning
explains the equal fall. But the relativity of motion is irrelevant,
true no matter how bodies fall.
The contended equivalence of falling of bodies and motion
of the earth or like frame of reference toward stationary ones,
the contention a staple of theory of relativity, does in fact not
hold. Differences have been noted by science. Since falling
objects tend toward the center of the earth, those equally dis¬
tant from it approach each other; and since the speed of falling
objects gradually increases, those of unequal distance from the
earth recede from each other. In other words when those ob¬
jects are held in motion, the distance between them changes,
and when the earth would be held in motion, that distance
remains. To maintain equivalence of the falling of bodies and
the opposite motion of the earth is hence in effect to maintain
that there is an equivalence insofar as there is one.
The more general thesis of relativity pertains to the above
question as to whether, apart from relative change in position,
a moving material object can be distinguished from one at rest,
and the thesis of the theory is that it cannot. The question is of
longstanding interest, and of interest is here motion in general,
because it appears as the physical world’s principal form of
change and therefore of causal events. Additional attention
may therefore be paid to it.
It may be noted that Newton’s famed laws of motion con¬
stitute a partial announcement of the discussed prevailance of
causal laws in the material world. The laws of motion are
1. Every body continues in its state of rest, or of uniform
motion in a straight line, unless it is compelled to change that
state by some force impressed upon it,
2. The change of motion is proportional to the force im¬
pressed, and is made in the direction in which that force is
impressed,
and
3. To every action there is on equal and opposite reaction; or,
the mutual actions of two bodies upon each other are always
equal and in opposite directions.
The first of these is sometimes interpreted as a definition of
force. Force would accordingly be that which acts on a body
when it changes from a state of rest or the described motion.
But then it is assumed that something always acts on a body
when it undergoes these changes, and this assumption should
be considered a principle, in the fashion of the principle that
every physical state has a cause. It is evidently a principle of this
79
kind that Newton declared, that a body will not undergo these
changes without a cause, although he did not necessarily
equate force with cause. He spoke of the force of gravity, for
instance, while owning to ignorance of its cause. He thus did
not perceive the mentioned synonymity here of explanation
and description. To look for o cause of something is to look for
some event resulting in it by some principle. At present it would
mean to find events that along with e.g. the freeing of dis¬
tanced bodies result in their gravitation toward each other. But
the freeing has already that result, and the force of gravity
needs no causal description beyond its own.
Because a force is determined by the effect, to speak of force
in the first law above can also be misleading, as it has been in
regard to inertia and is in regard to the second law.
The second law could by the meaning of force be understood
to redundantly say that if a force is such that it causes a certain
motion in a certain direction, then it causes that motion in that
direction. Regarding direction there is no escape from this inter¬
pretation. Force can be understood in the law to refer to the
causing event, and its direction is not always the same as the
effected one, as exemplified by billiard balls. Regarding the rest
of the law it can by its author’s comments be held to state that
if a certain event causes a certain motion then double the event
causes double the motion, and so forth. That is, the connection
between the causing event and the motion is constant. But this
is a form of causality itself, which is of constant connections.
The law accordingly asserts that there are causal connections
between certain events and certain motions, magnitudes
included.
An added interpretation is accepted, however, which can be
seen not to state a presumed law of nature but a definition,
one, unlike the first law, of force, this time by effect and
quantitatively. The law is held to state that the force on a body
equals the body’s mass multiplied by its acceleration, its
change of speed or direction, the equality symbolized as
F = ma.
What is in practice asserted by the equation is that if some
action has the force of accelerating by a certain amount a cer¬
tain mass then it will accelerate another mass in inverse pro¬
portion. I.e. the mass and acceleration, though varied, when
multiplied by each other equal the same. But in concord with
previous observations, this is a result of the definition of mass
as resistance to acceleration, beside numerical choice. To clar¬
ify, when at first the same action, as by a moving body, is con¬
sidered in relation to different masses, it is practical to assign
it the same numerical force, attained mathematically by mul¬
tiplying whichever acceleration by its reciprocal, the number
I
s
80
which if multiplied by it yields 1, and which is assigned hence
to the mass. Accordingly a certain force is numbered 1 by
numbering the same a certain affected mass and its corres¬
ponding acceleration, and if another acceleration is 2, its mass
is numbered V2. Of account is merely that these multiplied
result in 1, signifying the force as cause, with either mass or
acceleration signifying the effect. When a mass is measured by
volume, to be sure, there indeed apears the experience of an
inverse proportion, the changes of size and acceleration of two
masses the reverse. But with mass measured by acceleration,
the proportion is a mathematical convenience, defining in the
above equation quantity of force. The definition is useful
because when for instance knowing the mass of an object and
a certain acceleration is required, the needed force can be
computed.
The third law of motion can be viewed as but an indication
of whatever causal law applies in the pertinent event. As an
effect need not be in the direction of the cause, so need a
reaction not be in the direction opposite the action, or be equal
to it. That these should be thought otherwise is understandable
in view of the notion that resistance, rather than being a mea¬
sure of the force, is a separate and commensurate opposition
to it. Thus Newton cites a horse which by drawing a stone is
drawn by it, although its progress signifies merely the horse’s
power. What the law suggests, if flawed in particulars, is that
in an event among things should be considered effects in not
only some but all of them.
Although the three laws are about motion, they do not say
when motion of an object occurs. Whether no distinction can be
made in material objects between their relative motion and rest
was noted to have been a persistent issue, and it has involved
the concepts of space and time.
Newton argued absolute, independent, space, in relation to
which the motion of objects is determined. He alleged besides
similarly absolute time, which would determine the speed of
objects. Einstein asserted relativity of motion and acordingly of
speed, as well as of the measures of space and time. He none¬
theless described a separate though dependent universe of
space-time, with a curvature placed apart from material objects
and determining their course.
It will regarding space and time be recalled (p.68, second
par., and p.70, last par.) that they by their meaning are not
independently perceivable entities, but are delineated by
material objects and their events. As expounded in the first
chapter, in consequence, by speaking of space and time as
independent, absolute, entities though not perceivable, no
entities can be referred to. For the same reasons a universe of
81
space-time as a dependent, relative, entity cannot be char¬
acterized by a curvature separate from material objects.
NevA/ton acknowledged that space and time are commonly
conceived in relation to sensible objects. Yet he posited a truer,
mathematical, meaning of them by which motion and duration
is determined. These are determined in relation to physical
things, however, such as the earth in the one case, and the
phases of the moon in the other, not to say that a true meaning
is one in use. Einstein proposed similarly a geometry of space
and time, the mentioned curvature held to be the cause of
corresponding motions of objects near massive bodies like
planets. These motions, associated with gravity and whatever
their nature, are again, however, found connected with the
nearness itself to those bodies, and as before need no other
explanation, especially the impossible curvature as cause.
Despite the discussion in relativity of, as in the case of the
moon’s orbit around the earth, the path of one object near
another, in a geometry of all such paths as associated with
space-time, the theory does not establish that any of the objects
describes those paths, because it holds motion to be relative.
That it is was asserted first with regard to rest and constant
velocity, constant speed and direction. These are the two states
of Newton’s first law, and inasmuch as they remain unless acted
upon, objects in those states are called inertial frames of
reference. It is then maintained that no distinction can be made
between an inertial frame at rest and in motion. Compared to
these later were reference frames that are accelerated, as was
indicated when speaking of objects falling under gravity. These
frames are additionally then said not to be distinguishable from
inertial ones. It should be remarked that this can be true in spite
of the noted changing distance between falling objects, as it
can be true if they did not fall equally otherwise. The issue
is whether there is a difference due to their states of motion
alone. The contention thus is that there is no difference be¬
tween rest and motion in general, since the alternatives of
constant velocity and acceleration, the last the negation of that
constancy, exhaust all motion.
To support this main contention of relativity it is argued that
the speed of light, if not affected by other particulars, is in all
frames of reference, regardless of any of their motion, always
measured to be the same. If it were not, if the speed of light,
assumed constant if not affected by other particulars, were
found to differ in the frames, motion would not be relative, to
be gaugeable with respect to that speed.
The thus argued invariance of the measurement clashes with
common sense, for if, for example, relative to one thing
another is in motion in a straight line, and in the same direction
s
82
relative to the second thing o third, then the speed of the third
thing, which may be o ray of light, is token to be greater in
relation to the first than in relation to the second. In order to
therefore justify asserting the invariance of the speed of light,
relativity offers o definition of particular time, of when on
event, os is the onset or finish of o motion, con be said to occur.
It should be of value to remark that in various areas, as ear¬
lier suggested, attempts are made at demonstrations by way of
offering corresponding definitions, and that if the definitions do
not denote the entities at issue than the demonstrations are not
achieved. The definition presently discussed is among these.
It is taken into consideration that the time of an occurrence
is associated with simultaneity. For an event to occur at a certain
time is for it to be in some way simultaneous with another, for
instance with a designation on a clock. It is accordingly queried
how simultaneity is determined, and the definition by relativity
is as follows. It is taken into account that since the transmission
of light is not instantaneous, e.g. it takes years for the light of
distant stars to reach the earth, two events cannot be judged
simultaneous because they appear to be so to the observer.
Hence to decide that two events occur at the same time, a
synchronization of clocks placed at the events is considered. The
two clocks are held synchronized if when a light signal is sent
from one to the other and reflected back, the time on the
second clock when it reflects the signal is midway between the
times on the first clock when it emits and receives the light.
It may be noted that to speak of when these events occur is
question-begging. It can be asked what makes a time on a
clock and an adjacent stage of the light signal simultaneous.
But the present point is that this definition of synchronization,
of sameness of time, is meant appropriate regardless of any
motion of the clocks relative to something else. The clocks are
viewed as fixed in the same reference frame, but a little
thought will reveal that ordinarily if the frame is held to be in
motion parallel to the signal the definition does not apply. If the
unit is seen as moving in the direction of the second clock, both
of the clocks moving accordingly, the light is held to travel
proportionately farther to reach that clock then to return to the
first one. The second clock recedes from the light, and the first
clock approaches it. The times on the clocks would hence not
be synchronized as above, the light taking proportionately
longer on its first trip between the clocks than on its return trip.
What the above definition does correspondingly is to individ¬
ualize time for every reference frame, deciding by fiat that in
each of them the time of the trip is the same for the same
distance in the frame. In other words, the conclusion that the
speed of light is measured the same in all frames of reference
83
is circularly inferred from the premise that the measure is
decided to be the same.
By the foregoing it should become evident that to thus define
particular time separately for each reference frame is not in
accord with the meaning by which the time in question is con¬
sidered. By the individualized definition it was determined that
simultaneity, or measures of time, and measures of space are
relative, and it will be suitable to explain these contentions
before seeing the untenability of the concerned definitions and
of what is inferred from them.
As regards measures of time, if when one frame moving in
a straight line relatively to another passes it a light is flashed
nearby, then by the above different measures a ray parallel to
the motion reaches a place next to a given place in the one
frame at a different time according to it than according to the
other frame. For by then the place corresponding in the other
frame in distance from the light source has passed that place
in the initial frame, and since the same time is assigned to both
of these places in their frames, the times are not the same
where the frames are next to that stage of the ray. That is to say,
the same events would occur according to different frames at
different times.
It may again be noted that there is a question-begging in
speaking of when things pass or are next to each other, viz
when aligned perpendicularly to the motion. If time in the two
frames varied as supposed, the time when parts of them pass
each other could as above not be always the same for both,
although it is at a time as a single one when the ray is held to
be next to such passing parts. Thus relativity presupposes by the
instant of passing or adjacency a common supertime it denies.
As regards measures of space, or length, an additional defin¬
ition is offered. By it in two frames moving in a straight line rel¬
ative to each other the length of an object in one is measured
in the other by finding where with respect to it the correspond¬
ing ends of the object lie at one time. If then the length is par¬
allel to the motion, the same time for the ends is measured by
clocks passing them, and the length by the distance between
the clocks. And by the different measures of time, when one of
those clocks passes one end, a clock passing the other does not
read the same time. Consequently when a clock there does
read that time, the first clock is not passing the first end, and
hence the distance between those clocks differs from the length
in question as measured in its own frame. That is to say, the
same objects would according to different frames be of dif¬
ferent sizes.
Neither time nor space are meant to be measured according
to points of view, however, wherefrom different quantities of
84
\
each follow depending on the frame of reference.
In relativity it is often said that the differences depend on
what is seen by the observers. But os in the cose of light from
distant stars, informing of long post events os brought forward
by science itself, men continually moke allowances os to what
situations in the world obtain, rather than judging by subjective
appearances. As was observed regarding identity of physical
entities, judgements made about them differ from individual
perceptions on which founded, with the perceptions of for
instance size or tonality not necessarily representing what is
attributed to the objects (p.69, lost par.). The actual attributes
ore meant to be perceived under certain conditions only, and
it is these perceptions that ore comprehended os potential
through the others. All creatures hove the capacity for objective
appraisal, inferring facts from merely signs of them, an animal
knowing that the small perception of a distant prey represents
a full-fledged meal.
While the sign can differ in its nature, perhaps owing to the
standpoint, the fact cannot, lest it be contradicted. Accordingly
in measures of space, i.e. size in some respect, there may as
mentioned be varied appearances of it according to e.g. the
object’s distance, but only one size is meant to apply, one that
at least ideally is perceived if the object is appropriately placed
alongside the chosen standard. All other judgements of size are
inferences. Similarly in measures of time a certain one only is
meant to apply to an event, namely the occasion when by
discounting representative perceptions through such as distance
the event would ideally be perceived simultaneously with
the chosen standard. Simultaneous perception, it may be re¬
marked, need not be substantiated, because, as all perception,
it is not an inferred, but a known, state. One has a capacity to
distinguish between present and past perceptions. As in the
case of space, accordingly, other assessments of time of an
event than by that direct perception of its occurrence are made
by inference. To these belong inferences allowing time for the
travel of light, or for other production of signs of past events, as
in archaeology, these allowances made in relativity inconsis¬
tently for the speed of light but not the speed of relatively
moving reference frames, by which speed the arrival time of
light perceived may be shortened or lengthened. Through
these and other faulty premises more than one conclusion of
such as a time or a size may be reached, but only one of them
can be correct, the one corresponding to the mentioned ideal
perception. Differing designations of measures, as for time
zones, are of course a convenience of language only, in time
zones enabling similar hours for day and night.
It may be added that it has been questioned that ideal per-
85
ceptions are appropriate in considering reality. It has been
objected that Newton’s first law of motion does not apply to
reality, because no straight motion can be said to occur in it. It
could also be objected that if it did occur it could not be pre¬
cisely measured, as cannot be other worldly things despite
methods like magnification and slow-motion photography, by
which concurrence in size and time may be more exactly
observed. This exactness may also raise questions about
defining these concurrences by their perceptions, though ideal
ones. But straight motions or deviations from them, as well as
other measures in reality, are as through instruments deter¬
mined just in accordance with ideal perceptions of them, which
it is regarding reality the aim to more and more approximate
in accuracy. Laws of nature that concern these measures, as any
of motion, can be expressed in ideal form to facilitate their
use, and for the same reason ideal measurements, sometimes
expressed in mathematics as limits, are conceived as if per¬
ceived in full precision, which real measurements are held to
approach.
It may also be wondered whether the measurements one
person finds worldly things to have or approach are the same
ones other persons find. The answer lies in the referred to shar¬
ing of the experience of the same world by all, the induction
of which enlarged on later. One shares the experience of the
same laws of nature, from which the same consequences are
deduced, by likewise shared laws of logic. From the former
laws, from any known fact, can by the latter laws not be de¬
duced contradictory results. By being deduced the results are
likewise facts, which cannot be contradictory. That is why the
correctly inferred time or size of something cannot differ with
different observers. It may be added that should someone’s
perception of a worldly particular be abnormal, allowance is
likewise made for physical misdirection.
On the basis of its mistaken belief that time and space differ
in different frames of reference relativity holds in more
mistaken reasoning that these extents differ in an exactly
reverse manner with respect to two frames moving relative to
each other. It is argued that from either frame time in the other
appears equally slower, and space shorter in the direction of
the relative motion, the appearances called time dilation and
space contraction.
The unsuitability here of the concept of appearance was
spoken of above, and at the speed of light, of account now, the
concept becomes figurative. What is actually meant by time
dilation is that when relative to one frame another is
considered in motion then the clocks in the other are slower
than the clocks in the first. It may at once be noted to be
86
\
ludicrous that the pace of clocks should change merely by
considering one of the frames instead of the other as relatively
moving. If this relativity were to hold then certain clocks would
impossibly be slower than others while speedier.
How the dilemma is encountered should be evinced by
giving an instance of the argument used for this relativity. In the
diagram the horizontal lines depict reference frames A and 6,
with 6, having passed A, moving horizontally relative to it at
4/5 the speed of light. When 6 was passing A the clocks in both
of them read zero, and a flash of light, pictured by the circle,
occurred in the center of 6. In the diagram, 6 has moved 4 units
since, and with it the source of light, now the right bottom
corner of the triangle. At this stage a light ray, forming the right
side of the triangle, reached a place vertically above in 6. But
relative to A the ray travelled diagonally, forming the left side
of the triangle. Hence since by postulate light travels relative to
A 5 units for each 4 of 6, the diagonal is 5 units long. And by the
renowned Pythagorean theorem the square on the diagonal
equals the sum of the squares on the triangle’s other sides. In
consequence the right side is 3 units long. Then in accord with
the supposed sameness of speed of light for all frames, if units
on clocks are chosen to match units of length travelled by light
relative to the frames, A’s clock reads 5, and 6’s clock 3.
Accordingly the clock in 6 is slower than the clock in A, and it
is argued that the reverse would be true if the light had in like
manner occurred in A. There is more than one flaw.
By what was said the argument of another light is irrelevant;
it is enough that with the first light the clock in 6 is slower than
the clock in A, whence the opposite cannot be true of those
very clocks, contrary to the contention that all clocks of A would
be somehow slower from the viewpoint of 6. Further, since light
travels relative to A at 5/4 the speed of 6, when the base of a
like triangle in A is 4, signifying 4 units of relative motion by 6,
then the vertical, depicting the corresponding ray relative to A,
is 5; consequently the triangle is of different proportions than
the other, contrary again to contention. Furthermore even with
the flash as first conceived in 6, the time on 6’s clock need by
the theory itself not be less than on A’s, and for that matter if a
horizontal ray to the left is considered, the time is three times
the previous one; since as before relative to A the source of light
moved 4 units to the right while the rays travelled 5 units, the
ray in the left direction travelled 9 units relative to 6, whose
clock accordingly reads 9.
It is also maintained that by the argued measurement of time
Pythagorean relations as discussed, well known as of Lorentz
transformations, always hold. That this is not the case is
evidenced by the last described example. The horizontal ray of
87
9 units in 6 is in the same line with the 4 units of relative motion
of B, os well os with the 5 units of the ray relative to A. These
lengths form no triangle, nor do they have Pythagorean
proportions. The error of thinking all the lengths to have them
can in accord with the supposed measurement of time be
ascribed to the belief that light travels in either frame the same
distance in all directions. In the example diagramed, since the
verticie ray in 6 travels 3 units, it is supposed the other rays do
too, and since the other distances marked, those travelled by 6
and light relative to A, remain by postulate, the proportions
would always obtain. That the vertical distance be the one
attended to first can be held due to experiments gauging
speeds of light perpendicular to the motion of its source.
It may be additionally noticed that in the diagram, as
depicted by the vertical and base of the triangle, light travels
relative to 6 at 3/4 the speed of A, in negation of the contention
that from these believed relativities it follows that nothing
travels faster than light relative to any, at least nonaccelerating,
frame of reference.
In preceding paragraphs it was merely disclosed that things
thought to be the consequence of the measuring of time
proposed in relativity do not follow, the measuring itself seen
inappropriate, and there should be no need to make similar
observations on space. Some likewise undue derivations may
further nevertheless be surveyed, because they touch more
closely on whether or not motion is relative.
Certain of the derivations are that there are objective dif¬
ferences of duration and other measure in objects in accord¬
ance with their motion. Principally, events are said to slow
down in an object as it gains speed. The conclusions are firstly
uncalled for because the premises offered state that the
differences are subjective. They are posited true from a point of
view of one reference frame and not another, and to then hold
them true from the point of view of each frame is contradictory.
In addition, to infer from a lesser time on a clock, as done, the
slowing of other events in the frame is likewise without foun¬
dation. The time on the clock is by the theory less not because
it is so affected by the putative motion, but because it is so set
by men. Should motion have the effect on things, it should have
it on the clock apart from that setting.
The initially intended inference that light behaves the same
in relation to all frames of reference does, lastly, also not follow.
Returning to the diagram this becomes clear by merely con¬
sidering that the rays travel different lengths relative to A than
to 6. The clocks, that would also read different times, are
irrelevant, it sufficing that during the relative motion of the
frames the light travelled different distances relative to them.
88
\
Under the presunned constancy of the speed of light, its
noncapriciousness, nnotion is, accordingly, not relative, in the
sense that due to it there would be no distinction between
relatively nnoving frames, a distinction e.g. between the
distances light travels relatively to them.
That the speed of light is thus constant, not capricious, pre¬
supposes moreover, since speed is measured relative to some¬
thing, a, also perhaps ideal, reference frame relative to which
the constancy takes place. In the diagram such a frame is A, in
relation to which light is postulated to travel 5 units in all
directions, to have a constant speed. That frame can hence be
held to be at rest, by the customarily meant equilibrium, with
the other frame, relative to which the light was seen to travel
varying distances, in motion. More generally, under that con¬
stancy, assumed in the theory not to be otherwise affected,
there is a distinction between motion and rest.
There need be no such distinction merely because light
travels different distances for different frames, however. Should
the distance travelled be e.g. dependent on the one of the
source of the light, then the light might relative to the source
travel equal distances in all directions, as do due to inertial law
other things in moving frames when propelled with equal
force. In that case the same light can travel different distances
relative to other frames, but its source need not be at rest rela¬
tive to for instance another source of light, the light travelling
relative to that source equal distances for the same reason.
Contrastingly, if though impossible it were true after all that
the speed of light, the distance it travels, be the same with
respect to all relatively moving reference frames, the sought
after relativity of motion would not follow either. There may
be some other factor by which motion of an object can be
distinguished.
The assertion in relativity that events in an object slow
factually down with its motion (last p., fourth par.) has been
found ill-supported by some, not because falsely derived from
settings of clocks, but because it is in the same breath main¬
tained that motion is relative. The inherent contradiction had
been illustrated by the so-called twin paradox. In it it is by the
theory found that a twin space-traveler at high speed, upon
leaving the earth and returning after years, has as a result of
the speed aged much less than the other twin, who remained
on earth. But since by relativity the earth can be held in motion
and the traveler at rest, there should be no difference in age.
That there is a paradox is persistently denied, with many argu¬
ments advanced against it. But as observed in the first chapter,
if something is demonstrated then argument cannot confute it.
To say both, that there is nothing by which things can be con-
89
sidered to be in motion, and that there is something, the speed
of events in them, is a contradiction, which cannot be explained
away.
In effort to counter the paradox it is often adduced that
experiments amply demonstrate that physical processes slow
down in objects at high speeds. But aside from the argument’s
denied supposition that the motion is distinguished before that
difference is observed, if that difference among relatively
moving objects is detected, then the contended relativity of
motion is false. It will not do to ignore a contradiction in order
to by Procrustean means save a theory.
Newtonian mechanics already contained a principle of rela¬
tivity, by which motion could not be distinguished in either of
two bodies in relative motion to each other at constant velocity.
The principle has been thought incomplete, and modern
relativity holds to extend it to other, accelerated, motion, as
observed. But classical mechanics was, instead, deficient in the
opposite way.
According to Newton an attribute distinguishing motion is in
rotating objects centrifugal force, the force by which things in
circular motion recede from its axis. That rotation can be
thought relative is illustrated by the one-time view that, as
everyday experience still has it, the earth, rather than rotating,
is unmoving and the heavens, e.g. the sun, revolve around it.
But the centrifugal force, of receding from the axis of circular
motion, applies to certain only of things in that relative motion.
It applies to a centrifuge or its contents, receding from its axis,
but it does not for the same axis apply to the surroundings,
though relatively rotating about it. It is accordingly justified to
among two relatively rotating objects regard one as in actual
motion.
This inference of Newton, that there is an absolute motion of
rotation, has been, together with the like in other motion, de¬
nied under the influence of his corresponding claim of an abso¬
lute space. It being noted that space is about no observable
object, and that what are considered rotating ones appear so
relative to the fixed stars, it was proposed that centrifugal force
is causally connected with those stars. There is no need, how¬
ever, to postulate an absolute space for the motion to hold. It
suffices that the difference occurs to one object moving rela¬
tively to another. And for this reason it is also irrelevant what
the cause of the difference may be, or what else may be con¬
nected, in another needlessness of explanation (p.79, first par.).
It remains that the centrifugal force occurs to one of the objects
in relative motion, the motion itself nameable the cause with¬
out groundlessly the stars, and it is convenient to refer to that
object as in motion and to an object without the force as at rest.
in comparison to which absence of the force, in keeping with
said equilibrium, the force in other objects occurs in varying
degrees and directions.
The centrifugal motion of objects was observed by Newton to
be due to their property of moving in a straight line if not pre¬
vented by some force, in conformance with his discussed first
law. The rotating elements if able recede by that law from the
axis, to continue in a straight line at a tangent instead. He over¬
looked an underlying situation, however, extending to other
motion beside rotation.
It has to do with what can be said to be resistance to deflec¬
tion from a straight course in things that may be held in motion,
the resistance increasing with speed. Resistance in things was
observed before to be a manifestation of force on them (p.76,
second par.), and the usual naming of the above law as of
inertia is correspondingly misleading. Inertia was mentioned
to refer to the resistance (same paragraph), whereas the law
speaks merely of absence of force. A state of rest or uniform
velocity is thus not a state of resistance, and when presently
observing that things held in motion have a resistance, it is
meant that a force affects them to a lesser degree than it does
things at rest. That this difference exists is easily evidenced
through those circular motions, which signify restraining forces
exerted on objects moving otherwise in a straight line. The
mentioned centrifuge, as exemplified by the domestic washing
machine, can provide an illustration. Outside the machine if
clothing is pressed against an obstacle and released, elasticity
will push it back to some of its former state. The same will not
happen during a spinning cycle in the machine. Because of the
speed of the items, they, unlike those at rest, will resist the
elastic forces that would deflect them from outward motion in
a straight line. A more ordinary example is provided by a
whirling around of an object on a string. Should the string
while lying still be stretched by means of the object thereafter
released, elasticity would pull the object back. Not so when the
string is stretched by the object in whirling. The speeding object
will resist its deflection from a straight course. To further disclose
the widespread observability of this difference between motion
and rest, gravity can be considered. An object suspended what
would be held motionless above ground and then released will
fall to earth quickly. But when hurtling parallel to the ground
an object resists falling, to the point that at high speed, as in the
case of the moon, it does not fall to earth at all.
There is then a quite evident difference between what would
be regarded as motion and rest, although the difference
escapes notice as a clearly formulated phenomenon. This
phenomenon could side by side with other forces of nature be
91
termed motive force, inasmuch os more resistance in on object
usually translates into more impact on the object resisted, and
it should not be confounded with energy of motion, the kinetic
energy mentioned (e.g. p. 75, first par.). Without the present
motive force that energy is merely of the law of inertia, that a
moving body will as much keep its velocity as a still one its
station, without offering more resistance to the change of its
state. Since by the law of inertia a moving object tends to con¬
tinue in a straight line, it will upon meeting in that direction
another object exert on it a corresponding force, ascribed to
kinetic energy. But by that law the effect need be no different
if the other object is regarded as instead moving toward the first
one. There still would be no difference between the powers of
motion and rest. This was seen not to be the case in the above
examples, where the forces of the moving object were, in the
spoken of form of resistance, perpendicular to their motion. The
same resistance was seen not to occur when the objects are
what is normally viewed as stationary while the same, perpen¬
dicular, action is taking place. The perpendicularly depressed
clothing, similar to the above meeting objects, will normally
push apart, but not when speeding in a washing machine. It
should be remembered that the rotation, motion, can be
viewed as relative, and hence the difference in resistance
between it and what is viewed as rest.
With motion thus possessing added force, it should be no
wonder if clocks and other physical processes slow down within
objects at high speed. The parts of the clocks and other
elements, partaking in the motion of the whole, likewise resist
deflection from that motion. Their actions not of that motion are
retarded.
The retardation of action in moving objects does as made
apparent moreover not require in order to be discernible the
high speed supposed in relativity. As a matter of fact, as known
of centrifugal force but not known as hence true of all motion,
the motive force, the retardation of action, increases as the
square of the motion’s speed. A tenfold increase in speed, for
instance, means a hundredfold increase in the force. And the
force, the retardation, can accordingly be easily confirmed to
act in all directions. Clocks and other objects are found to
function at a constant rate in all directions while partaking in
the speeding of the earth through the universe at over 100 miles
per second, by which the solar system stays aloft. It could be
asked if e.g. a clock’s constant speed in all directions can be
illusory, in that one what seems an equal period on a clock
may not truly last as long as another. But if another clock is
positioned at other angles than the first, a sameness of time is
still observed. The motive force accordingly all-directional, it
s
92
should be likewise no wonder that it is found that the higher a
particle’s speed the more difficult its increase. The particle gains
in resistance, and it is in this regard falsely inferred that the
object’s mass, its size, becomes larger as well. Mass as resis¬
tance to force is again mistaken for mass as quantity of matter.
An effect similarly resulting from resistance was by likewise
erroneous argument determined about gravity. As spoken of
before {p.77, last par., through p.78, second par.), there is
contended a relativity between bodies falling due to gravity
toward the earth and the earth moving instead toward them,
each viewpoint held correct about the same event. The
relativity is further considered extended to a more general one
between gravitation and accelerated systems. What is viewed
as happening by gravitational force is thought to be viewable
as happening by corresponding acceleration of what is other¬
wise the object of the force, and vice versa. Hence since in
accelerated systems time is determined to slow down, it is
argued that time slows down under gravity.
That matters are not the same regarding what are found to be
gravitational forces and what are found to be corresponding
accelerations was noted with respect to falling bodies (p.78,
second par.). By the supposed relativity, in addition, time in
those bodies when considered at rest and free of gravity, with
the earth falling toward them, is unlike in falling ones not held
to slow down, in further negation of the equivalence.
Things do in fact slow down under gravitation, but simply
because as in the case of motive force, or any other, gravita¬
tional force counters opposing ones, and consequently their
action is impeded. This interaction of forces takes place in the
very examples of the operation of motive force listed. That force
impedes the action of others, e.g. the falling down due to
gravity, while the others prevent motion in a straight line.
Whereas action slows down in things under the gravitational
force of reference frames like the earth, the same was seen
above not requisite in things with regard to which a reference
frame is accelerating instead, the general equivalence claimed
on the basis of certain instances. Among them are those where
things seated on, or hanging from objects seated on, earth are
respectively compared to things pressed against, or stretched
from, a reference frame due to its respective acceleration to¬
ward, or away from, them. But in the later things action slows
down progressively more, due to the acceleration, not the case
in those on earth. Also compared are to things falling toward
earth ones that inside a rotating reference frame fall toward its
peripheral wall due to centrifugal force. But in these things, in
uniform motion by the law of inertia, action is slow at a con¬
stant rate, not the case in those falling to earth and hence
93
accelerating, not to mention the referred to changes of distance
between objects falling toward the earth, and the compound¬
ing of slowing in them due to increasing force of gravity.
The believed relativity of gravitation and accelerating
systems is thus persistently wrong as were others, specifically
the relativities of time or simultaneity and space or distance,
and most notably the main thesis, relativity of motion. The
mistaken theories, by way of which a number of correct conclu¬
sions was argued, illustrate the pitfall of consequently holding
the premises, the theories, right. Conclusions in relativity, as
often happens elsewhere, were seen also not to follow from
premises given, to as much as contradict them. Inasmuch as
these errors occur in the most exact sciences, they remind that
theory in general be treated with great caution, that knowl¬
edge be not presumed lightly.
Upon having determined the existence of motive force as a
correlate of other forces, all of them the expression of causal
connections, a more general inquiry can proceed as to whether
there exist other invariable connections in nature, whether, as
sometimes supposed, other laws of nature exist beside causal
ones.
The reason that the laws or principles treated are discussed
as connections was referred to previously. These laws declare
in the quest of what may be held true of things that something
is the case about some things, and thereby establish a connec¬
tion between these things and what is the case about them, be
it their existence alone. The continued existence of nature at
large can thus be considered a constant connection, and it
could be considered a law of nature. So could the fact that all
of nature is subject besides to laws peculiar to it. In seeking
laws of nature, however, it is these laws themselves to which
nature is subject that are the connections meant.
In endeavoring accordingly to find what other such laws or
constant connections beside causal ones, broadly ones of
causation, might exist, the interest is by the preceding not in
merely whether the existence of some matter, of the described
stuff which happenings in the external world concern, alto¬
gether is always accompanied by the existence of some other.
That connection would only be of an instance of the contin¬
uance of all matter. The one existence would be known as
always accompanied by the other simply because the other, as
well as the first, is already known as constant and thus accom¬
panying everything.
The connection sought in a law of nature must consequently
be about some condition of matter beside its existence, it must
be about some material state, as earlier referred to (p.74,
second par.). And since the connection would concern a thing
s
94
and something the case regarding it, i.e. it would concern the
accompaniment of the first thing by the second, at least one of
them should by the preceding be about some material state,
beside the existence of some matter, since the mere coexis¬
tence of matter was excluded.
If then it were the first thing which is a state and not the
second, the situation would be of no interest os before. The first
thing would be known os always accompanied by the second
—still only of the existence of some matter—simply because
the second is already known os constant, accompanying
everything.
And if the second thing and not the first were o material
state, perhaps o vicinity, it, os observed (some p.74), would
hove o cause, and hence it would be of o change, not constant.
But then os before observed differently (p.71, third par.) it could
not always accompany the first thing—which is constant—and
could in consequence not be of o low of nature.
Consequently not only one of the two things connected, but
each of them, must be o state and therefore the connection o
causal one, since both states connected, having causes, would
be of changes (e.g. p.71, fifth par.). It may be added that the
connection con be the reverse of the causation, by which one
event might be always preceded, rather than succeeded, by
another. In that cose, since nothing con bring about the later
event without the prior one, the prior one is at least o con¬
tributing cause of the later. The low would accordingly be that
o certain event always has o certain other os o cause.
As o result of the foregoing, causal lows ore the only lows of
nature, further searches into which lows need therefore be
conducted among them. At present the search con be directed
at any type of causal low presupposed by knowledge of more
particular ones.
It con initially be noted that oil facts about nature ore found
to be conveyed by means of one’s body os o living organism.
The issue of what defines life was mentioned at the
beginning of the first chapter, and later with regard to the self
(p.50, second par.), and it was observed that the definition, as
elsewhere, is a matter of choice. In the later mention it was
indicated that what is of concern when speaking of life is that
it is characterized by certain purpose, one, more specifically,
not in the sense in which purpose characterizes a what is
regarded as lifeless tool, but in the sense that purposive
behavior in the living is considered to stem from within.
It was indicated also (p.60, first par.) that by purposive
behavior is meant one that by appropriate adjustment accomo¬
dates whatever circumstance toward attainment of a particular
end. This has to do with living organisms inasmuch as they are
95
entities which whatever the circumstance are found to utilize it
toward the same ends.
These meanings of life and purposiveness may be thought to
be by some not appropriate. It may be proposed that to speak
of purposiveness in living organisms unjustifiably connotes
design. Or it may contrariwise be put forward that some man¬
made machines function purposively but would not be said to
be alive. Life should thus signify less than purpose in the first
case and more in the second. The important question, rather
than being what names to assign, is, however, what sort of dif¬
ferences can be discerned between organisms and other
material entities.
And there is discernible a fundamental causal difference, it
is that other material entities do not function purposively as
described. They do not utilize circumstances toward the same
ends, but respond in different directions according to different
circumstances. This is the case also with man-made machines
built to respond to different circumstances in a way resulting in
the same end, for instance the same temperature in an
enclosure. There are in general of course varied circumstances
resulting should they occur in the same ends, and that is a very
factor made use of by organisms, in adjusting their behavior in
order to through varied means obtain the same results. The
machine, however, will lead to constant results only insofar as
its user supplies conditions required. It will not do so under all
conditions even if unimpaired, and should it it may well be
called alive.
The causal laws by which machines and other lifeless objects
exclusively function are customarily called physical and
chemical ones, and it is, to be sure, very often proclaimed that
organisms, too, are subject only to these laws. The outlook is
termed mechanism, and it is so widespread that to contest it
should meet with especial opposition.
But the refutation is contained in the above, and it is similar
to the earlier demonstration of free will. The invariable finding
that unlike things solely subject to those laws organisms,
availing themselves of them, function in accordance with ends
as described, with what may be called final cause, cannot be
overturned by another finding, which can be no more then
likewise invariable and cannot contradict fact.
This presupposes that things acting only in conformity with
mechanical cause do not also act in conformity with final
cause. That this presupposition is pertinent is attested by the
frequent contention that the components of organisms have as
part of physical and chemical forces a built-in property to act in
accordance with ends. What is meant by these forces could thus
be all sorts of things, without distinguishing them from the vital
s
96
or life force that is spoken of by sonne and that mechanism
argues against; The point is that if the behavior of organisms
be wholly explainable by physical and chemical laws as
purported, it must be decided what is meant by these laws. If
they are meant to include activity occurring in accordance with
ends, and with it perhaps the gamut of laws of nature, then the
behavior of organisms is in fact explainable by them. But this
would not nullify the purposive activity. It would embrace it as
part of the laws. Neither is the activity nullified by calling it
functional instead, as has become the practice.
Without considering volition of sentient beings, there exists
in nature a distinction thus between things that act in
accordance with ends and those that do not, and in concord
with accustomed designations they may respectively be termed
animate and inanimate, or of, as said, final cause and only
mechanical cause. These two kinds of cause represent then
different laws of nature, in that the laws usually regarded as
physical and chemical are of forces not leading to described
purposive ends, whereas organic laws are of forces that do. The
last mentioned laws are in their general aspects known as
principles of preservation of self and the species, and in
particulars organisms strive to preserve diverse functions, each
in behalf of the, largely cyclic, goals of the entire unit.
In the body of a sentient being those functions include
transmitting of information to the mind, as well as ability for
subsequent action. The information transmitted is of course one
depended on for knowledge of the external world, and con¬
sequently that knowledge presupposes the purposive function
of the body, with other physical processes part. To wit, in con¬
formance with what has been said any reality learned of the
material world, e.g. its particular causal laws, presupposes the
reality in it of the purposive organism, namely of final, together
with utilized mechanical, cause.
With these types of cause presupposed by the body as
informant of material realities,
THEOREM 11.6. Final and mechanical cause in the physical
world are realities prior to other realities in it.
The presence in organisms of final cause, of purpose, has
been thought refuted in the last century by Darwin’s theory of
natural selection. It should be remarked that a theory, not being
proof, cannot refute. The one in question ascribes the adapta¬
tion of organisms so as to cope with their environment to
nature’s through evolution nonpurposive elimination of organ¬
isms not adapted. The minority thought to adapt is held to do
so by accident, by random mutation.
As seen, however, the disclosure of purpose in organisms,
with their acknowledged life itself distinguished by purpose.
97
does not require evidence extending over generations of
numerous members of a kind. Evidence of a single organism’s
behavior while alive was sufficient. The life and purpose re¬
vealed themselves as action toward self-preservation, consist¬
ing in, beside development to maturity and retainment of that
state, adjustment to new environments, as in acquiring resis¬
tance to the harmful. Such acquired characters are irrelevantly
argued not inheritable, the purposiveness occurring notwith¬
standing. Furthermore heredity itself, by which the kind con¬
tinues, was seen as manifestation of purpose, that of preser¬
vation of the species.
Even if these discrepancies are neglected, moreover, natural
selection is in itself statistically untenable, as some have ar¬
gued, if without success. When here discussing the everyday
purposive behavior of organisms, it was observed that the
behavior of other things differs by not leading toward the same
ends under all conditions. Their resulting in disparate ends is
as a matter of fact so extensive that results are statistically
determined to be progressively less specific to eventually have
an equal chance at innumerable possibilities. This diffusion is
sometimes referred to as entropy, and compared to an organ¬
ism’s preservative adjustment to the environment it is signified
by erosion. Since the diffusion increases with time, a nonpurposive entity’s chance of preservative change decreases, and its
long-term accidental adaptation, by which it would survive
natural selection, is therefore uncountably less probable than,
as in the individual behaviors discussed, a short-term act of
self-preservation by a likewise supposed unpurposeful object.
The accidental adaptation of one surviving lineage of an
organism, ignoring the single organism’s constant adaptation,
can alone by what was said be seen unlikely at the least. That
the same accidental adaptation in addition happen to the many
lineages of a species is yet the more unreasonably unlikely,
with the topmost unlikelihood the accidental adaptation in the
very many species of organisms extant.
While the preceding has to do with how untenable acci¬
dental adaptation is in kinds of organisms that have survived,
the same is the case regarding the kinds that have become
extinct and hence held to fall victim to natural selection. The
very fact that they once were alive, not to mention survival
through generations, signifies again that they were purposive
beings, and they were adapted to function correspondingly.
Their past existence lacks as much conformity with mechanical
probability as does the present existence of the others.
It has been proposed that probability depends on points of
view, in a contradictory relativity similar to those discussed. It
is posited as highly unlikely that a man have male descendents
s
I
98
in an uninterrupted line for very many generations. Yet every
living man is a descendant from such a long line. Ironically the
birth of any offspring was seen purposive preservation of the
kind and cannot serve to demonstrate lock of purpose. Pre¬
suming the improbability true, further, it is not contradicted.
Keeping in mind that only fathers of fathers ore counted, the
more ancestral the fathers the fewer they ore, since men hove
many common ancestors. The few ore accordingly the
statistically rare ones of their contemporaries. But this sort of
justification is not necessary. If something almost never occurs
then it is by definition improbable, and if not then not. If
mechanical survival of progeny is statistically untenable then
it cannot continually occur. In other words o statistic cannot be
both true and false if to ovoid contradiction.
The contended chance adaptations, meaning here by chance
not absence of cause but of purpose, ore accordingly not
merely improbable but impossible. Statistical probability is os
indicated based on fact, and if the fact regarding certain
entities, now organisms, is that something is never true of them,
now that they not be adoptive, then there is no probability, i.e.
no possibility, of it. And the constancy of the adaptiveness,
present even when inadequate for survival, renders the adap¬
tiveness of generations, in likeness to the behavior of single
organisms, of final cause, purposive.
It is of concern to add that the destructive potential of the
theory of natural selection, also known as survival of the fittest,
was noted by others, it asserting a hereditary inequality among
men, leading to haughtiness in some and hopelessness in
others, with many a tragic result in recent history.
Observing in the foregoing the purposefulness of organisms,
it can be further observed that specific purposive laws are far
more evident than are physical and chemical ones. Whereas
laws of an organism, especially where concerning its whole,
are comparatively easy to determine, other laws of nature, as
indicated in the introduction and elsewhere, are elusive. The
most primitive man soon learns of the general action of organ¬
isms toward fruition or preservation, and knowledge of the
function of particular organs is not far behind. But other causal
principles by which events ordinarily observed occur have to
do largely with interactions of minutest particles, and their
detection hence requires sophisticated instruments and may, if
quantum theory is right, meet insurmountable obstacles. The
interactions, the causal principles, are spoken of in terms of
forces, and the remaining force of acquaintance beside the
ones just referred to is gravitation, which waited millennia to
be found a principle. The motive force herein described (ca.
pp.90-91) was seen yet to be recognized.
99
The detection of realities by means of instruments, used to
ascertain larger phenomena about the universe as well as
smaller ones about its particles, is, as was suggested (p.67,
fourth par.), in addition subordinate to the detection of realities
by means of unaided senses, because the instruments
themselves are relied on by being perceived through those
senses. There is accordingly subsequent to the previous cases,
also an order in priority regarding these two mediums and the
realities perceived by way of them.
The combined discussed order in reality is depicted in Figure
II. At the end of it knowledge of reality perceived through the
aid of instruments presupposes knowledge of the instruments
as reality (the two realities circles in dotted line, the horizontal
line in one circle signifying means of transmitting knowledge);
the knowledge of instruments in turn presupposes knowledge
of reality known through sense perception, which knowledge
presupposes knowledge of the reality of the body as conveyor
of the perceptions (the last two realities circles in dashed line,
with again a horizontal line for transmission of knowledge); in
accordance further with the theorems of this chapter and as
made reference to in its headings, the body as conveyor of
realities presupposes the reality of final and mechanical cause,
this and in the preceding mentioned realities and their
knowledge presupposing the reality and knowledge of the
world, which presuppose the reality and knowledge of
voluntary and involuntary mind as determiner, by which are
presupposed the reality and knowledge of the self as ultimate
concern (these realities circles in solid line, with vertical lines
for appropriate divisions).
The solid lines thus represent the principal determinations in
this chapter, the other lines representing, in the form of the
body and instruments, successive extensions of the self as
receiver of knowledge, together with knowledge received
through them. Should the self be postulated of possible
disembodied existence, that knowledge, perception of external
reality, might also come about without those intermediaries, as
hinted by the subject matter of the first half of the diagram.
Returning to that subject matter, it may again be observed
that both processes in accordance with ends, the one of the
VOLUNTARY INVOLUNTARY FINAL MECHANICAL
!
\
100
final cause of unconscious organisnns and the one of voluntary
mind, represent additional forces beside the mechanical ones
dealt with, because they are not results of mechanical forces
alone. That an organism like one’s body should have its own
force of final cause and in sentient beings the added force of
voluntary mind is specifically of value in one’s purposes. In
one’s body the force was observed to strive for preservation of
one’s functions, including transmission of knowledge to the
mind and ability to act accordingly, to act in accordance with,
that is to say, one’s purposes, that action an expression of the
other, mental, force. Even in plants the force by which they grow
and propagate serves to replenish man’s sustenance to
therewith have a function in man’s purposes, toward which
other forces of nature are likewise applied.
Man’s purposes were highlighted at the beginning of this
chapter as ground for determining what is real, and for
identifying the self as primary reality. In the process the self has
been taken to apply to others beside this writer. That it does has
for the matter of that, as indicated largely in the last chapter,
been taken to be true from the beginning of this writing, in
supposing a readership for it. Similarly the existence of selves
other than oneself, of other conscious individuals, is presumed
in all ordinary human transaction and is thus taken for granted.
It is nevertheless the subject of debate, in what is called the
question of other minds, and its examination here should bring
out an instance of inference of entities not perceived, not at
least by each individual. As a matter of fact inferred entities,
because of the need for their inference, are unperceived as a
rule, at least temporarily, but because the act of inferring is an
everyday recourse in coordinating one’s manifold perceptions,
its foundation can be overlooked, and disputes on the order of
the present one arise.
It can first be noted that in raising the question of other minds
in discourse, as done in these queries, the words used,
including “mind”, are for the most part received from other
men. The question could accordingly be accurately whether
what other men mean by their mind corresponds to the entity
of one’s own one learned to associate with the word. The
underlying question would be whether they, too, possess that
entity. But since however the issue is put in language, under¬
standing of the meaning by such other minds is here presumed,
minds and the like may as well be spoken of, and the task is to
explicate why the presumption of that understanding consists
in knowledge, because of which the explication should be
comprehended by the reader.
Arguments denying such knowledge start from the justified
contention that inductive knowledge, to which the questioned
101
knowledge belongs, con be obtained only from the experience
of numbers of instances, on the basis of which a generalization
can be made, in this case that other human forms beside one’s
own are imbued with minds. It is maintained that the
requirement of the repeated instances is in this case not met.
One only has the experience of one’s own mind, and it is
proposed that one cannot accordingly infer the others. The
numerosity experienced, however, is here not of beings with
minds, but of attributes of oneself that have counterparts in
other beings. From the constant known accompaniment of one’s
attributes by counterparts in other beings, counterparts of other
of one’s attributes, e.g. consciousness, can be inferred.
The constant known accompaniment does not, of course,
hold for all of one’s attributes without exception. The color of
one’s eyes may not be repeated in someone else. What does
correspond are fundamental functions of even the most
primitive animal, functions in regard to which animals can in
fact be said to be distinguished from plants by having minds, by
perceiving, as in the form of feeling. Animals are often said to
in contrast to plants feel pleasure or pain, and should these
attributes be held not to distinguish the first of these beings
from the second, one need only be reminded that the matter
has to do with definition, with what is wished to be meant by
an animal compared to a plant. What is of account is that even
at the simplest level creatures usually called animals display
behavior sufficiently parallel to one’s own and, in continuing
induction from the more to the less complex, to that of other
animals to warrant the inference that they also perceive. From
the great multiplicity of shared functions in other men one can
induce their mind in corresponding function, as one can in
multitudes of progressively less complex but still very similar
animals, and from these many correspondences one can
induce the same of the simplest creatures that display the
appropriate behavior. They in the main react to injurious or
seemingly so events by through motion at once trying to avoid
them, thus instantly seeking their well-being, from which
behavior like that of other animals their corresponding feelings
of pain and pleasure is inferred. Should entities regarded as
plants be consistently so responsive, they may be considered to
have like feelings.
That also certain other living beings beside oneself perceive
can be determined by more than induction, furthermore. In
practice it is really other beings’ actions that determines their
consciousness. Their attitude like one’s own toward the world is
equivalent to their like consciousness of it as by demanding
treatment like that desired for oneself. These functions
accordingly not only signify in other beings consciousness, but
s
I
102
selves as possesors of it, in keeping with former observations
(e.g. p.50, second par.) on the self os characterized by a basic
purpose of satisfaction. The import is that other beings” attitudes
as analogous to one’s own cannot be ignored, whatever the
exact nature of others’ perceptions.
The nature of the perceptions even in beings otherwise most
alike need not be precisely the same, as illustrated by
colorblindness. Nonetheless the more similar to one’s own an
animal’s overt characteristics, the more justified one is to induce
that its mind likewise corresponds. As animals gain the ability
to communicate, furthermore, they disclose added similar
consciousness of, beside their environment, purely mental
contents. In man these mental contents are of course especially
abounding. They are communicated in the course of history in
recorded form through highly wrought languages and find
realization in the development of civilization. Beside their
actuality other minds in men can accordingly be induced to be
like one’s own to the utmost.
That they are can, as suggested before (p.lOO, second par.),
be held for the most part taken for granted, as can be the
associated induction of which mental content is communicated
by which linguistic expression. Words and languages pertaining
to given things are known to be variable, and it is accordingly
not particular usages that lie at the basis of why communica¬
tions are comprehended. It is rather the understanding that
communication is taking place to begin with. This understand¬
ing occurs before individual meanings may be comprehended,
as when not knowing the meaning of what someone says
though knowing that something is meant to be conveyed, and
accordingly one can induce it through not merely one kind of
sign but any variety of signs the speaker wishes to make, nonlinguistic ones called ostensive. The gist is that induction of
linguistic or related meaning goes beyond signs alone, to have
a more thorough underpinning. It is based on the deeper
learning of other selves as of, alongside other attributes,
perceptions like those of oneself, and one learns, in further¬
ance of this process, by use of the means of communication
more of the sameness of the inner lives of others.
The reality of, as also consequential as described, many alike
selves, of within their purview their voluntary and involuntary
mental contents, the external world, and its final and mech¬
anical causalities, all explored herein in relation to any one
person, should suffice as the worldly realities principally
determined in this chapter. To determine more particular
realities in nature, aside from occasional exceptions, is not the
intention of this disquisition; such determinations are the
province of specialized searches, not necessarily proceeding
103
from common observations, mentioned as the material for the
present inquiries. And the determination of any reality trans¬
cending nature as customarily understood will be considered
separately; to that end and for other possible inquiries what has
preceded should provide a foundation, supplemented by the
following investigation of matters of deduction.
s
104
Chapter III
LOGIC AND
MATHEMATICS
The difference between what is meant by logic and what by
mathematics is not well established. Logic is ordinarily
regarded as having to do with rules of correct inference, and
the kind of inference in question, as indicated, is here
deduction. As observed previously, however (e.g. p.l3, third
par.), the only way in which something, including an inferred
thing, is known to be true is by finding it to be so, and that by
whatever nature it is meant manifested, presently by being
intrinsic. For that reason logical principles are not about how,
but what, things can be deduced from given ones. But
mathematical principles, too, are about this, and the question
remains how do the two disciplines differ.
The generality of the above frequent description of logic may
suggest that it in the comparison somehow treats of all things
irrespective of quantity, whereas mathematics treats of them
with respect to quantity. But logic, too, treats of quantities, as in
the concepts of “aW” and ”some”, applied to things unless fully
negated, in which case their quantity is zero, a concept part of
mathematics likewise.
The concept of “all” in logic can in kinship to discussed
necessity be taken to represent the universality of a principle,
by which something is true of all of a kind. It is thereby inherent
in logical, mathematical, or other principles themselves, which
concern all instances of the conditions considered, or of the
things considered in them, as a causal law would be of all of
certain events regarding all of certain things. The observation
is in concord with the allusion in the preceding chapter (p. 67,
third par.) that principles, e.g. laws of nature, are expected to
be true of certain things, e.g. matter, without exception.
“Some” and negation can similarly be taken to concern prin¬
ciples. To assert some can in kinship to possibility be viewed as
denying an opposing principle, a necessity if not an impossi¬
bility, the last of which can be viewed as signified by negation.
105
In all, these quantitative concepts take part by that universality
in mathematical principles as well, apart from the quantities
those principles particularly speak of. Accordingly a distinction
between logic and mathematics may be easier to make.
Considering in accord with the preceding paragraph that
principles are about what holds or does not of certain things
under certain conditions, or perhaps unconditionally, these
things—^which may be referred to in terms of conditions, states,
themselves, as in propositional logic, or in terms of attributes,
as in predicate logic—can in logic be anything, whereas in
mathematics they are limited to various quantitative concepts.
How in diverse logics these things are not limited in concept
shall be elucidated later, and the distinction here made
between them and ones in mathematics will be appropriate for
the investigations ahead.
While considering that logic and mathematics as deductive
fields differ by the nature of their principles, it should be noted
that principles are not the only things determined deductively.
Most of deduction is for that matter of individual instances from
principles already determined, and these principles, serving
that purpose, can be of all sorts, from causal ones to merely
definitions by convention. The deductions are nonetheless
regarded as of logic, because performed in conformance with
it. However, since, as noted again at the start of this chapter,
something can be known to be true only if found to be so, it
must also be found that an instance results from a principle, be
the principle a logical one prescribing that result.
More about that requirement later, now it being of interest
that while questions have often been raised as to the validity
of logical or mathematical axioms, basic laws, there has been
little question of the validity of inference of instances from
them. Furthermore since by the foregoing these axioms must
likewise be found true to enter the sphere of knowledge, and
they are considered to be indemonstrable, it is in logic and
mathematics neither substantiated that their basic laws are
true, nor that the inferred instances, which may be further laws,
fo 11 ow.
The belief that initial principles cannot be proven can be
explained through the very notion that once these are accepted
other principles and single cases follow without need of more
confirmation, with deduction thus held to be inference of the
particular from the general. Similarly to the belief that certain
basic terms cannot be defined, it is argued that since a
deductive proof must follow from some principle, proof of
which must follow from some other and so on, initial principles
must be accepted without proof.
Some terms, as noted in the first chapter (p. 28, fourth par.),
1
s
106
are thought undefinable because what they name is regarded
as unanalyzable into components, and thus upon defining
things by their components, and these by theirs and so on,
undefinability would likewise be reached. Accordingly key
terms assigned no meaning, ones by which others are defined,
are indeed used alongside unproven axioms in logic and math¬
ematics, although these terms may not be unanalyzable.
It can be observed again that in matters of worldly reality
such omissions are, with justification, for science usually not ac¬
cepted. As spoken of in the introduction, existencies believed not
to admit of demonstration, as might be ones of metaphysical
entities, are repudiated, as are factual statements that are held
meaningless, among them ones believed unverifiable likewise.
Yet such failures, extending throughout due to the dependence
on them of deductions and definitions, are commended in
logic and mathetmatics, sciences accorded greater certainty
than are others.
That all language can be defined was expounded previously,
and, as indicated by what was said, deductive truths can
likewise be substantiated. The inference of fundamental deduc¬
tive principles from the material considered is at least as sim¬
ple a procedure as is inference of further things through them,
and to demonstrate these procedures and what some of the
inferences are is the object of this chapter, similarly to objects
in others.
In speaking in the preceding sentence of the material con¬
sidered, the reference is to the content of any language or sym¬
bolism used. That content has in much of mathematics and
logic been set aside as not pertinent, the concentration being
on form, on linguistic symbols and their combination. This in¬
attention to meaning is in accord with the above, as connected
with presumed undefinabilities. But the supposing of possible
deduction on the basis of language without content, content to
which the results are said to apply by accordingly interpreting
the language, may be ascribed to added reasons. Such a one
can be considered the extreme universality of logic and math¬
ematics, leading to the conception of them as of truths separate
from anything they may be applied to. A more contemporary
ground can be held to be the disposition not to allow for reflec¬
tively acquired knowledge, with reliance on the physicality of
language instead. Whatever the reason, the attempt to seek in
language solutions to problems outside it is as misdirected here
as it was seen to be elsewhere (e.g. p. 23, last par., through p. 24,
second par.). It need only be repeated that in order to know
whether something, e.g. a logical principle as pertaining to the
things it speaks about, is true, it must be found, observed, to be
so, in whatever manner what is in question is meant observed.
107
as by reflection. To instead contemplate the corresponding lan¬
guage will not do, unless it takes the form of its content (p. 26,
lost par.).
The dismissal of the meaning of language in logic or math¬
ematics is thus o hindrance in the search for appropriate
knowledge, as well as in the resolution of the paradoxes or
other incongruities that hove perplexed men in that endeavor.
These incongruities, introduced through the centuries and by
and large known os paradoxes, falsely affect these sciences, by
motivating elaborate restrictive theories, in effort to avoid
concerned contradictions. They display contradictions partly
because of faulty postulates, but they con oil be found to arise
from concealment by language. The language fails to disclose
contradictory premises, resulting in contradictory conclusions.
A seemingly consistent statement of the form “A is B” will
accordingly turn out contradictory when it is recognized that 6
designates o negation of A. In paradoxes their very disclosure
moreover constitutes ironically disclosure of the underlying
inconsistency, since the paradoxical ports noticed ore thereby
determined to hove been hidden under the language in the
first place. Those ports ore nevertheless sufficiently removed
from that language to obscure the reason for their notice.
The following section is about how the incongruities foil to
inform of the inconsistencies, how they con furthermore foil to
impart information altogether. The subsequent section will
include the substantiation that factual contradiction is not
possible, os o result of which language that harbors
contradictions con be disallowed os designating fact, os con be
disallowed language that locks informing content.
Section 1
PARADOXES
‘”Paradox” is often intended to refer to on utterance which
though reasonable on the surface appears to coll for both
affirmation and denial of the some thing. But some incon¬
gruities coll for neither affirmation nor denial, in opposition to
what appears required, and they may with some liberality also
be called paradoxes.
In the paradox perhaps most discoursed about, the ancient
“liar”, both of these situations take place. It may be phrased os
1. This statement is false.
As appears evident by its content, if the statement is true then
it is false, and if false os it soys then it is true. This is the situation
s
108
of indicating both affirmation and denial, which situation is in
this instance mainly acknowledged. Less acknowledged is that
the statement may allow of neither affirmation nor denial.
These two presently examined interpretations combine in il¬
luminating the vagaries of language, as it can pretend to say
either less or more than is the case.
The solution to the first and usual interpretation, that the
statement is both true and false, may be found surprisingly
simple when it is considered that it has for so long been elusive.
It has in the past been prefigured and rests on the hidden
premise
1. i. By the truth of a statement is meant what it declares.
This premise, in agreement with the aforesaid, is disclosed
in disclosing the paradox, by finding statement 1 false if true
and true if false, because of the very associating of its truth
with, as in other statements, what it declares, namely its falsity.
The conclusion is
l.ii. Statement 1 contradicts definition l.i.
and is accordingly illegitimate, as would be any that overtly
declares both the truth and falsity of something.
It may be noteworthy that statement 1, the liar paradox, has
been contended false, in keeping with certain presumed logi¬
cal principles, as based on to be looked into truth tables. By
these principles if a statement or proposition implies a contra¬
diction, possibly its own negation, it is false.
It should be mentioned again that (p. 65, second par.) when
it is said that a proposition or statement implies something, it
is not really they that are meant, but their truth. It is not the
statement “This object is a tree”, but its truth, which implies that
the object is a plant. And the truth, or falsity, meant to apply to
a statement, to a declarative sentence in accord with the
discussion on its meaning (p.l5 last par. through p.l6 third par.),
is the same one meant to apply to the, mental, proposition it
stands for. For it is the same supposed facts that are in question.
And speaking of supposed facts, that a statement or proposi¬
tion, in accordance with the above, imply a contradiction can
be true only if they represent such suppositions, since actu¬
alities cannot, by to be seen law, imply contradictions. In
statement 1, in the paradox (last p.), that supposition can be
said to be that the statement implies it is false, or by the
preceding correctly that the truth of the statement implies
the statement is false. But it does not follow that the state¬
ment is false. From the supposition that the truth of “Paris is
in France” implies “Paris is in France” is false it does not follow
it is false.
Interestingly by the same belief that a statement or proposi¬
tion implying a contradiction is false the paradox is again true
109
as well as false. Since it was also determined that its falsity
implies its truth, a contradiction, it should not be false, it should
be true. This presumes that it is either true or false, as held to
be under the believed principles above, which, however, were
seen to ensure neither.
That a proposition leading to a contradiction need not be
false invalidates as a method of proof the reductio ad absurdum
mentioned in the introduction (p.5 fifth par.). The method is
used for propositions leading not only to false ones, that
disagree with some reality, but to contradictory ones, in the
sense that disagreement follows from the first proposition itself,
or from it as a result of some other assumed proposition. In this
manner the method is perhaps most notably used in Euclid’s
geometry, and the invalidity does not mean that the con¬
clusions are incorrect. The same inferences can as a rule be
reached in accordance with transposition later herein explored.
By it again if A implies 6 then not-B implies not-A, with either
of the letters and their negations interchangeable, and an
affinity with reductio inferences is readily seen. When both A
and not-A are on the basis of assumptions true then since
actualities, including conceptual ones, cannot result in
contradiction, there is likely a conflicting assumption 6 which
implies one of the contradictories, say not-A, unless the last is
as in the liar paradox assumed independently. Accordingly A
would be held true first, and by transposition of the preceding
it implies not-B, which represents the denial of the assumption
of the reductio.
As an example can serve Euclid’s Proposition 6. It states that
if angles a and b in the large triangle at right, the entire figure
representing above A, are equal then so are its sides. For proof
it is assumed, 6 above, that instead the right side of the small
bottom triangle, or of an otherwise like one whose right side
extends above the large triangle, equals the left side of the
large one. It is then argued that the two triangles thus having
those sides equal, as well as the base and the angles contained
by these lines, the triangles, too, are, by Euclid’s Proposition 4,
equal, not-A above; but they are by initial construction
contradictorily unequal. It is via reductio inferred through
deriving by it the inequality, not-B, of the pertaining sides of the
two triangles that the right side of the large triangle is not
unequal to the left side, that it is equal. The inference rests on
the further to be considered principle that a quantity (the right
side of the large triangle) is either smaller than, equal to, or
larger than a given one (the left side).
The inference by transposition follows a simpler route, not
needing a false assumption, 6, and its result, not-A. Said
Proposition 4 transposed states that if two triangles are unequal
I
\
no
then they do not have on equal angle with equal lines con¬
taining it. Hence the small triangle or any unequal, A, to the
other but with an equal angle and base containing it cannot,
not-B, have the other containing line equal. Therefore as
before, the right side of the large triangle equals its left side.
Not only is either the falsity or the truth of paradox 1 (p.l07)
thus not ensured by reductio ad absurdum, but neither truth nor
falsity can be seen to, as mentioned, apply. Because, in
improvement of definition l.i (p.l08),
l.iii. By the truth and falsity of a statement are meant,
respectively, what distinct from its truth or falsity it
declares, and the negation of that.
Definition l.i was relied on on the assumption that every
statement is either true or false, whereas in conformance with
the last definition it should be clear that if a statement is not
about something distinct from its truth or falsity then, like an
incomplete or nondeclarative sentence, it cannot lay claim to
either. Like other nondistinct entities discussed, the truth or
falsity of a statement is meant to concern a content outside
themselves, and without such a content, with nothing said that
would be true or false, a statement is vacuous and may be
termed uniforming.
As with other words, it may be remarked, there exist diverse
theories of truth, and the present definition may be questioned.
It may therefore aid to be reminded that the meaning of words
is a matter of choice, and the present one is the one of interest
and presumed to be so commonly.
Accordingly,
l.iv. Statement 1 contradicts definition l.iii
and is neither true nor false but, for that reason as well, il¬
legitimate. Since calling for neither affirmation nor denial it
might also not be considered a paradox in its first interpretation
(p.108 second par.), unless because of the conditionality by
which the truth and falsity result from each other.
Solutions to the liar paradox and similar incongruities have
been attempted by rejecting the reflexivity in these utterances,
and by replacing it by some form of hierarchy. For instance the
use of a metalanguage, a language not the one in which the
utterances are made, is thought requried for stating their truth
or falsity. The problem is again, however, not solved by means
of language. No matter what the words used for concepts or
what language they are said to belong to, contradiction in those
concepts is not therewith eliminated. As long as the same
concepts are at issue, they, redundantly, retain their, perhaps
contradictory, identity. E.g. as long as any words of any lan¬
guage, however applied, have the above meaning of “True”
and “”false”, or of other words concerned, statement 1 remains
Ill
a paradox.
The use of a metalanguage, of an additional deductive for¬
mal system phrased in it, is believed requisite for deciding that
certain logical or mathematical statements are true. The
statements are considered unprovable in the language or
system in which they are presented, and they bear a close
resemblance to the liar paradox. They can be exemplified by
2. This statement is unprovable.
It is irrelevant again in the system of what language the
statement is tried to be proven, what it speaks of and what
hence is to be proven remaining whatever its language, which
can arbitrarily be meant to be that of the system, whose
difference in language namely does not count and whose
correctness is rather at issue. The statement is in fact thought
demonstrated true by using standard language in reasoning,
and it can be found illegitimate on more grounds than is the
first paradox.
It should at this point be seen without difficulty that the state¬
ment is uninforming in the sense explained. Like a statement’s
truth or falsity, its provability or unprovability is of something it
declares that is distinct from these, and therefore statement 2,
saying nothing that would be provable or unprovable, is neither
one nor the other and, with these concerned in it, hence
neither true nor false.
The recongition that such a statement is none of these should
be of pertinence to the foundations of mathematics, allied with
logic, to both of which the statement, since presumed true and
correspondingly unprovable, brought an above alluded to
indeterminacy, with protracted elaborations on whether truths
in varied formal systems are decidable. Since these statements
are furthermore uninforming, there is, as suggested before the
start of this section, no need to fear their insidious appearance
in calculations.
Despite the introductive observation here that statement 2 is
neither provable nor unprovable, it may be of interest that were
it one or the other, and with the additional premises in
arguments contending its truth, it would not be simply true, but
would, on the basis of a premise unnoted, become a paradox
similar to the starting one of statement 1. The argument that 2
is true or unprovable has proceeded thusly.
It is first supposed that
2.i. If a statement or proposition is provable it is true.
By this and 2 it is inferred that
2.ii. If 2 is provable it is unprovable,
and from this and the referred to (p.l08 fifth par.) presumed
principle that
2.iii. If a statement or proposition implies its negation
s
112
the negation is true
it is concluded that
2.iv. 2 is unprovable and hence by its content true.
Some remarks on the justification of these propositions.
The supposing that certain propositions, like 2.i or mathe¬
matical axioms, hold is often submitted to be correct intuitively.
Perhaps needless to say, the recourse to intuition again belies
in deductive sciences the confidence in their unquestionable
certainty. The approach does not even participate in inductive
generalization from worldly experience, preferable if all else
fails. As regards 2. i, it is correct by definition. In analogy with
an earlier noted meaning of the knowable (p.47 first par.), by
something provable is meant something true that can be
ascertained to be so.
The inference, further, of 2.ii from 2.i and 2 presupposes by
moving from true in 2.i to unprovable in 2.ii, from the truth of
2 to what it states, the meaning of truth in l.i, or l.iii, a
meaning, as indicated, often disputed by those concerned.
That 2.iii is not an acceptable principle was explicated.
And conclusion 2.iv twice more acknowledges that meaning
of truth, by moving from true in 2. iii to unprovable through 2. ii,
and from unprovable in its own statement to true.
Providing nonetheless that premises 2.i through 2.iii are all
correct, along with the assumption that 2 is either true or false,
by deriving conclusion 2.iv it is believed that further matters
that cannot be ascertained were found beside such as logical
axioms, to which by the foregoing incidentally statement 2
might have been added without ado about metalanguage. In
contrast to the other supposed impossibilities, unprovability and
corresponding truth of 2 is thought by the above reasoning to
be ascertained. And therein lurks the paradox. In that
reasoning the premise is omitted that, as observed with regard
to 2.i (this p. third par.),
2.V. If the truth of a statement or proposition can be
ascertained it is provable.
Hence since in conclusion 2.iv the truth of 2 is by the premises
provided ascertained,
2.vi. 2 is provable, and false.
Therefore and by 2.iv
2.vii. 2 is both provable and unprovable, and true and
false,
and correspondingly a paradox.
As in other cases it results from violating underlying prem¬
ises. By 2.iii and 2.v
2.viii. If a statement or proposition implies its negation
the negation is provable.
By this and 2. i
113
2. ix. The provability and hence truth of a statement can¬
not be made to be its unprovability.
In consequence
2. x. 2 contradicts 2.ix
and is accordingly illegitimate.
Statement 2 can lastly if assumed either provable or unprovable seen illegitimate on a deeper ground. As suggested
when comparing the provable to the knowable (facing p. third
par.), the provable can be regarded as identical with the true,
as could the knowable with the real. For as indicated when
considering that last identity (p. 46 last par.), what is true, not
merely hypothetical, can be held synonymous with what is real,
similarly to earlier synonymities with it of fact and the
necessary. And what is provable, as indicated throughout
likewise (e.g. p.3 first par.), can be held synonymous with what
is knowable. With what is provable correspondingly identical
with what is true, statement 2, meant like 1 and others to be
true, should accordingly be provable and is contradictory for
that reason as well.
But statements 1 and 2 were both found neither true nor
false, since not meeting the described requirement of
informing. There are statements furthermore that can be held
neither true nor false although they inform. Such incongruities
have been exemplified by
3. The present king of France is wise,
or the familiar
4. The husband stopped beating his wife,
it being understood that he never beat her.
Two versions are given, because thinkers have inclined to
advance theories that intend to explain a particular incongruity,
but are unrelated to other puzzles of the kind.
Statements 3 and 4 can be held neither true nor false, since
by the meaning of the attributes the present king of France,
who does not exist, is neither wise nor unwise, and the
husband neither stopped nor did not stop beating his wife. It
could in accordance with all these examples be thought
enough to reject the belief that a statement or proposition must
be either true or false.
Yet a conviction may linger that the principle must be
somehow sound—otherwise a myriad of apparently obvious
conclusions, in everyday life or in sciences as exact as those
discussed, could not be drawn. Indeed a statement or
proposition requires truth or falsity when, as set forth in the next
section, matters are in accord with what is conversely required
by the truth and falsity. Matters concerning 3 and 4, as in the
case of uniforming statements, should have to hence be in
accord with those requirements by the truth and falsity, which
114
requirements are as usual unspoken though, as evidenced
above, disclosed in the disclosing of the inharmonies.
With respect to 3 the pertinent tacit premise is
3.1. For the present king of France to be NA/ise or unwise
he must exist.
But it is a matter of fact that
3.11. The present king of France does not exist.
Therefore and by 3.i
3.111. The present king of France is neither wise nor
unwise,
and 3 contradicts 3.iii.
With respect to 4 the premise is
4.1. For the husband to stop or not stop beating his
wife he must have beaten her.
But it was postulated that
4.11. The husband never beat his wife.
Therefore and by 4. i
4.111. The husband neither stopped nor did not stop
beating his wife,
and 4 contradicts 4.iii.
It may be noted that the deductions take a form again like
reductio ad absurdum, but utilizing instead transposition in a
manner known as modus tollens, by which if A implies 6 (3. i or
4.i), and nof-B is true (3.ii or 4.ii), nof-A is (3.iii or 4.iii).
It can also be observed that the same unmet requirement for
truth or falsity underlies many statements alleged meaningless.
They may be exemplified by ‘The sound is blue”, in the literal
sense of the color. Such statements are not meaningless—
having concepts behind them—nor uninforming in the
discussed sense—since being blue can be true. They merely
give an attribute meant with its negation to apply to something
else, here the visual.
That the principle that a proposition is either true or false
may not universally apply has been suspected since antiquity.
Aristotle’s discussion on whether it can be rightly said that there
either will or will not be a sea battle tomorrow proposes that
the principle may apply to the past and present but not the
future. The problem hinges on holding a proposition to be true
or false when considered, it being said that it is, rather than will
be, true or false regarding a future event. It is argued that since
accordingly propositions that anything will or will not occur
would respectively be now true and false, all that will and will
not occur would be predetermined, with no room for contin¬
gency. Since the determinism appears unacceptable, it is
concluded that propositions about the future may be neither
true nor false.
It is again the deceptiveness of language, however, which
115
gives credence to the argument. The present tense of “‘is” when
speaking of whether a proposition is true or false does not
concern the state at issue—^which can be of the past as well—
but the present time of the proposition regarding that state.
When it is accordingly said, in the present tense, that a
proposition about an undetermined future may be true, it is
merely meant that it is presently supposed that the future in
question will occur. This meaning in general is certainly the one
herein when saying that a proposition about the future or any
time is either true or false. It is meant that at that time the state
has been or will be present or not, though neither may be
determined, the principle applying in this sense when the
spoken of qualification is met.
That the principle may not apply in all cases has in a vein
similar to the one of the present tense of statements been more
recently ascribed to problems of existence. As illustrated by
statement 3, the proposal is that those propositions are true or
false whose subject exists or, in an extended sense as may
concern infinities, would sometime be encountered. But as
attested by statement 4, propositions can be neither true nor
false despite conformance to such requirement. And that Don
Quixote is the hero of a romance is true or false without that
conformance.
The appropriateness of the principle, as suggested in the
preceding, is in fact determined by the intended meaning itself
of the particular affirmation and denial, to dissolve the
uncertainty. By “even” and “odd” in the mathematical sense
are meant affirmation and denial regarding a property of
whole numbers. Of other things neither of the two can be
predicated. While it will in the next section be substantiated
that either affirmation or denial is required within their
intended range, it should at present be enough to say that
inasmuch as their range is determined by their users, there
should be no difficulty knowing where the principle applies.
As regards incongruities which are subject to—instead of
neither—both, affirmation and denial, and which might
accordingly be called true paradoxes, the next may be cited.
The recently perhaps most discussed is Russell’s paradox,
named after Betrand Russell, its enunciator and a chief
expositor of paradoxes. It may be given in two known versions,
the second one closer to everyday life.
5. There is a class which has as its members every
class which is not a member of itself, and none which is
a member of itself.
6. There is a barber in a village who shaves every man
living there who does not shave himself, and none who
shaves himself.
s
116
As before, two examples are furnished because of attempted
solutions of one of them that are not suited for the other.
What is paradoxical is that in 5 if the initial class is a member
of itself then—by the second clause—it is not, and if it is not
then—by the first clause—it is, and in 6 if the barber shaves
himself then he does not, and if he does not then he does.
The neglected premise explaining the paradox in 5 is
5.1. A class is a member of itself if and only if it is a
member of that very class.
This, it will be noted, is an application of the meaning of
“‘itself” and the like.
But since in 5 being a member or not in the initial class is
considered of all classes, including that initial class, and those
that are members of themselves are not members of it, and
those that are not members of themselves are members of it,
5.11. Statement 5 contradicts premise 5.i.
Similarly, the neglected premise in 6 is
6.1. A barber shaves himself if and only if he shaves
that very barber.
But since in 6 being shaved or not by the barber is considered
of all men of the village, including that barber, and those that
shave themselves are not shaved by him, and those that do not
shave themselves are shaved by him,
6.11. Statement 6 contradicts premise 6. i.
The concealed premises can again be seen disclosed in the
paradoxical findings about 5 and 6. E.g. the inital class is found
to be a member of itself if it is not because of the association
of “itself” with that class.
The positing of that class in statement 5 is thought justifiable,
because both, classes that are members of themselves and that
are not, can be held to exist. The first and stranger ones have
been illustrated by the class of nonmen, itself a nonman, and
the second by the class of men, itself not a man. Accordingly
the class of all the last mentioned classes, of those not members
of themselves, might be said to likewise exist, in seeming
agreement with 5. The class of these classes, their totality, can
indeed be said to exist, but not to the full exclusion of the other
classes. The second relation given in 5, or 6, cannot hold if the
first does, or the first if the second does, if contradiction is to be
avoided. With either relation, e.g. shaving the nonshavers,
there is at least one exception to the other, by shaving a shaver,
himself.
The attempts at previously solving these two paradoxes were,
as in the case of 3 and 4, primarily directed at the first of them.
Answers were sought in hierarchies mentioned with 1 and 2,
more specifically in restricting the meaning of membership in
a class to things other than the class. While this can be done by
117
linguistic fiat, however, it does again not prevent the underlying
situation, which, however worded, con be used for the
paradox. That the class, the totality, of oil things not men is also
not o man is o fact, whether or not the totality be called o
member or other of itself. And if the totality of oil and only
totalities not the kind of thing they ore totalities of is considered,
the paradox remains. If the initial totality is the kind it is a
totality of, then, since those kinds are not the kind they are
totalities of, it is not, and so forth.
The needless through hierarchies prohibiting of the reflexivities, as of self-reference of a language and self-membership
of a class, should for 6 result in no one’s shaving oneself. This
elucidates that the last paradoxes do not, as theory presumes,
depend on the relation between a class and things that are
members or not of it. In 6 the relation is instead between a
barber and those who are shaved or not by, not members or not
of, him.
That the last paradox is in fact one was in avoiding equating
it with 5 attempted to be denied, by again recourse to
language. It was suggested that the barber is a v/oman or not
a resident of the village, and the language herein in 6 is to
accommodate this question. There is no obstacle to expressly
postulating that the barber is a man and resident, however, and
the paradox is in effect.
Other paradoxes leading to both affirmation and denial have
been put forward, and it should suffice here to adequately
resolve ones of a wide interest. As with the other incongruities,
since actuality admits of no contradiction, there should be no
problem with these in practice. The rather laborious debated
antinomies of the 18th-century philosopher Immanuel Kant will
accordingly not also be discussed here. They cannot, further¬
more, be regarded as paradoxes, since the contradictory results
alleged do not obtain.
Analogous to the preceding two is the paradox known as
Grelling’s. It concerns the word “heterological”, intended to
refer to all and only those words not referring to themselves.
“English”, for instance, does not qualify. Accordingly if “hetero¬
logical” refers to itself then, since it refers only to words not
referring to themselves, it does not, and if it does not then,
since it refers to all words not referring to themselves, it does.
The premise that makes the condition contradictory is again
that a word refers to itself if and only if it refers to that very
word. Hence since being referred to or not by the word
heterological is considered of all words, including that word, it
cannot be defined to refer to words not referring to themselves,
and not to words referring to themselves.
Analogous reflexive paradoxes can evidently continue to be
s
I
118
constructed. Reflexivity is conducive to incongruities, because
normally when something has to do with something it is not
expected to have to do with itself. Thus when regarding shaving
by a barber one may not think of him as shaving himself, or
when regarding assertions of truth one may not think of it with
respect to the assertions. Accordingly another such paradox
may be considered, differing by being viewed mistakenly as
concerning number. Known as Berry’s paradox, it can be
paraphrased as
7. The smallest thing not nameable in fewer than twelve
words.
“Smallest thing” replaces a phrase such as “least natural
number”, which number is groundlessly thought to exist,
because nameable thus by description 7. The thing need not
be the smallest either, but anything specified, and it appears
both nameable and not in fewer than twelve words. The first
since description 7 in fact names it in only ten words, and the
second since the description says so. The clue to the solution
is in the first of these, corresponding to which the tacit
premise is
7.i. All things are nameable in fewer than twelve
words.
This is the case simply because anything can be arbitrarily
named by a single word.
Consequently
7.ii. Description 7 contradicts premise 7.i.
The notion that the numerical form of the description refers
to an existing number can be explained by the role played by
numerals. Numbers are most often named in terms of
numerals, even if put into words, as they can be held if spoken.
And in accordance there can indeed be a least natural number
not nameable in fewer than a given amount of words—or
syllables, as 7 is often presented. The least not nameable in less
than two syllables is seven, and in less than two words, com¬
pounded, twenty-one. But the issue is naming numbers by other
than numerals. And any number can be assigned as short a
name as desired, as is “pi” in comparing a circle to its diameter.
The mistaking of the varied nature of other language for that
of numerals enters also into what is known as Richard’s paradox
and consists in, rather than a paradox, a misunderstanding. The
postulation is of
8. The list of all finite natural-language descriptions
naming a real number,
a number expressible in decimals. It is then observed that in
accordance with set theory a description can be added naming
a real number not on the list. The description may for present
purposes by phrased as
119
8.i. The next real number not listed.
It is concluded that by 8 and 8.i
8.ii. The list both does and does not include oil those
descriptions of real numbers.
The conclusion does not follow, however, even with pertinent
set theory accepted. While description 8.i may name on
unlisted number, it need not be on unlisted description. Once
listed, it con be used repeatedly, each time naming o new
number.
The assertions of set theory on which the preceding is
founded, however, ore likewise erroneous. It is argued that the
set of real numbers is larger than the set of natural numbers. In
support it is assumed that the sets ore of equal size, and on
believing to derive a contradiction, it is by reductio ad
absurdum inferred that they are not. With reductio seen not
reliable or necessary, the presumed proof can be as follows.
1 ^ .1 • • •
2 ^ .2 ‘ • ‘
3 ^ .3- • •
As pictured, with every natural number is paired a real
number in decimals, the mutual pairing assumed with reductio
in addition possible throughout. It suffices that it is argued that
after a real number is paired with every natural one, another
real number can be added by the method known as diagonal.
If the decimals were as in the left arrangement, then in ac¬
cordance with the diagonal there, the number on the right
could be produced by changing the first digit of the first
number, the second digit of the second number, and the like in
the third case. That is to say, there should always be a number
differing from previous ones by at least the one digit under the
diagonal.
As should be evident, if the digits were limited to three they
would soon be exhausted, and the point is that a new number
can be produced indefinitely since a new digit can always be
added at the end. The same is true of natural numbers, how¬
ever, in the opposite direction. An opposite diagonal could be
used, and in fact when previous digits are exhausted, the next
number is produced by adding a digit in front.
The reasoning flaw in set theory lay in supposing completion
of these sets, so that by asssuming completion of one of them
the other can be continued since infinite. Neither can be
complete, for their infinity allows continuance by definition.
As a result it is also not required that the list referred to in
120
preceding 8 {p.118), and held to be the size of the set of natural
numbers, be exhausted, although no paradox was seen
regardless. The list has to merely by digits name those natural
numbers, themselves real, to be endless, giving no occasion for
8.i, much less 8.ii, there being no list reaching all those
numbers.
The notion that certain infinite sets are, as in the preceding,
of the same size stems from a concept of countability, by which
a set in question is paired with the set of natural numbers in a
manner like the above. The set of rational numbers, numbers
expressible in fractions, is held countable, i.e. the same size as
the set of natural numbers, by arranging it for example as
pictured, pairing the fractions with natural numbers in for
instance the sequence designated by the arrows.
1/1-^1/2 1/3
i T
2/1 ^-2/2 2/3
i t
3/1-> 3/2-^3/3
By such a sequence any given fraction, any given com¬
bination of the digits, is seen reachable. But this is not true of
every fraction there is, it should be noted, since there is an
infinity of them. By a fraction to be given is meant that it had
been conceived, not that it is reached via that sequence. And
this suggests that there is no reason to because of that
sequential listing decide that the sizes of the set of rational
numbers and the set of natural numbers are equal. Given
fractions may as well be listed haphazardly so as to be
reached, without in either case implying the equality.
As a matter of fact the same sort of enumeration last dis¬
played is, though denied, possible with every infinite set of
numbers. The above set of real numbers, held larger, can
serve as example, revealing in fact a mirror image with respect
to natural numbers, as did the spoken of diagonal.
.1 1.0 0.01 10.00
.9 9.9 9.99 99.99
Depicted are merely samples in the process, which con¬
sistently returns to every former column with every further
one. In the first one depicted the digits will be listed from 1
through 9; by the second depiction thereupon the same but
starting with zero since considered is repeated in that column
with 1 before the decimal point, then with 2, through 9; after
by the third depiction likewise listing all used digits in the
121
second column after the decimal point, the same is repeated
there with 1 in the first decimal column, then with 2 etc., to
repeat the whole process with 1 before the decimal point, and
so forth; thereafter by the fourth depiction all the preceding is
repeated with 10, with 11, through 99, to continue the same
with the third column after and then before the decimal point,
and so on with more columns. Accordingly any given real num¬
ber, any given combination of the digits, is by this sequence
again reached.
It might be objected that real numbers such as pi, only ex¬
pressed in infinite decimals, are never reached. These numbers
can by above, however, not be regarded as given. Because of
that infinity they are not conceived in their entirety, and to the
extent that they are given they are indeed reached by that
sequence. More pertinent is that that infinity compared to its
lack in rational numbers, so as to distinguish real ones from
them, is known without resorting to these counting methods,
and still more pertinent, this difference does not imply a
difference in the size of the sets, but merely in the way specific
numbers can be written. There still is an infinity of them in each
set.
Similarly are the counting methods irrelevant to the size of
any other infinite set of numbers, aside from the countability of
each of them by a method like the last. As can be subsets of sets
in set theory, which are likewise held uncountable, any of the
sets can be counted by, as in the last case, repeatedly
exhausting, with all required additions, all digits from 0 through
9 where they occur.
The faultily thought sizes of inifinite sets gave rise to para¬
doxes in set theory itself. For their survey it will suffice that the
paradoxes, known as Burali-Forti’s and Cantor’s, assert
9. The size of certain infinite sets is both largest and
not largest of a kind.
However, by the size of something, by how large it is, is
meant the measure of its complete extent in question, and
since the infinite is meant not to reach completion,
9.i. An infinite set has no size.
Therefore, in another case in which neither affirmation nor
denial holds,
9.ii. The size of an infinite set is neither larger nor not
larger than another,
and 9 and the theses leading to it contradict 9.ii, as well as
9.i.
Although the infinite is never complete—so as to consti¬
tute a whole, all of an extent—language need not be so inflex¬
ible as not to permit reference to, for example, simply all
numbers instead of all numbers that may be dealt with by man
in the course of time. But the lock of completeness, of size,
should nevertheless be kept in mind, for avoidance of more
error than the preceding. With the preceding it is also argued
that the whole is not necessarily greater than the part, con¬
trary to longstanding acceptance. As reason it is stated that,
for instance, the set of even and odd numbers combined is
equal in size to the set of either of the two alone. The claim
is based on shared infinity, despite the differences seen
argued regarding its size. As infinities, however, cannot be
unequal in size, possessing none, so can they not be equal.
And when any portion of them, any size, is taken, becoming
a whole, it is indeed greater than its part. It is so in fact by
the meaning of part, as less than all, the reciprocal confirmed
later.
A contradiction like the one between the infinite and the
complete is also the cause of ancient paradoxes regarding
infinitesimals, as set forth by Zeno of Elea. The most popular of
them, the ^Achilles”, recounts
10. In a race between Achilles and a tortoise the
tortoise has an early start. As a result Achilles, although
much swifter, cannot overtake the tortoise, for when he
reaches the place it held at the time of his start the tortoise
will be at a farther place, and when he reaches that place
the tortoise will be farther yet, and so forth.
In other words, supposing the distance covered by the tortoise
discounted for both runners after Achilles starts, in order to
reach the tortoise he must first cover an intermediate distance,
then an intermediate distance of what is left, to continue the
same without end. He accordingly would as suggested by
another paradox of Zeno, the ^’race course”, remain stationary
altogether, since to reach any place he must reach an
intermediate one, to be reached through an intermediate one
etc., none of which would by the preceding be hence reached.
The contentions were to sustain the thesis of Eleatic
philosophers that there is no motion. “Motion” is a name for an
experience, however, and therefore cannot be refuted.
In explaining the paradoxes it should be brought out that the
distances spoken of are finite. It is not expected that the tortoise
be overtaken by Achilles if it is infinitely far from him. No place
can for that matter be at infinity, because by it is meant that it
reaches beyond any place. The incongruities thus of course
reside in inconsistency with experience, by which finite
distances are traversed. Yet the assumption of infinite
divisibility, or reducibility, of a finite extent, leading to the
contradictions, is commonplace. That assumption, however, is
not only inconsistent with experience, but with finitude itself.
As the opposite of the above meaning of infinity as never
123
complete, by the finitude of something is meant that it reaches
completion. And by the infinite divisibility or reducibility of the
finite is meant that beyond any part of it reached there is an
infinity of other parts, that, in contradiction of the last sentence,
completion is never reached.
Accordingly,
10.i. A finite distance is not infinitely reducible,
and in consequence
10.ii. Paradox 10 contradicts determination 10.i.
The virtually universal belief that a finite extent can diminish
indefinitely can be once more attributed to the capacities of
language. A fraction can in notation be continually decreased
by increasing its denominator, as is ’/2 to ’/a. The availability
of always enlarging a denominator or other number is a
linguistic convention, but, despite the usefulness of setting no
limit to the minimum or maximum sizes of quantities in
mathematics, the language, as always, does not establish fact,
and can contradict it.
The observation that there can be no endlessness to a finite
extent nor an end to an infinite one has bearing on geometry,
including the much disputed 5th postulate of Euclid, which
resembles the Achilles paradox. Before proceeding to it and to
those preceding it it will be well to consider certain basic
concepts initially defined by him, some of which belong to
entities held indefinable.
He has been held not to make use of his definitions of the
last mentioned kind, but this will be seen not quite the case,
even if the use was not explicit. What matters especially is that
such definitions, as indicated, can be factors in deducing
propositions in question.
Among the concepts presently at issue are point, line and
surface. Euclid suggested that they can be extremities, and the
words are indeed used for that, for limits, as connoted by sur¬
face. They are defined by him and generally understood as in
effect lacking some customary dimension, some quantity by
which they would be perceived independently, if only as con¬
cepts. It is hence no wonder that they are thought indefinable,
and if without perceptual attributes in some connection, they
would by the in the first chapter observed criteria not be
concepts at all. But they are held to lack some dimension for
good reason. They are used to mark the end of one thing and
the beginning of another, often by choice without preexisting
difference between the things, or with either thing possibly a
void. They can thus be regarded not as independent entities,
but as limits of things as determined by the adjoining of further
things, either a possible void again.
The situation is peculiar as regards surface, however.
Although as surface it is not independent of the volume which
it concerns, as perception it is independent, and in fact objects
in the world are perceived by surfaces. For that reason, and
because of the parts played in defining these three concepts,
the usual order of defining them will be reversed, to begin with
surface.
It is defined by Euclid as that which has length and breadth
only. These attributes, however, imply the vertical and hori¬
zontal lines at right angles to each other by which something
is held to be of two dimensions, and lines at these angles had
not yet been defined. The like could be said if volume, a solid,
were now defined by three dimensions. However, volume,
depth in things, is detected without geometric knowledge of
dimensions, as indicated when discussing matter in the last
chapter (e.g. pp. 68, 69, last pars.). Acquaintance with things of
volume is by such as diminishment of their parts through dis¬
tance, and they can correspondingly be used in defining
surface. In keeping with the above informal definition of limit,
then,
a surface is a limit of a volume,
which, though considered only of the external world, can be
conceptual. To proceed, in place of Euclid’s definition of a line
as breadthless length, i.e. as what has one dimension, it can
be said that
a line is o limit of o surface.
And in place of his definition of a point as that which has no
part it can be said that
a point is a limit of a line.
It can be seen that as used it is an error to regard points as
constituents of lines, lines as constituents of surfaces, and
surfaces as constitutents of space, as has become the practice.
None of these presumed constituents have the required extent
possessed by the things of which they are limits and of which
they would be constituents. It may be added that, as an
instance, the concept of line has been used to represent single
extents such as of time. In that case, likewise, a point, as when
marking on a clock the beginning or end of a period, is a limit
of it and thus is not its component.
In continuing definitions in accord with what can be held
understood meanings,
an angle is a shape formed by two lines meeting at a point
and of a distance at next points.
Distance from a point is normally conceived as measured by
the length of a straight line, so that if the line starting at that
point is rotated all around it, the path described by the other
end marks all those distances from the point. But should the
straight line, not yet defined and perhaps bending, if prolonged
125
describe a path inside the first, the new path would be held to
be nearer the starting point, not farther. Hence distance from
a point, a center, can more generally be defined by the circular
or spherical path described by a point on any ideally constant
shape, as might be a compass, rotated at another point of it
around that center. Since any constant shape can describe those
paths, it will also be more fitting to soon define a circle and
sphere themselves by way of that general shape rather than, as
done by Euclid, a straight line.
But particular angles above are distinguished by more than
accordingly determined distances between points on the
meeting lines. Most angles have a reflex angle, a certain larger
space on one side of the angle than on the opposite side. The
difference is measured by the sum of uniform smaller angles,
degrees, between the two lines on a side. It is similar to the
rotation above and thought of again in terms of straight lines,
as well as planes. The last entity also not yet defined, a more
general conception than usual is again suitable.
A plane, as a surface that may not be seen frontally, is part
of the concept of three dimensions, although it is regarded
as of two dimensions technically. That it is so regarded is
understood by considering that an individual perception, as
through one eye at one moment, does not determine depth. It
only has the semblance of what is held to be a flat picture,
although like it it may display such as perspective. A single
perception, viewpoint, is thus the basis of area relationships
assigned to flat surfaces, to planes, in space. A circle on a
plane, as often noted, is seen at best as an ellipse if not
frontally viewed. Accordingly those relationships stricity hold in
accordance with a viewpoint, not a plane. To go farther, most
standard geometric concepts, as suggested when speaking of
motion in a straight line (p.84 last par.), are ideal, it being
unlikely to find in nature a true circle, square, or the plane in
which these are to lie. These concepts are thus authentically
conceptual, and the descibed viewpoint for those in a plane
concerns the mere form of their conception. In asserting things
that would take place in a plane, it is hence, as frequent, not
required to speak of it, the assertions alone of those things
informing of conceptions more precise than seen in an actual
plane, assuming it possible. In this vein and in accord with the
preceding it can be stated that
o circle is o closed line with every part equidistant from a
point within,
the definition informing of the conceptual relations without
reference to a plane. On what would be held an actual flat
surface a figure would be a circle if by suitable measures it
compared with that conception.
s
126
In consonance with these observations, when speaking of
degrees as marking the size of angles, the conception does
likewise not require a plane. Nor does it here require straight
lines. Similarly to distance, degrees can be conceived through
rotation of any constant line about a point on it, and they
become uniform by conceiving equal distances between stages
of a next point on the line. The distances or degrees are by a
later equality principle the same at all other points as well,
since there is by postulate no differentiation.
The next definition can be accordingly that
a straight line is a line with angles at every point equal on
all sides.
By all sides is meant, conceiving the line to be in a world of
depth, the sides of the line from each viewpoint as explained.
This meaning of straight line, in concord with its common
understanding as not deviating on any side, has vaguely been
shared by science. When the horizon, as demarcated by the
ocean for example, was found to represent the curvature of the
earth, it was inferred not to be straight, rather than holding a
straight line to be curved. Yet there has been an inclination to
redefine a straight line in view of physical phenomena, such as
the path of light. A straight line is sometimes thought of as the
shortest distance between two points, and an interpretation
offered of this distance is that it is the shortest route by which
two points in nature can in some manner be connected. The
path of light in a vacuum would accordingly be regarded as
straight although due to gravitation it should curve near a
massive body like the earth. Or travel on a globe like the earth
would be seen as straight if along a great circle, around the
center of the globe. But these curvatures vary, and it can be
asked what amount of curvature should serve as standard of
straightness. A more general flaw is the attempt to make a
deductive science dependent on physical observation. Such a
science is, as was indicated, built on what follows from general
concepts of so-called all possible worlds, independently of the
things in nature to which the results are applied, and therefore
to seek in nature answers to the questions is misdirected.
Doing so can be viewed as a form of spoken of bypassing the
mental and focusing on the material (e.g. p.2 second par.), as
can be the opposite tendency by which a straight line, called
only a line, is alongside other words left undefined, seeing
the word only as a physical utterance. In continued physicality
and conformity with a previous comment straight lines, to¬
gether with points and planes, are also though invisible held
to in infinite number be components of actual space, and
straight lines, planes, as well as space to stretch into infinity all
around.
127
The unserviceability of meaningless words was pointed out,
and so was the nonexistence, conceptual or real, of invisible
things, things without perceptual attributes. As regards the
infinite number of those points, lines and planes, it is partly
meant present in the infinitesimal direction, in which it was
seen impossible even if the entities existed as proposed.
Regarding the infinite extent of the lines, planes and space, i.e.
volume, these entities are usually in concept, let alone in
reality, considered as of finite extent, and therefore they and
others are as common meant here to be of some particular size.
Concerning any meaning the question can be raised whether
the entity defined is possible, and as made clear in the
foregoing the issue here is conceivability, not actual existence.
With regard to the first four italicized definitions (p.l24), of
point, any line, surface, and angle by not necessarily straight
lines, the referents can be discerned in nature. But straight lines
in nature, as seen, are queried, and circles were likewise noted
as ideal (defs. pp.125,126). However, as to the conceivable,
although like the physically possible likewise not required if it
serves the deductive purposes, the field is much more open.
Aside from the logically required, mentioned unconceivable
otherwise, it admits of the numberless products of fancy man is
capable of. More specifically and to the point, one can by
whichever perceiving faculty conceive next to any of its
elemental objects any other. It is the conceptually extrinsic
compared to the intrinsic, the latter governed by logical
requirements, but the former not even by the restrictions
imposed in the world by laws of nature. The visual or tactile
faculties, not referring now to the senses but their mental
counterparts, in addition allow these conceptions all around
the elemental objects, in what are known as two and three
dimensions.
Accordingly, a circle is possible, because one can conceive
a constant shape continuously turned all around a point.
And considering that
a sphere is a closed surface with every part equidistant from
a point within,
it is possible on the same ground.
Correspondingly a straight line is possible, because one can
conceive an element of a line at equal angles to a next one on
all sides. The possibility is more fully elucidated through the
meaning of equal angles as their coinciding if appropriately
superposed. Accordingly when from a viewpoint the opposing
angles of a line are unequal then if superposed at one arm of
each, both angles on the same side, the other arms part. But
since they can rotate continuously, they can close that distance.
Namely the angles can be equal. And they can be so from
128
\
each viewpoint, because each is distinct. The angles of a line
can further be equal on all sides at each point, since one can
by the preceding conceive at each the same.
Considering further that
a plane is a surface with angles at every point equal on each
side,
the viewpoints meant to be ones seeing the surface as a line
dividing two sides, a plane is possible because one can
conceive an element of a surface at equal angles to a next one
on both sides and from all viewpoints, and one can conceive
the same at any point.
The angle of a straight line or plane is called a straight angle,
and by definition
a right angle is half a straight angle.
Its possibility should in the light of the preceding be manifest
and not necessary to substantiate here, the present depiction of
superposed vertical arms of unequal angles, combined
equaling a straight angle, viewable if wished like the previous
one as closing the other arms.
But Euclid’s first three postulates likewise have to do with
possibility, in effect asserting it of certain things, and hence
warrant examination. They state
1. To draw a straight line from any point to any point,
2. To prolong a finite straight line continously,
and
3. To describe a circle with any center and distance.
Beginning with postulate 3, the distance can be taken to be
any separation, however determined, of the center from the
point from which the circle is to be described, and by the above
the point can be conceived to rotate all around the center.
Therefore the circle is possible.
The postulate may also be understood to say that a circle, or
a sphere, can expand indefinitely, viz that the line or surface
describing them will upon going beyond previous ones not
somehow return to them. This is questioned with regard to a
cosmological model of the universe as spherical, by which it
has no boundary not exceedable, but by which there is an
eventual return to a former one.
That one can conceive outside a circle or sphere a further
point, and thence describe another circle or sphere around the
center by a constant shape, was decided by what went before.
And as before, because all stages of the rotation are
undifferentiated by postulate, the new circle or sphere extends
past the old one throughout, i.e. it does not return.
It may be remarked that the return has been held possible for
such as a circle on a sphere or cylinder. But these surfaces are
not planes, represented by viewpoints as described. An
129
expanding circle on a transparent cylinder may from a
viewpoint appear as pictured at right, to eventually in parts
return to itself. The expansion in those parts does not signify
extension past the previous circle, however, those parts not lying
outside the circle. It is conceivability—and accordingly absence
of a contradictory return—of extension outside a circle, and
correspondingly a sphere, that was noted in the preceding.
That the continued straight line of postulate 2 is possible
should be likewise evident from what was said on the
possibility of a straight line and before. But the postulate is
similarly to the third one sometimes understood to say that a
straight line can extend indefinitely without returning to a
previous part, as is likewise questioned.
In answer it will help to conceive an initial part of a straight
line to rotate around its starting point, describing a sphere by
the other end. Let some initial extension of the line fall on or
inside the sphere. This can be conceived from a viewpoint by
which the extension is on one side of the starting line. It cannot
lie on that line, by the meaning of extension, and it cannot also
be on the other side. The two sides are by the foregoing
different meant areas, places, divided by the starting line, with
difference of things in perception synonymous with difference
of place. Accordingly if an extension were also on the other
side, not being of the same place it would not be the same one.
But with the first extension on one side and not the other, there
is a differentiation. Should an extension continue in a straight
line, however, there would be no differentiation between the
sides. Hence the extension, not falling on or inside the sphere,
must by mentioned principle (p.109, end of third par.) continue
outside the sphere. Further, upon describing there another
sphere, an extension will continue outside it as well. Therefore
since by the last postulate expanded spheres do not return to
former ones, a straight line extended will neither and will not
return to a previous part.
It is by the preceding not surprising that straight lines can
measure distance. And if postulate 1 is right then the measure
can be applied anywhere.
By the reasoning for the second and third postulates a straight
line, beside its rotation all around any starting point, allows
extension to the next distance beyond the sphere described,
with like rotation there, and so on. Therefore, filling all volume
concurrently with the spheres, it can reach any point, and the
line of the first postulate is possible.
The postulate succeeding those listed is
4. All right angles are equal,
and it follows quickly from the definition of straight line (p.l26).
Inasmuch as a straight line has at any point equal angles on all
s
130
sides, and there is no differentiation among parts of the same
or different straight lines, all their angles are equal, i.e. all have
the same number of degrees. And since right angles are half
those angles, they all have half that number of degrees and
therefore are equal.
The lack of differentiation in angles of different straight lines
can again be illustrated by superposition. If upon a straight line
is placed a line of some angle with one arm coinciding with
one of the straight angle, and the other arms part, then the
second line is not straight, since on the parting side its angle is
by that amount larger and on the other side smaller than the
straight angle. Therefore another straight line must coincide
with the first, their angles equal. The equality of right angles
can evidently be illustrated similarly. If on assuming a line
dividing a straight angle into two right angles another line
starting at the same point of the straight angle extends on the
same side of it as the dividing line while not coinciding with it,
it will again make different angles with the straight angle. To
make equal ones as right angles it must hence coincide with
that dividing line, making all right angles equal.
Such equalities are in geometry today often called congru¬
ences, and they evidently do not concern selfsameness, but a
sameness of things as compared to each other. Both concepts,
of selfsameness and of sameness by comparison, apply also to
the nondifferentiation spoken of, as, in the first case, to same¬
ness of a line rotating about a point,, and, in the second case,
to sameness of the angles of straight lines. Selfsameness should
need little say here, since simply characterized by the postu¬
lated entity. But the other sameness, congruence, requires an
added criterion mentioned, by which forms not only have a
resemblance, but are held identical. The identity is known to be
decided by upon placing one thing appropriately onto another
the concerned things coincide.
Notwithstanding that congruence is meant to be in geometry
decided by such superposition, it is inconsistently objected that
Euclid cannot make certain of it for proving a proposition. It
is maintained that in the process of applying one shape to
another it may change. The shapes, however, are meant to stay
constant by postulate, the process of superposition again ideal.
In nature it is no more likely that shapes may not stay the same
than that their coinciding may not be made certain. Similar
comments are made on propositions in Euclid that speak of
ways of construction. They are taken to involve the use of tools,
although they can like other proofs be understood as of what
follows from given assumptions.
Proposition 2 thus informs how given a certain straight line,
separate point, and other geometric assumptions, a straight line
131
equal to the first and starting fronn that point will follow. It has
been argued that a tool like a compass, presumed to be used
in other parts of the construction, could be used instead of the
geometric reasoning to transfer the line. But it is not tools that
are at issue, but what is deductively implied by what. However,
the transfer of that line from one place to another could well be
merely assumed, as was its rotation to describe a circle in the
same demonstration.
And Proposition 4 asserts what will result if two straight lines
forming an angle are superposed on an equal, congruent,
shape. The objection was that the shapes may change during
the procedure, and, as observed, they would by postulate
remain the same. There is no need, furthermore, for super¬
position, since they, as noted, would by definition of the equal¬
ity or congruence coincide, which is in question. Merely results
of the coincidence are needed.
A result stated in the proposition is that since the end points
of the two angles would coincide, so would respective straight
lines joining those points. In support it is argued that if these
lines do’ not coincide then, impossibly, two straight lines
enclose a space.
It should be observed that the two lines, as well as the
resulting triangles and their angles as also stated, will coincide
merely because the shapes are as before undifferentiated.
Could two straight lines enclose a space, they might do it with
each of the angles. That they cannot is with the proposition
informally supposed, and it will now briefly be proven for pre¬
sent purposes.
Let two lines at least one of which is straight enclose a space,
and let with one of the meeting points as center be with the
other described an arc of a circle. Then the straight line forms
equal angles with the arc on both sides, because there is again
no difference between them. But the other line does not form
equal angles with the arc there, because the enclosed space is
added on one side and subtracted on the other. Therefore it
cannot be straight, and two straight lines cannot enclose a
space.
Coming to the mentioned fifth postulate, the last, it in cor¬
respondence with the diagram at right states
5. If a straight line (horizontal) crossing two straight lines
(slopes) makes the interior angles (a, b) on its one side less
than two right angles, then the two lines, if prolonged
indefinitely, meet on that side.
A recent modification also asserts, as does a Euclid proposition,
that if the sum of those angles equals two right angles, then the
two lines do not meet on either side, they are parallel. It may
be added that the lines lie on the same plane or, in accord with
132
\
what was said,, are seen from a single viewpoint. The original
postulate will be proved at present.
If the distance between the slopes is reduced to zero on the
horizontal, they interchange above it their left and right
placements. For since a and b equal less than two right angles,
and either, since on the horizontal, equals with the, exterior,
angle on the other side of its slope two right angles, a e.g. is
smaller than that exterior angle of the other slope. Hence that
slope, both these angles being on the right side of their slopes,
extends now on the left side of the slope of o, the side opposite
the previous one, and the slopes do not meet again if not to
enclose a space. Let the slopes gradually return, to again be
distanced at the horizontal, interchanging placements in
conformance with the postulate. They cannot pass each other
all at once, namely coincide at one stage, since contradicting
the less than two right angles; since the angles concerned are
on opposing sides of the slopes, they would through the
coinciding equal two right angles. And should the slopes pass
each other not all at once but still at multiple points, they could
not both be straight; either because, if not coinciding between
those points, they enclose a space, or because, if coinciding,
they part elsewhere to differ in angles at the point of parting.
Hence the slopes pass each other at only one point at a time.
But since they can be prolonged indefinitely on the side
previous to passing, there always remain points not passed. In
consequence wherever on the horizontal, the slopes cross each
other and therefore meet.
The postulate resembles the Achilles paradox (p.l22) by the
through the prolongation approaching of one slope by the
other, indicating that a finite distance does not take infinity to
be traversed, as do the points beyond any one passed indicate
that an infinite line cannot come to a finish. These issues do not
come into account in the mentioned addition to the postulate,
when a and b equal two right angles. That here the slopes do
not meet is proved as follows.
By a method of Euclid angles c and d in the diagram at left
are found to equal a and b respectively. Since a and d, like a
and b, equal two right angles, b and d are, on subtracting a
from each pair, found equal. The same holds for a and c, on
subtracting b, and hence the lower slopes are but the upper
slopes turned about. If two such once more undifferentiated
lines meet in one direction they meet in the other, enclosing a
space. Since the present lines are straight, they cannot enclose
a space and therefore do not meet.
To find e.g. a and d to equal two right angles Euclid assumed
a further line, at right angles to the slope and dividing d into
two angles. He then observed those two angles combined to
133
equal d, and the one of them next to a to equal with it a right
angle, drawing via additional steps the conclusion, that a and
d equal two right angles. But the premises of the lengthy proof
use the sort of evidence that can be used directly. Instead of
finding that a right and other angle total d, or that a right angle
is totaled by the other and a, one can find at once that a and
d total the straight angle of the slope, its two right angles, it
being exactly that straight angle which a and d were meant to
fill. Also of note is that to know that an angle divided by a line
into two angles equals their sum makes alluded to tacit use of
defining a line as without breadth, and the like applies to other
meanings.
Meanings play instead a misleading part in non-Euclidean
geometry, founded on the rejection of the just proven fifth
postulate or its companion. These are replaced by postulates on
which are built new geometries, which correspondingly, while
thought to be consistent, engage in contradiction.
In the geometry known as hyperbolic is posited together with
the lines of the fifth postulate (p. 131) a line or circle at infinity.
If a and b, though less than two right angles, are sufficiently
large, it is then maintained, the two lines in question, the
present slopes, do not meet. As expounded, however, no
limiting line can be imposed on an infinite extent without
contradiction.
In the geometry known as elliptic a spherical surface is
assumed for the proposition in which a and b are right angles,
if not merely equaling two (facing p.). If the horizontal line is
conceived as the equator and the intersecting lines as
meridians, then the last two meet at the poles. However, the
surface is not a plane and the lines accordingly not straight,
contradicting preconditions.
It is argued in defense that certain terms must be suitably
redefined. It should be reiterated, however, that different
definitions do not escape problems of the content behind. The
infinite or the straight may be given other definitions, but it can
still be asked whether a proposition holds under the previous
ones.
The last two geometries are held to be as consistent as the
first, because if the original meaning of their terms is retained,
e.g. if a curved line is after all not considered straight, then
non-Euclidean theorems are found part of Euclidean ones. In
that case the assumed conditions considered in the older
geometry are not contradicted, but neither are its in the new
ones intended to be denied postulates, for the very reason that
the new geometries become part of the older. There appears
indeed another paradox, wherein, through again conflicting
use of language, non-Euclidean geometry is at once in discord
s
134
and concord with Euclidean.
A more general error is to suppose that if upon appropriately
interpreting the language of a deductive system its propositions
are consistent then so is the system. As observed, initial terms
are often left undefined, and the validity of the system is
tested by thus defining them. The terms could be defined in
contradictory ways, however. The validity would hence be
guaranteed only for a qualifying interpretation, not for the
system.
That the validity of a deductive science be held dependent
on its consistency can be attributed to the longstanding notion
that those statements are deductively true whose denial is a
contradiction. But a denial of any fact, including worldly ones,
is merely false, not a contradiction, which has to do with
opposition in statements, and a fact, e.g. a logical one, still
must be established to determine its denial false. The supposed
contradiction in the denial of a deduction is further confounded
with its consistency whereby no, internal, contradiction is
derived through it. Contradiction is of course one of the things
barred by logical law, but there are other requisites. A
conclusion may not follow for other reasons than that it harbors
a contradiction. For instance in mentioned affirmation of the
consequent (p.77 third par.) it does not follow that if A implies
6 then B implies A, but although such an inference would be
invalid, it would not be a contradiction. I.e. mere consistency,
noncontradiction, is not the test of validity. Contradictory results
were in paradoxes in fact seen possible through correct
inference, their cause lying in some premise. Accordingly
consistency rather than being the test of validity of a deduc¬
tive science, is a test of, possibly nondeductive, consistent
premises from which conclusions are through the science
derived.
Furthermore emphasis on the, rightly or wrongly found,
validity of a system of e.g. geometry, to the dismissal of its
particular content and accordingly the truth or falsity of its
postulates, merely upholds deductive ways, logical principles,
not geometric or other more particular truths. Since the last are
in those cases the interest, the postulates should be known true,
so that what is inferred be also, and to make the postulates and
the rest certain, knowledge of the meaning of the language
was observed required. It was also pointed out that both,
defining the words and ascertaining the basic theorems is
possible. Some of this was done in the preceding regarding
geometry, and more general, logical, factors admitting of the
same were part.
Section 2
LAWS OF THOUGHT OR
OF COMPLEMENTS
The three so-called laws of thought, taken into account
severally in what has preceded, are traditionally regarded as
fundamental to all logic. They have in recent times been often
criticized, sometimes justifiably, but have nonetheless been
adhered to, since inseparable from reasoning.
They have been believed indemonstrable, and in accordance
with what was said, they are to be here demonstrated, insofar
as valid principles. Their formulation is frequent in terms of
propositions, such as
LAW OF CONTRADICTION. A proposition is not both true and
false.
LAW OF EXCLUDED MIDDLE. A proposition is either true or
false.
LAW OF IDENTITY. A proposition implies itself.
Propositions, especially as associated with statements, were
observed herein as doubtful objects of logical investigation. A
statement may not inform of anything, contrary to the expected.
Or, by speaking of a proposition as true or false, no distinction
is made between it as a whole and what is predicated of the
subject, a distinction that will matter. But even if what is referred
to is clarified, by speaking of propositions or statements the
inquiry is distracted from its object. Since it is their truth or
falsity that is of interest, that is to say broadly facts or realities,
realities in question can be referred to directly.
Standard logic does make a distinction between whole prop¬
ositions and what is predicated of things in them, there being
a distinction made between propositional and predicate logics.
A similar distinction will be made at present, but in con¬
formance with the preceding it will be about fact or reality, or
the absence of the same. As regards predicates, it correspond¬
ingly will be about facts of particular attributes or lack of them
of things, and as regards propositions, of pertinent existence or
nonexistence of things altogether. The truth or falsity of a prop¬
osition as a whole, concerning a state, is thus an instance of the
existence or non-existence altogether of a thing, the state, as
adumbrated earlier (p.45 third par.). The appropriate study
about existence can accordingly be named existential logic,
and the like study about attributes attributive logic.
This division can begin with the laws of thought, and related
mathematical principles can be adjoined. The adjoining is
fitting here because these principles are equally about basic
s
1
136
complements, and there are comparable analogies later.
The complemental character is not present in the law of
identity, which, however, need not be considered a law,
because it consists in obvious redundancy. It is given also, more
in keeping with the emphasis herein on things instead of
propositions, in a form like
A equals A,
noted early (p. 17 second par.) to be found uninformative, while
held on to as a law. Whether expressed through implication,
when the inferred may be anything associated within given
bounds with the inferred from, or through a form in which the
inferred is something the inferred from is, the contended law
repeats that, granted that what is considered is a certain thing,
it is the thing.
It has been maintained that the same takes place in all of
logic and mathematics, that all of it consists of tautologies, as
mentioned in the introduction {p.2 first par.). That it does not
was explicated there (p. 2 fifth par.) and is recognized in finding
“A equals B,” where things though logically equal are not
spoken of alike, informative, as, with the opposite on above
displayed ‘A equals A” likewise noted early. As mentioned
also, it has been argued that deductive principles are true by
virtue of definition, and that they are tautologies for that
reason. The view is aided by the almost sole reliance for proof
of logical theorems on mentioned truth tables. The tables do in
fact depend on definitions and can be said to for the most part
repeat them. As indicated, however (p.l08 fifth through seventh
pars.), they will be seen not to be relied on. Of relevance in
point of fact is not whether a principle is true by virtue of
definition, but whether it is an inference, deriving, perhaps
merely as a name, something not posited before, unlike done
in the law of identity, circularly repeating the premised in the
concluded.
In discourse all things are substantiated by virtue of, beside
else, definition, meaning, because the things are substantiated
of whatever is meant by the words used, hence relying on the
definitions. But once it is known what is meant, how it is
defined, anything else known or established about it is not of
that definition. Furthermore, even if the premises, things
known, are true as a matter of definition, conclusions can be
drawn not previously recognized. Mathematical learning in
general can, in fact, be said to result from definitions.
As an example can serve a proof that 2 + 2 equals 4. Proof
of such basic equalities—ones involving single digits and that
are, as in the process of addition, assumed true—has been
evading detection, in further instance of how hidden premises,
here definitions again, can prevent solutions. At present the first
137
definition is
i. 2 equals 1 + 1.
Hence
ii. 2 + 2 equals 2+1 +1.
The next definition is
iii. 3 equals 2+1.
Hence and by ii
iv. 2 + 2 equals 3+1.
And the last definition is
V. 4 equals 3+1.
Therefore and by iv
vi. 2 + 2 equals 4.
The proof also conforms to later equality principles, and what
matters now is that although the basic premises here are
definitions, the conclusion, with the other inferences, supplies
new information. In this case the information was likely
acquired already, although one may be hard pressed to say
how, but the same does not, of course, hold for more complex
equations. What happens is that when considering something,
one may not be conscious of all that one, perhaps by definition,
knows about it. When considering 2 + 2, one may not be
conscious of definition i, only after recalling it to infer ii, and
when considering ii, one may not be conscious of definition iii,
to infer iv, and so forth. This act of inference illustrates men¬
tioned (p. 18 first par.) drawing upon two senses of meaning,
one as attributes in consciousness, and one as attributes that
define. And while in this process what is known and deduced
seem, as in mathematics, to be mainly diverse wordings for the
same things, other things known to be true under those names
can be connected. By the above example two children can
divide four pieces of sweets equally. That the mathematical
deductions only inform of other wordings is accordingly
likewise not the case. That 2 + 2 equals 4 informs that one
amount of 2, however named, added to another of the same
equals an amount of differently known 4, however named.
Ironically, while it is maintained that logical and mathe¬
matical principles are true by definition, they are at times
unduly proclaimed not to be. Whereas above definitions of
particular numbers might not be disputed, it is denied that
certain axioms accepted about in general, natural, numbers
define them. The axioms state
1. 0 is a number,
2. The successor of any number is a number,
3. No two numbers have the same successor,
and
4. 0 is not the successor of any number.
All numbers but 0 are also, it was failed to be stated, meant
s
138
to be successors.
Granting that 0 is a number, a definition in consonance with
common understanding can, however, be that
a number is, on starting with 0, any quantity counted by
successively adding a new unit.
Should 0 not be conceded, the starting number can be 1.
Conjointly with the definition numbers are thus said by axiom
1 to include 0, by 2 to include any added unit, by 3 to be new
in counting, and by 4 to start with 0. The forgotten axiom, that
all numbers but 0 are successors, is in the definition, since
definitions are identities.
The axioms are thought not to implicitly define number
because, in likeness to non-Euclidean geometry (ca. p. 133),
they are held to admit of other interpretation. They can by
changing meanings of words in them, it is found, apply to other
things. To repeat, however, by redefining words one does not
speak of the same things, even if the changed meanings, as in
the present case, are merely broader ones. Then spoken of are
the things thus referred to, not specifically the others, as in the
above definition, holding of all and only what is here meant by
numbers.
In additional irony, a last axiom, though unknown to most
who think with numbers, has been held to define them. Known
as the axiom of mathematical induction, it states
5. If 0 has a property, and the successor of any number that
has it also has it, then all numbers have it.
The axiom can as a matter of fact be simply deduced fromthe others, viz from a usual meaning of number. By it and the
if clauses of the axiom, since 0 has the property, 1 has it, hence
2 has it, and so forth for all numbers.
As expounded early herein, something can be technically
defined by anything true of, and only of, it. For instance the last
above axiom might define number if it meets those conditions.
However, aside from the axiom having to be confirmed, that is
it could not be used as definition until proven, of interest in
general is, rather than all that may define something, how it is
initially conceived. This initial character of the concept is
normally meant to define the thing, with other attributes
findings. And the question at this time is how are the findings
made when intrinsic, deductive.
And as indicated at the beginning of this chapter, deductive
findings to be made, if certainty be attained, are requisite in
places unsuspected. To wit, whenever one thing is said to
follow from another, one is justified to ask for the reason. This
does not obtain, naturally, if the same thing is in consciousness
in each case, even though put in different language. Then the
same thing is, circularly, expressed, rather than something yet
139
to become of awareness. But otherwise, even if the deduced,
the new in awareness, is a name, the question is why the
deduction.
A bit more on the meaning of consciousness and awareness.
The words are presently used somewhat freely, their intended
sense doubtless evident from the context. Earlier in a strict
sense things were meant to be sometimes of consciousness but
not awareness. They are the things that though part of con¬
ceptions, need be inferred. When now speaking of things
already in consciousness, meant are evidently things in con¬
scious awareness, in thought and not needing to be deduced.
What matters at this point is that when it is said that one thing
follows from another, there should be a reason why the other,
differing from the first, can be inferred, in agreement with the
need for finding that something is the case (e.g. p.26 last par.).
And it will not do to state a further, likewise differing, ground.
There should be a reason why the conclusion follows from this
premise, and if a further one is offered, there should be a
reason for the inference from it, and so on. A like need was
recognized by the 19th-century author Lewis Carroll, and it is in
accordance with it believed that any statement that, as may a
logical principle, licenses a conclusion cannot be considered a
premise. But what a premise is is again of choice of definition,
and if the principle have another name, the need to see why
the conclusion follows remains.
This need appears as mentioned strange as regards the very
inference of instances from logical or other principles. In accord
with the above, at issue are principles regarding which, as
loosely the definition’^3 + 1 equals 4″ instances of the subject,
e.g. broadly “2 + 2”, are not in consciousness, and therefore a
reason is wanting for the deduction, e.g. “2 + 2 equals 4”. And
the situation in practice always applies, because the instances,
far from merely not in thought as part of the principles, appeal
to them for the answers.
Accordingly in seeing what deduction consists in, it is not
enough to determine the basic principles, all else decided by
complying with them. There might be forms of deduction
holding only for things more particular than the general ones
of logic, or mathematics, but, especially, the principles do not
take care of, as seen, the preponderant deductions from the
general, including those principles, to the particular, the
procedure by which deduction was noted distinguished often
from induction, inference from the particular to the general.
To in keeping with previous observations accordingly say that
all things deduced must be appropriately perceived to follow
is to oddly equally assert a principle. A reason can still be
required why as a result a deduction must be so perceived to
140
\
follow in a particular case. Fortunately or not, ascertained
deductions of cases from principles are constantly made
regardlessly, and they are founded on the same kind of to be
demonstrated conceptualization as are deductive principles.
The demonstration would be awkward if the reader were
asked to envision the concepts concerned, so as to mentally
observe the results. It is advisable therefore to put the concepts
in physical form, in which as diagrams similar substantiations
were submitted by men in the past.
Largely known as Euler diagrams, after the fecund 18thcentury mathematician, the forms were used to verify Aristotle’s
syllogisms, but, misapplied and seemingly confined in use,
they gave little assurance. They are as much as looked upon
with disdain, for which both rationalism and empiricism may
offer cause. The one because the use of diagrams seems to
diminish the prowess of reason, somehow thought to function
without conceptualization, and the other because they are at
odds with the conviction that the only principles not tautologies
are laws of nature. They are thus used at best for confirming
some theorems in a schoolboy environment, without recogni¬
tion that confirmation is tantamount to proof, as noted in the
introduction. In consequence of these, deductive truths remain
undemonstrated.
In keeping with the expounded, in order to demonstrate
them they must be observed with regard to the concepts
concerned, and in physical proof the simplest form, as offered
by diagrams, is welcome. They are hence made use of here to
that end, and it may only be added that they are representa¬
tions, with the rest supplied by the imagination.
How an instance is deduced from a principle, a process
explained to require the same perceptual validation as do
principles, is represented in Figure III. In it all things 6 (small
circle) belong to things C (large circle). Thus, being a C can be
viewed as a principle regarding every 6, since a principle is
meant to be about all of something (B) as a thing (C) of which
something is true. And A (dot) is a 6. Therefore, as disclosed by
the figure. A, also, is a C, a thing of which what is at issue is
FIGURE III
141
true. It is easily visualized that the like obtains in the case of
identities of A with B, or B with C, as exemplified by the earlier
numerical equations.
The peculiarity that deduction from a rule calls for the same
ascertainment the rule does has bearing on the mentioned
syllogism ‘”All men are mortal; Socrates is a man; therefore
Socrates is mortal”. This inference of an instance, the con¬
clusion, from a rule, the first premise, is in its entirety often
furnished as an instance of a syllogistic rule prescribing the
inference. And as should be clear now, applying the syllogistic
rule calls for the same kind of mental act as applying the rule
stated by the first premise.
The former is in fact much more complex. Instead of the “B
is C” that can symbolize that premise as a principle, the general
syllogism as a principle is symbolized by “If A is 6, and 6 is C,
then A is C”. In the simpler case, in accord with Figure III,
Socrates (A) need by being a man (6) merely be perceived as
mortal (C) without more specifics. But as to the logical
principle, parts of the two Socrates premises (the two now the
figure’s A) must first be perceived as cases of the principle’s A,
B and C in the two assumed relations (the two the figure’s B),
thereafter perceiving that another such relation is implied of
certain of these cases (the implied belonging to the figure’s C),
whereupon that relation is concluded to hold. The added
conclusion, saying that Socrates is mortal, is needed because,
unlike before, it is separate from in the figure the connection
of A with C If it is yet considered that the logical law yielding
an instance by such complication is a rather ordinary one, it will
be appreciated that, as indicated, proof of basic logical laws
can be simpler than proof of instances from them.
With these observations the two laws of thought, those not of
discussed redundancy, can again be turned to. But their from
the onset misleading titles as of contradiction and excluded
middle are still less suitable presently.
The first law is regarded with some justification as mis¬
named, because it, rather than requiring contradiction,
excludes it. The second law is also known as of excluded third
and contrariwise requires, instead of what it says, inclusion. Its
generalized meaning is that
At least one, A or not-A.
It does not exclude the third possibility that both be the case,
done by the other law by asserting that
Not both, A and not-A.
In addition, because the two laws, mentioned as of basic
complements, will be formulated here for, beside the described
existential and attributive logics, mathematics, they will be
about more than two alternatives and their contradiction. But
they retain the concepts of inclusion and exclusion and con in
their several versions therefore be called laws of inclusion and
laws of exclusion.
Taking the traditional laws of thought for existential logic
then, the law of exclusion is
THEOREM lll.i.i. A thing does not both exist and not exist at
the same time.
Proof. In Figure lll.l the circle stands for the times a thing A
exists, the existence signified by the letter alone. All other area
stands for other times, not those when A exists, signified by ‘(A).
The meaning of ‘”other area” and correspondingly “other times”
and “not those” can be held ostensive (p.l02 third par.) in the
figure. And by times when A does as straight negation of
existence not exist, signified by A, are meant not those when
it exists. As disclosed by the figure, therefore, not both A and A
are the case at the same time, proving the theorem.
The distinction between A and ‘(A), which may be read
“non-A” and “not A”, is made because, as was suggested (p.135
third par.), what is denied of something directly, as in “when
A does not exist”, need not be what is denied indirectly, as in
“not when A exists”. Parentheses are familiar as demarcators,
and the apostrophy, often used for omission, offers a compact
negation sign placed, as in speech, before its object. It should
also be observed that “when” and “time”, as well as “exis¬
tence”, are used here extremely broadly. The when or time
need not be temporal only, but can also pertain to a particular
place; in that case the theorem means that a thing cannot occur
and not occur at the same time and place in usually nature, the
figure then depicting places at one time. Differently and more
generally, the meaning of place can be extended from part of
a momentary perception to one of a continuity of them, as
understood to occur in time and expressed when saying that
something takes place; the theorem means accordingly that a
thing cannot be and not be of the same place in perception, be
the place part of a momentary perception or a continuity of
them, the figure then depicting places thus extended into
FIGURE lll.l
143
perceptions through time. Existence con consequently likewise
be either about presence in the world sometime perceived, or
presence anywhere in consciousness, in which case A can for
instance mean the existence of a logical principle, its place
everywhere, and A the nonexistence, its place nowhere, or the
opposite. In short, concerned is any sort of being, in any
territory.
It has been said that the dividing line between affirmation
and denial is not as definite, that sometimes both may apply,
that there may be an intermediate area where they overlap.
Considering not only existence but anything that may be true
about a thing, it might be said that a zebra is both white and
not white. What delimits a given thing is, however, again a
matter of definition, meaning. For instance something may be
meant to be white if more than 50% of its surface is, otherwise
not, and it cannot then be both. A meaning of yes and no by
which no is not the complete absence of yes, as would black
stripes not be absent on a zebra, is not one here considered. By
the present meaning if it is too difficult to know that e.g. a zebra
is over 50% white, then whether it accordingly is white or not
is unknown, but only one alternative is true.
The law of inclusion which is companion to the preceding is
THEOREM lll.l. ii. A thing either exists or does not exist at any
time.
Proof. As disclosed in Figure lll.l, either A or A is the case at
any time, proving the theorem.
It should apart from again the preceding broad application
of the diagram be noted that, in keeping with the remark (p.l40
fourth par.) that the diagrams only represent, the line
describing the circle is of course meant to lack breadth, one that
might be taken as a place of neither A nor A. Any place, even
a limit if wished, other than of A is, to repeat, meant to be of A.
The preceding two theorems can also be expressed as certain
implications between the states of A. As will be seen, for
logical purposes all propositions can be put in a comparable
form, as indicated by past observations that all states concern
something associated with something else.
The general forms, not necessarily speaking of A with A,
used above for the theorems would be “Not both, A and B” and
“At least one, A or 6” similarly to the laws as displayed gen¬
eralized before (p.l41). These forms are held to be respectively
equivalent to “A implies not-B, and 6 implies not-A” and “Not-A
implies 6, and not-B implies A”, and they can be seen to be so
by their meaning. For the concept behind “not both” can be
held to be ‘To find one is not to find the other” and the concept
behind “or” “Not to find one is to find the other”. Correspond¬
ingly the theorems can have the form
144
‘A /s equivalent to ‘(A)
and
A is equivalent to ‘(A).
Equ ivalence is regarded as the same as mutual implication,
which sameness will by a certain meaning be found an
inference.
The implications from left to right represent above Theorem
lll.l.i, and those from right to left Theorem lll.l.ii. The first
formula was, further, seen as a definition, with only the second
an inference, obtained, as suggested in the preceeding, by
upon viewing, finding. A, not finding A, and upon not finding
A, finding A. That this formula is not somehow likewise a
conscious part of the definition, a part of it at all in fact, is, more
than in previous absence of some part from consciousness
(p.l37 second par.), evidenced by the mere lack of reference in
the definition to an area other than of A and ‘(A). Seeing the
place of A to be that of ‘(A), it is not seen, obvious though it
may be, that the place of A is that of ‘(A).
The second formula is known as the law of double negation
mentioned (p.32), without the distinction here made by
parentheses. Since the two kinds of negation are at present
equivalent, the parentheses could indeed be omitted without
change in validity. In consonance with former observation (e.g.
p.l42 last par.), however, the same is not the case in the other,
attributive, logic.
Adapting the laws of thought to it, the law of exclusion is
THEOREM III.2. i. A thing does not both have and not have a
given attribute at the same time.
Proof. In Figure III.2 of a given time the small circle stands for
the things that are A, or simply every A; all other area of the
large circle stands for every non-A, symbolized A; all area
outside A stands for things other than A, for everything not A,
symbolized ‘(A); and the large circle stands for every 6, thus
comprising, whether of a further.attribute or not, every A and
A. An attribute A can accordingly be meant to apply to only
certain things B, and its negation. A, to only the rest of those
FIGURE III.2
145
things. This was noted of, among other things, even and odd,
applying only to numbers. By ‘A is nonetheless meant o thing
other than A. By the figure, therefore, o thing is not both A and
A at the same time.
The letters used for attributes or their absence are the same
as those used for things existing or not because, as was
suggested (p.l05 second par.), the concepts will in the two
logics be found easily merged. Since a thing of attribute A can
be spoken of as an A, the existence of the same A can be
spoken of as well. That it or its negation is spoken of is likely
understood by the context, indicated to {p.l43 sixth par.) mainly
involve association of one thing with another, and specifically
implication in one logic and attribution in the other. It ought to
be further said that, similarly to the existential case, ^’thing”
and “”attribute” are used extremely broadly here, so as to make
time again diversely depicted in the figure. The theorem was
given as about a time of a thing since, like existence, most
things are thought of as worldly, with a discussed continued
identity through time. But some things can again merely belong
to consciousness, and as perceptions of only a certain time, they
need not be specified as of a time by the theorem; whenever
a perception occurs, that and only that will be its time, and the
figure can correspondingly depict all perceptions regardless of
time, with 6 perhaps visual qualities, and A blue ones. What is
more, a thing can be a certain time, or length of it, itself,
making the mention of time in the theorem once more
unneeded; in that case the figure again depicts all perceptions,
along with those of, as on a clock, times, perhaps as durations
signified by 6, and seconds by A. Concerned are things of any
attribute at any of their time.
To the last theorem the, qualified, companion law of inclu¬
sion is
THEOREM lll.2.ii. A thing either has or does not have a given
attribute at any time if and only if it belongs to things
comprising what has and does not have the attribute.
Proof. As confirmed by Figure III.2, a thing is either A or A at
any time if and only if it belongs to things 6 comprising A and
A.
As seen in the figure, the principle applies, together with the
one preceding, without the qualification if A is replaced by
‘(A). In ordinary language, the ‘fA) counterpart of Theorem
111.2. i can then state that A thing is not both one that has and
not one that has a given attribute at the same time, and the
counterpart of Theorem lll.2.ii that A thing is either one that has
or not one that has a given attribute at any time.
As before the two theorems can be expressed in a form like
implication, specifically so that a thing can be held to imply an
3
s
146
attribute. The simple forms used previously (p.144 displays),
ho\A/ever, ore now one-directionol, suited only for the first
theorem.
‘A is ‘(A)
and
A /s ‘(A).
The attributions do because of the qualifications evidently not
hold in the opposite direction, for the other theorem.
As with the previous theorems, the first formula is true os o
matter of, partial, definition, only the second inferred. That
formula, as indicated, would change in validity without the
parentheses, to in that event in truth apply in both directions.
The negation would be the limited one, which indeed implies
A as the alternative. It would seem that the full form can hence
be used for both theorems after all, with the first formula
redundant without the parentheses. But as suggested by the
redundancy and the discussed controversy regarding the law of
excluded middle, the usual interest is in whether if either, a
certain attribute or its negation, in a limited sense, does not, in
a limitless sense, hold, the other does.
Appropriate formulas combined would be
A equals (‘(A) & B)
and
A equals (‘(A) & B)
if and only if
(A & A) equals B,
with division, as before, into definition and inference. Equality,
or identity, will be seen to be the same as attribute in both
directions, as in the first chapter observed regarding definition
(p. 41 last par.).
For the above counterparts of the two theorems, with ‘{A) in
place of A, the formulas, no longer qualified, return to simple
form. Here, too, the first one, that ‘(A) equals ‘(A), is redundant,
and the second would state that A equals ‘(‘(A)).
Adapting the two laws to mathematics, the one of exclu¬
sion is
THEOREM Ill.S.i. A quantity is not more than one of the
following, a given one, a smaller one, or a larger one.
Proof. In Figure III.3 circle Q stands for a given quantity; circle
S and all other area less than Q stand for smaller quantities;
and circle L and all other area more than Q stand for larger
quantities. The asymmetry indicates that quantities apply to
specific things, extending from a (left) point in a certain (right)
direction. The theorem is at all events seen in the figure to hold.
The law of inclusion is for mathematics
THEOREM III.3. ii. A quantity is either a given one, a smaller
one, or a larger one.
147
Proof. The theorem is substantiated by same Figure III.3.
It should not be required to list the implicational forms of
these theorems. It may merely be remarked that they take the
form “If one then not another, and if not any of two then the
third
More notably, the theorems can, on the basis anew of
underlying meaning, be deduced by logical principle.
That one quantity is less or smaller than another can be held
to mean that while all of it belongs to the other, some amount
outside it also does. And that one quantity is more or larger
than another can be held to mean that not all of it belongs to
the other, nor does some outside it. That in the meaning of
“less” further, some amount not of that quantity belongs to the
other is logically equivalent to not all of the other belonging to
the first. And in the meaning of “more” the denial that some
amount not of that quantity belongs to the other is equivalent
to all of the other belonging to the first. Thus with quantity A
less than quantity 6, and quantity C more, all A is 6, not all 6
is A, all 6 is C, and not all C is 6. And if two quantities are equal
then by the above alluded to (facing p. third par.) equality
principle all of either is of the other, or, by transposition, if not
all of either is of the other then the two are not equal. Hence
A does not equal B, 6 does not equal C, and since all A but not
all C is 6, A, for fear of contradiction, does not equal C. If
something A equals C then by referred to (p. 33 second par.)
later principle not all of it is 6 either. With none equaling
another, therefore, if a quantity equals either one, it does not,
by same noncontradiction, equal another, in compliance with
above Theorem lll.3.i.
To proceed in correlation with the preceding, if one quantity
does not equal another, that is if it is not true that all of each is
of the other, then by logical law not all of at least one quantity
is of the other. This can in fact be true of one of the quantities
only, because by “not all” is here meant an amount additional
to the other, meaning at once that the other, all of it, is part of
the first. Then if a quantity does not equal a given one then
either, all of it is of the other and not all of the other is of it, or
FIGURE III.3
148
\
the opposite. But these alternatives were seen to make the
quantity either less or more than the other. That is to say, in
compliance with Theorem ill. 3.ii, a quantity is either equal to
a given one, or less, or more.
The last two theorems are manifestly easier to establish by
means of a diagram, should even that much be needed, rather
than by in the preceding manner inferring them via others. That
they and others are logically deducible should nonetheless
prove of interest.
Among other theorems are further ones that can be regarded
as of complements, this time concerning states usually
expressed by means of two variables, e.g. A and 6, signifying
things identified by different attributes. The theorems are akin
to the familiar square of opposition in syllogistic, which deals
with logic here called attributive. But they also have an
existential form and can apply to mathematics, although
having to do with attribute for the most part. In its connection
there is dispute over the meaning of the designations “a\\” and
”some” regarding a subject, the validity of the principles
accordingly questioned, and a related issue extends into the
other logic. It should at this moment suffice to repeat that what
is meant by words is a matter of choice. Addressing the mean¬
ings at issue afterward, the principles will now be indicated as
pertains to attribute. For exclusion there is, when there is only
one A,
THEOREM lll.4.i. Not both, A is B, and A is ‘(B),
and with more A
THEOREM lll.4.ii. Not more than one of the following, all A
/s B, some A is B and some is ‘(B), and all A is ‘(B).
Both theorems also apply to ‘6 (“non-B”), as previously (p.l44
last par.) distinguished from ‘(BJ (“not B”). The same is not the
case with inclusion, in, treating separately the single and
multiple A again,
TntoREM III.4.iii. Either A is B, or A is ‘(B),
and
THEOREM III.4. iv. Either all A is B, some A is B and some is
‘(B), or all A is ‘(B).
As elsewhere, these two theorems apply to ‘B if and only if
A belongs to things comprising 6 and ‘6. The simpler version is
chosen because in this form the theorems represent the other
logic as well, the qualified version easily enough supplied
mentally. It should nevertheless be noted that, here and sub¬
sequently, principles often thought valid may again not be
because of the limited range of a negation.
Postponing proof of the preceding theorems, it may first be
noted that they, too, are of truths commonly apprehended, as
in an awareness of the trichotomy of “all”, “some”, and “none”.
149
Instead of the trichotomy, however, logic usually states that, for
exclusion.
Not both, all A is B, and some A is ‘B,
and
Not both, all A is ‘B, and some A is B,
and for inclusion.
Either all A is B, or some A is ‘B,
and
Either all A is ‘B, or some A is B.
In conformance with the aforesaid, the last two formulas
must really be qualified, applying instead to ‘(BJ. Above
Theorem lll.4.ii, analogous to the present other two formulas,
is ironically at the same time rejected.
It is specifically proposed that all A can be both 6 and ‘B. In
support is cited the like of the referred to (p.l43 last par.)
meaning of ‘”Not both, A and B” as “To find A is not to find 6”
and the opposite. In the present case the partial meaning, of
similar “A thing is not both A and B” and expressed in the last
above displayed line as the negation of “Some A is B”, would
be “All A is “6”. It is then argued that if nothing is A then each
“A thing is not both A and 6” and “A thing is not both A and ‘6”
is true, and that hence by the preceding meaning all A is
respectively both ‘B and 6.
It may be remarked that here, too, the limited range of ‘B is
not taken account of. The denial of “A thing is not both A and
6″ would by that interpretation not mean that all A is ‘6, but
only ‘(BJ. Likewise for changing from denial of “A thing is not
both A and ‘6″ to 6. But more pertinent is that “All A is 6 ” is
commonly not meant to be true if nothing is A. If if nothing is
A then “A thing is not both A and not-B” is meant true, then the
last does not mean the same as “All A is 6”. Neither does “A
thing is either not-A or 6″ if, as maintained also, true if nothing
is A.
The present meanings of “not both” and “either, . . or” are,
as to be elucidated, ones dependent on a connection between
the two things at issue, not on the existential status of either, the
meanings otherwise not now of concern. A like situation
obtains with “Some A is 6”. In contrast to the other case, it is
said to mean that something is both A and 6, and hence that
something is A. As a result it is also held false that either some
A is 6 or some is not-B, alternatives likewise ordinarily
acknowledged and here implicit in Theorem III.4.iv (facing p.).
If nothing is A, it is considered, then by that meaning neither
is true. But as before, the meaning at issue of “Some A is 6” has
to do with a connection between A and 6, irrespectively of their
existential status.
As did the laws of thought pertain to any imaginable thing or
s
150
existence, so do in logical principles here statements about all,
some or other of their subjects pertain in them to the same
unlimited spectrum. The truth of the statements, as in the
variety of them in everyday life, is not dependent on some form
of existence or nonexistence of their subjects, in what is called
existential import, but has to do with whatever may be known
of something of whatever nature. A statement can also declare
that a certain thing has worldly existence, or that another has
not even a conceptual one. In the first instance the subject, as
observed earlier (p.39 fourth par.), is of a still only conceptual
status, and in the second instance of none at all, represented
only linguistically. Mentioned argument advanced that
existence is not an attribute (p.38 fifth par.), or that its assertion
is a tautology, with a subject held at least in singular
propositions to exist by its mention alone, are of no avail. What
an attribute is is, again, a matter of definition, and it remains
that existence is affirmed or denied of things, with under¬
standing of what is said, and without previous knowledge of
the facts.
In the light of these remarks the contention that universal
propositions, with ‘”all” are true if the subjects do not exist can
run into contradiction. A statement like “All these assumed
situations exist” would be true if they did not. To assert that
singular propositions, with a single subject, presume its
existence is likewise inconsistent with the arguments on
universal ones. When there is no more than one A it can by “A
is 6″ be equally meant “A thing is not both A and not-B” and
that this is true if A does not exist. Both that existence is not a
predicate and that single subjects of statements are assumed to
exist are, by holding that to speak of an individual’s existence
is somehow senseless, propounded in defense of, as to be
brought forth in the last chapter, arguments against particular
existences, serving to illustrate how premises can be strained
to accommodate an aimed at conclusion, not rather seeing
what findings can be made. As observed earlier (p.25 last par.),
existences, particular or general, are concerned in all searches
for fact, statements made accordingly. Of interest here is what
latitudes indeed can be present in statements, or states, as logic
applies to them.
In the recourse to the form “A thing is not both A and not-B”
for “All A is 6” its truth is clearly thought also decided if nothing
is not-B, namely if everything is 6. That all A is 6 does sound
more reasonable if everything is 6 than if nothing is A. But if the
translation of that form is the referred to opposite (last p. third
par.) which now is “All not-B is not-A”, then its truth if everything
is 6 seems odd again. With less attention to that form, it may
be observed that as “All A is 6” is here and ordinarily not meant
151
to follow if nothing is A, it is also meant unrevealing if every¬
thing is 6. And this is indicative of the alluded to connection
intended between A, the subject, and 6, the thing connected
with it, in any statement.
The intent is to assert about the subject something not the
case with regard to everything else as well, within whichever
area concerned. An example was furnished when seeking to
find laws of nature beside causal ones, in seeing it of no
interest that something in nature is accompanied by the
existence of matter, which accompanies all things (p.93 fifth
par. through p.94 second par.). Such an absence of a revealing
connection is in current logic part of, beside above predication,
implication.
The now treated principles were mentioned adaptable to
existential logic, and the connections regarding A and 6 in it
would as before be of implications. As suggested (p.l36 second
par.) when discussing the law of identity, implication is herein
distinguished from attribution by concerning something, 6,
associated with the subject. A, within given bounds, whereas
in attribution 6 is something A itself is. Attribution can thus be
considered a case of implication, when A is one with the
associated 6. And implication, in likeness to attribution, is held
present if A is false or if 6 is true.
Truth and falsity is spoken of in place of existence or non¬
existence because, as explained, such as implication has been
viewed as holding between propositions, although, while
really holding between their truth, they therein hold between
actual states, which are but some of the things that can be
associated. “A implies B” was in any event noted (p.l43 last
par.) equivalent to ”Not both A and not-B”, regarded as true in
the absence of A or presence of 6. This is, however, not the
customary understanding of implication, which, as held in logic
also, frequently connotes deducibility of 6 from A. There is no
need to go so far, since the principles about the connections
pertain not only to logical and even causal ones, but to any
within any chosen area. Nevertheless the connections are
meant to be revealing as observed, 6 to be associated with A
but not with all other things in that area also. How common this
meaning is is disclosed by considering that if one thing, e.g. in
nature, does relative to another merely coexist, which all things
in nature do, then there is held to be no connection between
the two.
With these observations in view. Figure III.4 (next p.) can
serve to validate the present theorems (p.l48) with regard to
either logic, and with as full a versatility as before. Whereas in
attribution the diagram, as in previous Figures III and III.2
(pp.140, 144), depicts an A as being a 6, in implication instead
152
the time or place of A con be looked upon as a time or place
of 6. Since a thing would be regarded as identical with another
if occurring at the very same time and place, though not if only
occurring at the same time or the same place, the large circle
can represent times or places of certain wider limits within
which 6 occurs, the small circle, the times or places of A, falling
within those limits. In a causal connection accordingly A, the
cause, is accompanied by 6, the effect, within a proximate
place or time. These proximities make the connections
revealing as explicated, since accompaniment anywhere and
anytime applies to everything and makes for no connection.
Should the connection depicted by the diagram be viewed as
causal, then the area for the figure would thus all represent
nature, with negations of events A and 6 as before outside their
respective circles. In that case the discoursed existence, in the
world, is indeed presumed of A and 6, or ‘(B) as alternative in
the theorems, causality concerning the discussed very coming
into existence of the effect upon the cause. This can be viewed
as justification here for the title existential logic, since in it
dominantly treated implication, much applying to causal
connections, is accordingly conspicuously of existence. But the
connections can, as observed, be of anything else, deductive
ones examples in having one concept entail another, and that
is why attribution can be considered to belong to them. The
whole area in the figure can hence represent again any specific
area of consciousness, from a particular place and time in the
world to perception in general. The question then is whether in
the limited or unlimited area when, for instance, all A in it is 6,
can some not be.
Without the dependence thus on existential factors, par¬
ticularly worldly ones, with only the question whether 6 is
associated with A in its area, which can be of the world as well.
Theorems lll.4.i through lll.4.iv can be seen true by means of
Figure III.4, For the first one of the theorems compared to the
second it will be recalled (p.l43 second par.) that by the
negation of something, B, of a, single, thing. A, is meant
complete absence of its affirmation. The reader will also find
FIGURE III.4
153
it easy to confirm that the lost two theorems hold for ‘6 if and
only if A (e.g. o prime number) belongs to things (e.g. whole
numbers) that, perhaps given by o third circle C, comprise 6 and
‘B (e.g. even and odd numbers).
It may be observed that in syllogistic the three alternatives for
all A in relation to 6 or ‘(B) are viewed by way of as many as
seven diagrams. “‘All A is 6” is, along with the diagram of figure
III.4, depicted by a single circle with A equaling 6; “All A is ‘(B)”
is depicted by discrete circles for A and 6, and, when A and 6
exhaust the universe of discourse, by a circle for it of two parts,
one for A, and one for 6; and “”Some A is 6 and some is ‘(B)”
is depicted by two circles for A and 6 partly overlapping each
other, by a small circle for 6 in a large one for A, and, when A
and 6 exhaust the universe, by a circle for it of three parts, one
for A, one for 6, and one for both. The exhausting of the
universe is added to provide for all connections of “”all”” and
“”some”” between A and 6, including their negations, the
connection “”All ‘(A) is B” being thus pictured. Even if so many
diagrams are used to verify a theorem, however, there is no
hindrance to having circles stand for negations, so that three
kinds of diagrams will suffice, without the complication of
containing the universe in some cases within a shape.
The discrete circles for “”All A is ‘(B)” can be replaced by a
small one in a large one, the succeeding divided circle, for the
additional “”All ‘(A) is 6″”, by a circle alone, and the last, twice
divided, circle by also a small one in a large one. There could
yet be a circle picturing, as an instance, 6 as exhausting the
universe, as would be thing, although the connections would
not be revealing as explained.
But regardless of these, not only do the multiple diagrams fail
to aid a principle”s perception, to rather obstruct it, but that they
represent all the alternatives itself requires proof. These
alternatives, as is not too difficult to ascertain, are in their
various forms in fact established by the theorems presently at
issue. That these theorems hold does, further, not require more
than, with the aid of Figure III.4, viewing under “”(All) A is B”
A as confined to 6, whether to a part or whole of it; under “”(All)
A is ‘(B)” comparably; and under “”Some A is 6 and some is
‘(B)” A as extending over both, 6, whether part or whole of it,
and ‘(B) comparably. These choices become in the added
diagrams perplexing to see, and they no doubt need moreover
not be enunciated to enable perception of the theorems from
the single diagram.
It should be equally simple to perceive from the same
diagram the similar displayed principles following them (p.l49),
as applying to ‘(B) and qualifiedly to ‘B, it being understood that
“”some”” can extend to “”all”” as “”possible”” was understood to
extend to “‘necessary” (p.37 second par.). These principles can
be of utility in logic, since by them when “all” applies, “some”
can at once be denied of the opposite, and when “some”
applies, the like can be denied of “ail”; and when “not all”
applies, “some” can of the opposite at once be affirmed, and
when “not some” applies, “all” can be affirmed likewise.
While viewing of all of these principles by means of diagram
offers no problem, they can in a lengthier manner be deduced
from others in more than one way.
Theorems 111.4. i and III.4. iii (p.l48), when there is a single A,
can be deduced straightforwardly from laws of thought. Since
A is a thing, and by those laws a thing is not both 6 and ‘(BJ and
is one of the two, the same is by transitivity true of A.
For another proof it may be considered that by denying that
A is 6 is meant that A is something not B, that it is ‘(B). For, the
conception in the denial is something regarding A as a thing,
namely that it is a thing other than 6. One could go even so far
as to cover by alternatives such as 6 and ‘(B) also mentioned
things not existing even conceptually, but only by name (p.l50
first par.). In that case A may be just such a thing, and by
denying that it is 6 it would be hence likewise meant that it is
something not 6, ‘(B). What counts is that 6 and ‘(B) are meant
to be inclusive in any sense, and by the preceding thus half of
each. Theorems 111.4. i and 111,4. iii, can as in laws of thought be
held true by definition, put in, again, implicational form as
(A is ‘(B)j is equivalent to ‘(A is BJ,
From left to right the implication is half of “Not both . . .”, and
from right to left half of “Either.. . or…”. The other halves are
stated by
(A /s BJ is equivalent to ‘(A is ‘(B)J.
And by the first displayed preceding formula the second of
the equivalent units of the second formula is equivalent to “A
is ‘(‘(B))” and hence, by laws of thought and transitivity, to “A
is 6″, the first unit of that formula, the formula therefore
likewise true.
From the first formula the second can, in a third proof, be in
fact deduced directly by transposition, by which if one thing is
equivalent to the negation of another, then the affirmation of
the other is equivalent to the negation of the first.
For Theorems III.4. ii and III.4. iv (p.l48), and the similar four
displayed principles succeeding them (p.l49), all on more than
one A, proof in a manner like the preceding second kind
should suffice. It can here, too, first be considered that by
denying that all A is 6 is meant that some A is ‘(B), and by
denying that some A is 6 is meant that all A is ‘(B). For, the
conception in the first denial, unlike in “All A are things other
than 6″, is only that some A is a thing other than 6, and the
155
conception in the second denial, unlike in only ^’Some A is a
thing other than B”, is that all A are things other than 6.
Half of the last mentioned four principles can thus, with ‘(B)
in place of ‘6, again be held true by definition, put in
implicational form as
(All A is ‘(By is equivalent to ‘(Some A is B),
and
(Some A is ‘(By /s equivalent to ‘(All A is B).
In the same form, Theorems III.4. ii and 111.4. iv can be put as
(All A is ‘(By is equivalent to (‘(Some A is B and some A /s
‘(By and ‘(A//A/s Bjj,
(Some A /s B and some A /s ‘(By is equivalent to (‘(All A is
‘(B)) and ‘(All A is B)),
and
(All A is Bj is equivalent to (‘(Some A is B and some A is ‘(By
and ‘(All A is ‘(B))).
Proving in the first of these three formulas the implication
from left to right, the first half of the “some” unit, ignoring the
negation of the whole, is false by the first above displayed
definition, and hence the unit, with its negation, is true, the
last inference holds because here, unlike earlier (p.l51 fourth
par.), “Not both A and 6” is true if and only if either A or 6 is
false, since meant to be the denial of “A and 6”, meant true if
and only if both A and 6 are true. And the last unit in the
formula is true because the first unit implies by aforesaid
meaning of “some” (p.l53 last par.) that some A is ‘(B), which
by the second above definition implies the last unit.
Proving the implication from right to left, the last unit implies
by that definition the second half of the “some” unit, and hence
to make that unit a negation its first half is false. The last
inference holds once more because here “not both” is true if
and only if either is false. And because that first half is made
false the first unit is by the first above definition true.
In the second formula the second halves of the two large
units are equivalent by the second of those definitions. And by
that definition the first half of the second unit is equivalent to
“Some A is ‘(‘(B))” and therefore to “Some A is 6”, the first half
of the first unit.
The proof of the third formula is the same as that of the first,
but for a reverse treatment of the “some” unit and deriving the
equivalencies as in the last proof.
The same method can be used for, regarding ‘(B), the four
principles succeeding Theorems 111.4. i-lll.4. iv and half of which
were seen to comprise those definitions (this p. second par.).
The other half can be stated as
(All A is B^ is equivalent to ‘(Some A Is ‘(B)),
and
s
156
(Some A is BJ is equivalent to ‘(All A is ‘(By,
and it actually comprises the last mentioned equivalencies
in the third above formula. Hence to prove it the second halves
of the present formulas will be found through the defini¬
tional ones equivalent to the first halves with ‘(‘(B) and there¬
fore with B.
To see how any of the eight principles con be deduced from
others for ‘B it may be recalled that half of them, those that by
former remarks may be termed of inclusion, were by diagram
seen qualified {p.153 first and last pars.), namely true if and
only if A belongs to things C comprising 6 and ‘B.
If it does, for the first half of “if and only if,” then by Theorem
lll.2.ii (p.l45) and transitivity any one A, since a C which thus
is either 6 or ‘B, is either 6 or ‘6, confirming for ‘B the
counterpart of Theorem lll.4.iii (p.l48). In consequence follows
also the second half of the displayed principles (p.l49)
succeeding the last theorems. If ” ‘(All A is 6)” or ” ‘(Some A
is 6)” is true then, since these were noted (p.154 last par.) to
respectively mean some and each A is a thing other than 6, and
here as a C an A other than B is ‘B, respectively “Some A is ‘B”
and “All A is ‘B” is true; by transposition if either of the latter
two is true negated then the respective one of the former two
is true unnegated. These four implications are counterparts, as
forms of last mentioned principles, of displayed ones with ‘(B)
(last p. second and last pars., and this p. above), viewed from
right to left. From these counterparts follows in addition the
counterpart of Theorem III.4.iv (p.l48), the last principle now
concerned, as, also, displayed as implications (last p. third
par.), viewed from right to left. The counterpart can from the
preceding ones be derived in the same manner as were those
implications.
That only if A, for the second half of “if and only if”, belongs
to things C comprising B and ‘B do the concerned principles
hold, to signify that if they hold then A belongs to C, follows
more simply because all these principles assert either B or ‘B
of the A. Hence if the principles apply, if the A are either B or
‘B, then since both B and ‘B are C, the A are by transitivity C.
This of course is hardly of interest, the question being under
what condition the principles hold, not that if they hold then the
condition does.
The other principles in the last pages discussed, those that
may be termed of exclusion, were to be seen by diagram to
hold for ‘B without qualification, because ‘B is an instance of
‘(B).
To prove via others that they do, it must be true that by the
counterpart for ‘B of Theorem III.4. i (p.l48) if A is ‘B then it is not
true that it is B. Since by the preceding it is by transitivity ‘(B),
157
the inference holds by that theorem; by transposition if A is 6
then it is not true that it is ‘B, confirming that counterpart.
Similarly for the first half of the principles succeeding the last
theorems (p.l49). If “”All A is “6”” or “”Some A is “6”” is true then,
since hence respectively all and some A is ‘(B), by partial
counterparts of those principles (p.l55 second par.) respectively
“” “(Some A is 6)”” and “” “(All A is 6)”” is true; by transposition if
either of the latter two is true unnegated then the respective
one of the former two is true negated, confirming the
principles. From them follows the counterpart for ‘6 of Theorem
lll.4.ii (p.l48), the last principle concerned, displayed as
implications (p.l55 third par.) viewed from left to right. The
counterpart can be derived from the preceding in the manner
they were.
The difference between ‘B and ‘(B) was seen not to enter
existential logic and accordingly implication. The recent
principles would hence apply to implication with standard
negations, should there not be some problem with linguistic
usage. Implication between A and 6 is customarily confined to
principles, to exclude the single A as well as some A only. Since
principles were seen to consist of no other than, revealingly,
constant accompaniment of one kind of thing by another,
however, the difference between them, i.e. that all A is joined
by 6, and that some A or the single A are is merely in those
quantities. With some liberality and serviceability, therefore, all
the accompaniments, as suggested (e.g. p.l51 fourth par.), may
be spoken of as implications, restricting, as in necessity, the
narrower meaning to the constancy. The theorems can for
implication accordingly say, for example, “”Not more than one
of the following, (all) A implies 6, some A implies 6 and some
implies ‘6, and (all) A implies “B””.
That “”some”” be less associated with implication is under¬
standable in view of the word”s mentioned (p.l04 last par.)
denial of a principle. With attributes the word has often value,
because, although an attribute may not be possessed by all of
a kind, it can characterize some for a substantial period, as may
a color some animals of a kind. There is hence a degree of
certainty that the attribute will repeatedly be observed, if only
in some cases, in consonance with earlier observations on
worldly things as retaining their identity through time.
Correspondingly it can also be of more account that a single
thing has a certain attribute than that a single event is
accompanied by a certain other. To speak of “”some”” in regard
to implication thus has mainly the significance of denying a
principle, normally done by simply denying an implication.
Similarly the accompaniment of single things by others is cus¬
tomarily only spoken of as implication when the connections
1
\
158
are instances of constant ones.
The usage notwithstanding, accompaniments of some or the
single A by 6 are of logical consequence and therefore can
together with constant ones be considered as outlined. That the
preceding principles can be applied to these connections can
beside diagrammatically be ascertained in the same ways as
above. And being very general, they concern connections
between quantities as well, as in the referred to ones between
prime numbers and their attributes of even or odd.
Application of the principles to numbers as well as to other
things was indicated a matter of course, and the principles of
this section can as others indeed be regarded as presupposed
in all walks of life. When something is discovered to be fact it
is disallowed that it is fiction, one’s very pursuits being guided
by these fundamental distinctions. It is for the present sections
to demonstrate that these presuppositions, though not any, are
well founded.
Section 3
OTHER THEOREMS AND
OBSERVATIONS
The principles investigated in the last section had much to do
with what is called external negation, alluded to at the start of
that section when mentioning affirmation or denial of whole
propositions, rather than of what is predicated of their subjects.
The negation was symbolized by an apostrophy before the
parenthesized proposition, e.g. as ‘(A is 6), with a similar
notation, ‘(A), for a similar indirect negation of an attribute.
Externally denied propositions were found equivalent to
certain, externally, affirmed ones, which may in general be
regarded as of more interest, as one would feel closer to one’s
goals knowing which propositions are true instead of which are
false. As was observed in relation to the liar paradox,
propositions are in fact meant to be true, even when asserting
falsity, and external negations are mainly denials of,
affirmative, propositions.
The emphasis on affirmative propositions, which may or may
not include internal negations, should prove of special value in
deriving conclusions from premises, as already illustrated when
seeing how instances are derived from principles. These
propositions will also correspondingly be concerned in sub¬
sequent further principles.
Beside the laws of complements there are other fundamental
159
principles in logic, and they are largely accepted as undemon¬
strated. For that reason and because demonstrations may be
faulty the principles may be false. Therefore since, as ex¬
pounded, substantiation of any truth is made by perceptual
finding, basic theorems will continue here to be proven through
diagrams. This is not to contend that every deductive theorem
must be apprehensible through a diagram, or that any truth
whatsoever is determined solely by perceiving it in accordance
with the criterion by which the occurrence in question is meant
perceived.
As indicated before (e.g. p.4 fourth par.), fine distinctions are
not always at once perceived, and many other things beyond
immediate reach are ascertained by inference. As brought
to attention in the discussion on deduction of instances,
however, inference is itself a form of mental perception. While
conclusions may thus not be perceived in the subject matter
considered, they may be perceived as consequences of
premises.
The present theorems are ones that represent the last men¬
tioned sort of perception, and they can hence be simply viewed
by diagrams, even if in part themselves deducible from others.
In the next two they are in their varied forms known as the law
of transposition, contraposition, and modus tollens, these re¬
ferred to variously before (e.g. p.114 fourth par.). The first is of
existential logic.
THEOREM III.5. If when A exists B exists then when B does not
exist A does not exist.
Proof. In Figure III.5 the small circle stands for the times A
exists, and the large circle for the times 6 exists; when A exists,
accordingly, 6 exists. As disclosed by the figure, therefore, when
6 does not exist A does not.
As previously, that the times of A are times of B would be
thought to require another diagram, a single circle when both
times are identical. But as in other cases, it is enough to
conceive A as confined to 6 to see the theorem hold. The
converse of the theorem is usually also given, to say that ‘M/hen
A then B” and ”When ‘B then A” are equivalent. By A and 6 is.
FIGURE III.5
160
\
however, meant anything, including negation in one of the
things or both, all of which, as said, can be represented by the
circles. Hence the converse is true as an instance of the
theorem. The theorem, it may be remarked, incorporates by
that comprehensiveness laws of thought, since denials of
denials in it are by those laws equivalent to the asserted
affirmations. These results can be discerned, however, in the
present figure as well.
It may be evident that the relations here between A and 6,
or their negations, are of the spoken of implications, and as
elsewhere they are about any revealing accompaniment except
that every A, including a solitary one, is involved. ”When” can
accordingly in this case, too, refer to a place as much as to a
time, and perhaps to a conception, as when the connection is
a logical principle. The revealingness of interest was also seen
(p.l52 first par.) to in “when” require a certain limit within
which A is accompanied by B, and that limit can ironically, in
further extensiveness of application, encompass in a concep¬
tion of such as a logical principle all that may be apprehended.
The present theorem as a whole itself is an example. If the
condition in it, “A implies 6”, is a logical principle also, then
what it implies, ” ‘6 implies ‘A”, can take effect everywhere the
condition does not. In other words the limit within which the
condition is accompanied by what it implies could be said to be
anywhere. The limit here is the conception itself, however, as
represented by the figure. What is conceived to accompany the
assumed in the theorem is not conceived to accompany
everything.
In the like theorem for attributive logic the accompaniment
is conceived as more limited.
THEOREM 111.6. When A is B then ‘B is ‘A if and only if it
belongs to things C comprising A and ‘A.
Proof. In Figure III.6 the small circle stands for every A and
some of C; the medium circle stands for every 6; and the area
outside it and of the large circle stands for every A and the rest
of C. The figure accordingly complies with the conditions of the
theorem and demonstrates it.
FIGURE III.6
161
As in other diagrams, A might also extend to the rest of 6, as
may instead A, and C accordingly. It is only needed to conceive
A as confined to 6, A to other than A, and C as comprising A
and A.
The situation may appear somewhat involved, but is less so
in practice. As elsewhere, one usually knows what sort of things
A and A concern, viz C, as well as whether something, viz
some 6 or ‘6, belongs to them. Therefrom one easily infers
whether ‘6 is A knowing that A is 6. Facts about referred to
even, odd, and prime numbers, suited for the restricted
coverage of A and A in the figure, can serve as example.
For it A in the diagram is the number 2, divisible only by 1
and itself and hence named a prime number, C; A is every
other number divisible only by 1 and itself and hence a prime
number C; and 6 is every even number. Accordingly since the
number 2, A, is 6, even, an odd number, ‘6, is A, another
number divisible only by one and itself, if and only if it is
prime, C.
The example also illustrates how the theorem, like the one
preceding it, applies when a connection has only a single
subject, A. There can in point of fact also be a single 6, identical
with the A. That of what is predicated, 6, of the subject. A, there
may only be one has been denied in logic. Suffice it to say at
this time that, as with affirming a subject’s existence (ca. p.l50),
single things are indeed found with unique attributes, e.g. as,
beside what distinguishes them otherwise, the ones of their
placement, and those attributes are asserted of them
accordingly. Aside from these arguments the last theorem is
accepted without the qualification as to C, although it is
without it that the theorem does not hold.
It does hold without it if applied to ‘(A), however, as in former
theorems. I.e. when A is B then ‘(B) is ‘(A), as pictorially easily
confirmed. ‘6 will also do, since included as subject, but A will
not, since not assured as attribute of either ‘6 or ‘(B).
Associable with these principles, in particular when about
causality, can be a mathematical one of similar form.
If quantity A is by an amount less than quantity B then a
quantity by that amount less than a quantity not B is not A.
The same may be said with “more” in place of “less”. The by
now familiar diagram at right provides confirmation, with
letters interchangeable.
The principle can again also be deduced by means of logic.
The condition in the principle can be stated as, equating
quantities, A = 6 – C, which implies by transposition, with
equality to be seen as mutual attribution, that'(6 — C) = ‘(A).
The implied can be seen to be the same as that of the principle,
since — C remains by postulate, and hence ” ‘(B — C)” is
162
\
identical with, as in a previous case (p.l55 fifth par.), ” ‘(B) —
C”. With “‘more” in place of ‘less” the inference is obviously
analogous. “A = 6 + C” implies ” ‘(B + C) = ‘(A)” or ” ‘(B)
+ C = ‘(A)”.
An example of transposition incorporating the mathematical
one, as well as displaying the common comprehension of
either, is furnished by the derivation that if for a food to be
cooked at a given time one must begin at a certain other then
if cooking is not begun at the required time the food will not be
done when desired. Notwithstanding the everyday decisions
made in accordance with these principles, they, too, are subject
to uncertainties. This is not surprising in the light of the qualified
nature of Theorem III.6, and Theorem III.5 is likewise a
participant object of misunderstanding.
Along with the names “transposition”, mainly used for the
latter of these theorems, and “contraposition”, used for the
former, was mentioned “modus tollens”. The formula so named
is viewed as a separate principle and may be stated as
If when A exists B exists, and B does not exist, then A does
not exist.
What is asserted by it, however, is the same as what is as¬
serted by Theorem III.5. Each has as conditions that when A
exists 6 exists and that 6 does at some time not exist, with the
consequence that A does at that time not exist.
It can certainly be stated that by certain two expressions are
meant the same thing, for possible inferences as a result,
because, as noted earlier (p.l37), something may be known
true under the one expression and not the other. A similar
likening of two statements is done in the accepted equivalence
of “A implies that 6 implies C” and “A and 6 implies C”, of
which statements in fact Theorem III.5 and the last italicized
formula are respective instances. Even this equivalence
neglects that the contents of the two statements not merely
imply each other but are the same. But while these statements
are at least in some manner identified with each other,
transposition and modus tollens are stated as independent
facts, as if to contend different things. Even if two facts merely
imply each other, as any under the two sides of transposition
itself, is it awry to state them independently.
As companion to modus tollens is known modus ponens,
which is of a parallel identity more treacherous and may be
stated as
If when A exists B exists, and A exists, then B exists.
In accordance with the preceding this is the same as saying
that If when A exists B exists then when A exists B exists.
There is revealed thus a redundancy within modus ponens
alone. Indeed it is superfluous to say that if A implies 6, and A,
163
then B, since for A to imply 6 already means that if merely A
then 6. The presumed principle has in the capacity of a rule of
inference been adapted to formal logic, logic whose symbolism
is intended not to require interpretation, meaning. Part of that
symbolism is a sign for what would ordinarily be ^’implies” and
which can here take the shape of an arrow. The preceding
modus ponens would then be put in a form such as
If A B, and A, then B.
The rule, however, only results in allotting the arrow or like
symbol a meaning after all, stating it regarding the connection
of A with 6.
With this rule is often paired one of substitution regarding
variables like A and 6, by stating such as
If a variable in a theorenn is uniformly replaced by any other
designation then the resulting proposition is true.
This rule equally supplies, however, a meaning to variables,
one in conformity with the standard one.
Instead of being rules of inference these statements thus
manage to define generalities that, expressed with symbols,
characterize logical principles. Because utmost generalities,
they contribute, as said, to the notion that they need not be
about anything at all. But they concern the characteristic that
in logical principles something is generally said to at all
times—signified by the implication sign—of all things—
signified by the variable—be true.
The adequacy of knowledge of a principle in order to apply
it in inference, without the redundancy of the preceding rules,
arises also in some of the following, in concord with earlier
observations on deduction of instances.
The next theorem is a transitive one, known as the law of
syllogism, and it has to do with conclusions from two premises
rather than from, in what is called immediate inference, former
single ones.
THEOREM III.7. If when A exists B exists, and when B exists C
exists, then when A exists C exists.
Proof. In Figure III.7 the circles stand as before for the times
the lettered entities exist, substantiating the theorem.
FIGURE III.7
Similarly to other cases, in transitivity the circles, mostly used
for attributes rather than present implications, are thought to
require the three additional diagrams shown. It suffices again,
however, to conceive A as confined to 6, and 6 to C The
implications can, further, again be comprehended in the broad
sense described, with possible single entities. The comparable
is the case with attribute.
When A /s B, and B is C, then A is C.
Same Figure III.7 serves to substantiate, the circles now
standing for the things named by the letters, and it serves as
well for a transitivity in mathematics, the circles accordingly
standing for quantities.
If quantity A is less than quantity B, and B less than quantity
C, then A is less than C.
The place of “less” can be taken by “more” and the principle
can once again be deduced by logical means.
As was observed (p.l47 fourth par.), that quantity A is less
than quantity 6 can be said to mean that while all A is 6, some
A is 6 also. Hence the principle holds for “less” if all A is C and
some A is C. The first of these conditions is true directly by
preceding transitivity, since all B is C by the sarne meaning.
And the second condition is true since those A that are 6 are
by same transitivity C.
Concerning “more” it was likewise observed that for quantity
A to be more than quantity 6 means that not all A is 6, nor is
some A as well B. These were in the last section seen
respectively equivalent to some A being ‘B, and all A being ‘B.
Hence the above principle holds for “more” if some A is ‘C and
all A is ‘C. Both of these conditions follow from the preceding
ones by transitivity, since all ‘B is ‘C for the reasons just given.
The principle can combine with other transitivity law in
causation, similarly to transposition. If A is, counting time as
quantity, followed by B, and B by C, then A is by C, the
successions viewable as implications, the time, within a
latitude, of one thing being a time of another.
That implication can be spoken of in terms of the time of A as
a time of B brings to view that transitivity has to do with infer¬
ence of instances, a procedure seen (e.g. pp.139-141) to like
other inference require justification beyond any principle. With
the time of B a time of C, the time of A is inferred to be a time of
C as an instance of a time of B. But to follow this principle, or
any other prescribing an inference, it must itself be ascertained
that an instance of them applies. Transitivity laws, instructing
that from a joining of a thing of B by one of C follows the same
joining of a thing of A as an instance of the first, would accord¬
ingly seem pointless. That an instance is at issue is, however,
not always obvious. It is in implication obscured as about
165
a time or place, since these are not explicit, and in the above
mathematical case the connections as about such logical ones
are still less evident. The same is not true with attributes,
although there, too, arise uncertainties. There is no problem in
deducing that one shall die because everyone does, as in the
example about Socrates (p.l41), but there may be a pause in
inferring that the sun is a star because every glowing heavenly
body is. When a premise is unfamiliar or the conclusion
implausible it may be more difficult to discern the connection
directly than through the transitive principle.
The transitivity of implication and progressive quantitative
inequality, or equality, is of course as inadvertently applied in
life as is the one last spoken of. One knows that if one thing is
followed by another, and the second by yet another, then the
first is followed by the last.
As a matter of fact one knows that this result ensues however
many the connections between the first thing and the last. The
general confirmation can easily, if wished, be made by adding
circles in Figure III.7, or by regular deduction. Since the
principles establish the connection between A and C, if it also
exists between C and D then by repeat transitivity it exists
between A and D, and so forth. Such prolonged transitivity is
well known regarding mathematical equations, in which it
functions alike in both directions. The equalities concerned are
not, however, customarily treated in logic as of the connections
here discussed, as should be apparent from the spoken of (ca.
p.39) undue differentiation between subject and attribute.
How equality or identity does belong to these connections,
and how it results in dual consequences is indicated by the
next theorem.
It is one mentioned (p.4T last par.) to be in some form known
as the axiom of extensionality, and it speaks of a reciprocity
between A and 6.
THEOREM 111.8. When A is B, and B is A, then A /s identical
with B, and conversely.
Proof. Figure III.8, the circle standing for A and B, demon¬
strates the theorem.
FIGURE III.8
The theorem can be demonstrated through others as well.
Similarly to the meaning of ‘less” or “more” (e.g. p,164 sixth
and seventh pars.), the conception that A is the same as,
identical with, 6 can be held to be that one sees all A to be 6
and not some A to be 6 as well. The first of these relations thus
makes half of the last theorem true by definition, and since the
second relation is by complement law equivalent to all A being
‘B and hence by transposition to all 6 being A, it makes the rest
of the theorem true.
A note may be added regarding the use of “if and only if”
or “all and only” in the earlier formulation of the theorem (p.41
last par. again) and elsewhere. As indicated (e.g. p.l56 fourth
par.), the “only” part is taken to signify an opposite direction,
as in “If 6 then A” or “All 6 is A” instead of “If A then 6” or “All
A is 6″. That it does can once more be held the result of mean¬
ing, by which “only” is understood as “not some other”. By the
last, as presently in “Not some A is 6”, was in turn understood
“All A is ‘6” found equivalent to “All 6 is A”. Therefore the
opposite direction. The same happens for “if” in compliance
with the like laws for implication. The frequent use herein of
“when” instead of “if” as in the theorems, should be clear from
its explained universality.
The last theorem also holds if the first two relations in it
between A and 6 are implications and the third is an
equivalence. This can be considered to be so in consequence
of the theorem, since, as explicated, implication can be taken
to speak of the time or place of one thing as a time or place of
the other, with equivalence denoting the identity of these times
or places.
Mathematics does not of course have a corresponding
principle relating to the previous postulation of A or 6 as
smaller or larger than the other. Each of A and B cannot at once
be smaller than the other, or larger, let alone also identical.
Theorem 111.8 has bearing, however, on well known math¬
ematical principles.
Anything true of 6 in the theorem can be viewed as attribute
C, 6 thus being a thing C of that attribute. Then since A is 6, by
transitivity it is C, which is to say that if A is identical with 6,
then anything true of 6 is true of A. And anything true of A is
true of 6. This, seen inaccurately interpreted by Leibniz’s law
(p.32 last par. through p.33 second par.), applies accordingly
equally to mathematics, specifically to selfsame quantities, as
when considered in numerical equations independently of
things which they concern. Hence the transitivity in the equa¬
tions, since what is true of one of the numerical forms is true of
another, and for the same reason when an operation, such as
addition, is performed on one of them, the result is the same
167
as when performed on another.
Equality often has two senses, however, in mathematics, as
noted (p.l30 third par.), as elsewhere. Beside the one of
selfsameness there is one of certain alikeness, as in equating
sizes in geometry. In those cases it can likewise be asked
whether something true of one of the equal things is true of the
other. Everything clearly cannot be the same for each, for
instance their location. The question presently is rather whether
in substantially mathematics, in analogy with the equality in
the last paragraph, if things are equal in the second,
nonidentity, sense, anything entailed by one of the things is
entailed by the other. By something entailed Is meant, as
common in logic, something deducible, something intrinsic as
explicated.
These equal things, things distinct but regarded by equal,
quantitative, attributes, it should be considered, are, as noted
about some geometric entities (e.g. p.l26 first par.), undif¬
ferentiated, in being conceived without individual character¬
istics that, as may be location, would distinguish one from
another. As in superposition of those geometric entities, they
are upon their comparison, whatever its manner, postulated
indistinguishable. And as in that superposition, if upon the
comparison things in question differ in some part, they are
contradictorily not equal. Consequently since anything entailed
by a thing is, as something intrinsic, part of it, anything entailed
by one of the equal things is, as an indistinguishable part,
entailed by the other.
Therefore whether quantitative things are equal because the
selfsame or because the same by comparison, it can be stated
as a general principle that
If quantitative things are equal then what is entailed by one
of them is entailed by the other.
This determination furnishes proof for further of Euclid’s
axioms, known as common notions. The concerned are
1. Things which are equal to the same thing are equal,
2. If equals be added to equals, the sums are equal,
and
3. If equals be subtracted from equals, the remainders are
equal.
The equals and equal things can be understood to be quan¬
titative ones, and matters tantamount to these axioms were
indicated above with regard to identity (facing p. last par.).
While identity allows the likes of these axioms to apply to all
things regarding anything, however, the above displayed
principle assures that they apply to quantitative ones even if the
equality is not selfsameness.
Before specific confirmation, a word on some entailment. As
s
168
noted earlier (p.33 fourth par.), definitions are sometimes
thought to be logically true, of entailment, because held to
speak of properties a thing by its nature always has. But things
were seen not always so defined, and the more general issue
instead is that properties can be entailed by things by postulate,
through definition or otherwise, rather than by inference. One
example was the definition that 4 equals 3 + 1, intrinsically
true as long as the digits stand for the numbers in question.
Another, nondefinitional, example was that one geometric
figure, e.g. an angle, equal another.
How these remarks pertain can be seen regarding the first
above axiom. The axiom is true by the before italicized princi¬
ple since by it, if both A and C are equal to 6, what is entailed
by 6 is entailed by equal A, namely being equal to C. If
‘A = B = C” represents ‘^2 + 2 = 3 + 1 =4″ then “B = C”
is a definition. Yet in accord with the preceding it can be re¬
garded as entailment, although since concerning selfsameness,
it is not an issue here. But a more pertaining situation occurs if
“A = B = C” refers to angles assumed in that order equal.
In the other two axioms the additions or subtractions can be
viewed as parts of equal assumptions, from which by the above
principle follow the equal sums or remainders.
The remaining axioms of Euclid can be regarded as
definitions.
4. Things which coincide ore equal,
and
5. The whole is greater than the part.
The equality in the first of these has to do with geometric
shapes, noted to be meant equal by their appropriate coincid¬
ing (p.l30 third par.). As to the last axiom, it was likewise
indicated that a part is meant to be less than a whole (p.l22 first
par.). There is likely no dispute that correspondingly the whole
is more, greater, than the part, although it can be justifiably
asked whether ‘less” and “more” are reciprocal.
This because the thoughts behind the use of the words may
differ, and, as in other inferences, it is to be determined
whether the things thought of imply each other. With respect to
“less” and “more” such a difference was already indicated (e.g.
p.l64 sixth and seventh pars.). For “Amount A is less then
amount 6″ one can be held to view A first, to find it to be
contained in 6, and some amount outside it to be contained in
6 as well; and for “Amount 6 is more than amount A” one can
be held to first view 6, to find not all of it to be contained in A,
but not some amount outside it to be contained in 6 either. The
equivalence of the two relations is without saying quite
evident, as made by the already found very serviceable
diagram at left, but it can nevertheless be regarded as a
169
deductive principle.
If amount A is less than amount B then B is more than A, and
conversely.
As is clear from all said about these relations, the principle
can also be deduced by usual logic. ‘”All A is 6″, half of “less”,
is by transposition equivalent to “All ‘B is A” and hence by
complement to ” ‘(Some ‘6 is A)”, half of “more”; and “Some
A is 6″, the other half of “less”, is by complement equivalent to
” ‘(All A is ‘6)” and hence by transposition, since “All 6 is A” is
by it equivalent to “All A is ‘6″, to ” ‘(All 6 is A)”, the other half
of “more”.
The reciprocity between “less” and “more” is only one of
those that can be seen as deductive. These are best known
when the relation is the same in both directions, the mutuality
called symmetry or commutativity. That of addition and
multiplication in mathematics is the most familiar.
a + b = b + a,
and
a X b = b X a.
By addition being commutative is more particularly meant
that whichever of two numbers is counted first, the total will be
the same. This is a consequence of the more general principle
that a given number of things, perhaps that total, will be the
same whatever the order of counting them. This in turn holds
because of the very postulating of a given number of things.
The amount of units is postulated to remain during the count¬
ing, in whichever order. It is not meant that during the counting
some unit is added or subtracted, to result in a different count.
Any quantity is more generally determined in accordance
with some standard, which in counting is the number se¬
quence. To count things, they are considered in what is called
one-to-one correspondence with the numbers, and to postulate
the things to remain of an amount, in whatever order counted,
means just that they so correspond. While the number se¬
quence as a standard enjoys quite an unchanging status since
a mental entity, other standards, such as measuring tapes, may
undergo change, if only in direction used, similar to the order
in things counted. Nonetheless any standard as well as the
quantity it is used for stay conceptually the same if postulated
to, in whatever manner the first determines the second. If in
reality a quantity determined should differ according to that
manner, then a change occurred in the standard or quantity.
The preceding is germane to the in the last chapter investi¬
gated relativity of motion, when considered only as change in
position (p.77 last par.). The position means the placement of
one thing relative to another, this determined by distances in
usually three dimensions. If then there is a change in those
s
170
ooo
ooo
distances for one object relative to another, the question is
whether the some change occurred for the other object relative
to the first. The distances are regarded as measured from one
of the objects in one case, and from the other object in the
other case. But by the above the same distances as well as
measuring means are assumed in question, the sameness
decided by the very correspondence of one to the other. There¬
fore the same change in distances, to wit motion, is measured
relative to either object.
Whereas it may be held self-evident that addition or meas¬
urement is commutative, that the same is true of multiplication
is in textbooks sometimes supported by diagrams. Thus if a
horizontal row of three things is multiplied vertically twice, the
depicted reveals that a vertical column of two things is mul¬
tiplied horizontally thrice.
The requirement of proof is in this regard, too, attempted to
be skirted by redefinition. Multiplication has been defined in
terms of the product of two, or more, numbers. But it is not
answered whether the product is the same for a x b and b X
a. And as in other cases, the proof can also be obtained
logically, on the basis of, rather than a numerical diagram,
meaning behind langudge. By “2 times 3” it is not meant that,
perhaps, only the third of three things is counted twice, but that
each of three things is. Generally, by a times b is meant a of
each unit of b. That is to say a is counted as many times as units
of b, namely a is counted b times, or, there is counted b times a.
Symmetries like these are in logic also held deductive for
what are called conjunction and disjunction, “A and B” and ‘A
or 6″, as equivalent respectively to “B and A” and “B or A”. The
latter equivalence can in accord with earlier observations
indeed be viewed as deductive. ‘A or B” can be taken to mean
A implies B”, equivalent by transposition to ‘B implies A” and
hence “B or A”. By ‘A and B” is, however, normally meant no
implicational connection, it being merely stated regardless of
sequence that both things signified by the letters are the case.
“B and A” is thought to be equivalent deductively in accordance
with truth tables, which, as will be seen concerning other
principles as well, only restate, however, that both expressions
are meant true when both A and 6, in either sequence, apply.
A lack of implicational connection does not pertain, however,
to the identity in the last theorem (p.l65), with identity, mutual
attribution, a form of implication. Whether
If A is identical with B then B /s identical with A
could correspondingly be questioned. Indeed by what was
observed (e.g. p.l66 first par.) the two halves of the symmetry
represent different thoughts. The first is that upon viewing A, by
the attribute concerned, one finds it to be 6, and not some A
171
to be 6 also; and the second thought is the reverse. It can hence
be asked NA/hether one implies the other, and the affirmative
answer is of course manifest without such analysis, or even a
diagram like that of Figure III.8 (p.l65). If desired the implica¬
tion is available by usual deduction, through complements and
transposition.
The symmetry thus applies to equivalence as well, seen as
an instance of identity (p.l66 third par.), and it does to math¬
ematical equality likewise. For when that equality is not of
identity but of comparison, it is still conceived in the described
manner, by which e.g. in angles through superposition all of
the one amount and not some of another is seen as concurring
with the second, and so forth, as would a corresponding
inequality be conceived similarly to described ‘less” and
“more”.
About symmetry or, as called in this case, conversion, is also
the last theorem to be in this chapter featured. If is principally
of attributive logic and does not speak of previous “all” in¬
cluding the unitary, but of “some” alone.
TFIEOREAA 111.9. When some A is B then some B /s A.
Proof. Figure III.9.
The circles are believed again to require four more diagrams,
depicted, the last one here adjusted as previously (p.l53). But
it is enough to consider that at least one A, among any number,
is a 6, among any number, to see the theorem hold.
Once more, the theorem can be deduced by way of others.
“Some A is 6″ is by complement equivalent to ” ‘{All A is ‘6)”,
which by transposition, since “All 6 is A” is by it equivalent to
“All A is ‘B”, is equivalent to “‘(All B is A)”, which by com¬
plement is equivalent to “Some 6 is A”.
The theorem is likely not to need a counterpart in existential
logic, in which “Some A is joined by 6” would be expressed by
“A does not imply ‘6”. The comparable, as expounded, is not
strictly the case in attributive logic, where the denial of “All A
is ‘6″ need not lead to “Some A is 6” (e.g. p.l53 last par.). If this
seems ignored here in recent notations, it is because either the
needed qualification was met, or because A or ‘B was used
FIGURE III.9
s
172
for the exhaustive ‘{A) or ‘{B). ”Some” as explained (p.l57 last
par.) has besides a more independent significance in
attribution than in implication, the present theorem having
utility accordingly.
The theorem has been argued to, aside from mentioned like
arguments on “some” altogether, require the existence of its
subiects. It was rightly maintained that from “Should any A exist
then some is 6″ does not follow “Should any B exist then some
is A”. If no A exists but 6 does, the premise can be true, as when
A is always 6, but the conclusion cannot. But it was wrongly
maintained that the meaning of the theorem cannot be on the
order of “When A can be B then 6 can be A”. To say that A can
be 6, it is contended, is not to say that some A is actually 6. The
issue is again, however, not the meaning that expressions may
in certain usage have, but what sort of meaning is of concern.
And as already indicated regarding propositions used in logical
principles, “some” is meant to have, like “all” and other things
concerned, an extremely wide application, for all of which the
principles hold.
The crux of that meaning of “some” is its herein oft men¬
tioned denial of an “all”. “All” was meant, further, to apply to,
from all conceivable things of logical principles, to worldly
things in causal laws, to particular things of a place or time of
any limit. “Some A is 6 need accordingly not refer to the denial
of “All A is ‘6” in an existing situation, although it may. It can
refer to the denial of a supposed timeless law of nature. In that
event there may be no A which is 6, and it is as appropriate to
say “A can be 6” as “Some A is B”.
This broad equating of “some” with “can”, with possibility, is
in accord with the earlier description of possibility in terms of
“some” or “sometime” (p.36 last par.), alongside the descrip¬
tion of necessity in terms of “all” or “always” (p.35 third par.).
The respective identifying of these meanings with each other,
alluded to later also (e.g. p. 104 third par.), is indeed suitable for
logical purposes.
The second of these identities, one between “all” and the
necessary, has likewise been argued against by adducing
existential import. An example offered of the necessary has
ironically to do with causal rather than supposed logical law.
It is that all brakeless trains are unsafe, which, although none
exists, can be held true, with the denial false. It is proposed that
the same cannot be said of an ordinary “all” proposition. As
one reason is cited the mentioned (p.l49 third par.) interpreta¬
tion of propositions as true if the subject does not exist. In that
case anything asserted of it is alleged true, and hence a break¬
less train would be both unsafe and safe. It is further asserted
that should propositions be true only if the subject exists (p.ll5
173
second par.), the train would be neither safe nor unsafe. It may
be remarked again that the alternative meanings of truth or
falsity thus submitted seem quite remote from the ordinary
meaning, the more so since they are multiple, and hence each
meaning cannot be the accepted one. Most of all, what is
meant by ‘”all” depends on the intent, and the present one
appears much like the customary one.
“All” is commonly used as broadly as found feasible herein,
frequently relating to timeless laws without regard to the
existence of instances, as in the case of the brakeless train.
Correspondingly there is no need to differentiate in logic be¬
tween such, universal, propositions and necessary ones. As a
matter of fact universal ones, as exemplified by causal laws, are
customarily about timeless connections, above associated with
the necessary. Even singular ones can be regarded as belong¬
ing to the necessary, which was observed in the first chapter to
be spoken of with reference to any fact.
Likewise observed was that the possible refers to the nega¬
tion of the negative forms of these, of the impossible, or, as the
unnecessary, to their own negation. This is the identity with
“some” in the preceding, applying even to limited negation
discussed, since “Some A is ‘B” like “Some A is ‘(B)” denies
mostly timeless principles, confined to certain things C. For
example some numbers (A) are odd (‘6) rather than even (6),
all confined to whole numbers (C). “Some” understood thus as
the denial of also timeless principles, the last theorem (p.l71)
refers in the relation between A and B, as in previous ones, not
only to actual states, past or present, but to potential future ones
as well. When the relation concerns actuality then “some”
indeed speaks of existing things, not merely possible ones,
since denying the opposite actual “all”, but when denying a
constant principle then the relation need be possible only.
This broad understanding of “some” along with “all” of
course also enters the laws of complements (pp.148, 149), which
is why the possible itself was explicated (p.36 last par.) through
those laws. In its wide sense it had to do among them with
both, Theorems 111.4.iii and lll.4.iv. The first by equating the
possible, if something is not impossible, with its having been
the case; and the second and more pertinent by equating the
possible, if something is not impossible, with its sometime,
including always, being the case, or with its simply not being
precluded in the inclusive sense here of “some”.
In this sense “Some A is B” as depicted in now concerned
Figure III.9 (p.l71) may merely state something not precluded
by a principle in the future, and the figure discloses that should
the stated become true, then as in other cases “Some B is A” is
true likewise.
174
\
The figure can also serve for elucidating a matter in the first
proposition of Euclid. In it two circles ore so constructed that the
circumference of each passes through the center of the other.
It has been put forward that the proposition assumes without
support that the circles intersect. That they in fact do is con¬
firmed by the diagram, and it need only be repeated that all
deduction is confirmed perceptually. The intersecting can also,
as was the case with other principles in mathematics, be
deduced by logical means.
Each of the circles extends inside the other, since passing
through its center. And each circle extends outside, since
its diameter as a straight line does from that center (p.l29
third par.). Since both circles are by definition, further, closed
figures, of closed circumference with the inside meant to be on
one side and the outside on the other, each circumference,
being continuous, passes the other somewhere, to therefore
intersect it.
In view of the deducibility by logical means of the preceding
and earlier mathematical propositions it can be asked, as it has
been in the past, whether all of mathematics is so deducible.
In efforts to prove that it is, it has been tried to define num¬
bers in logical terms. It is not necessary, however, that some¬
thing be a logical concept in order that inferences about it be
logical. Logical inferences are made about anything. In this
light it should not come as a surprise to find those definitional
attempts unsuccessful. For example it is thought that to state,
through a connection herein discussed, that every A implies
one and the same 6, i.e. that there is exactly one 6 concerned,
is accomplished by. saying “A implies 6, and if A implies also
something C then C is identical with E>”. The 6 need not be a
single one, however, the statement amounting to saying that A
implies only 6. Another effort to subsume mathematics under
logic is depending on the omission of definitions, rather than
on unavailing ones. Algebraic notations for addition and mul¬
tiplication, for instance, are left open to other interpretations,
e.g. disjunction and conjunction in logic (p.l70 fourth par.).
Finding that the different contents share properties such as
commutativity, it is felt that they consequently consist in the
same logic. The shared properties like commutativity, however,
need themselves be substantiated, and the question is whether
they can logically.
Indeed all mathematical, and other, deduction is performable in accordance with principles of logic. The fact is sug¬
gested by apprehensions that appear more common than ones
of those principles. They are that certain inferences, as
determined by single and often amusing counterexamples
(p.77 last par.), are fallacious, in awareness that conclusions
175
that do not follow from premises logically do not otherwise. To
examine the matter it will be well to first see better what com¬
prises logical deduction.
In this treatise as in other works a distinction has been made
between a logic based fundamentally on implication and one
based on attribution. Having noted, however, that attribution is
an instance of implication (p.l51 third par.), it may also be
observed that insofar as the attributional principles parallel the
implicational ones, they are likewise true as instances of them.
this is not obvious, for because one kind of thing implies
some other it need not be as in attribution one of them. But
since implication can be so broad as to be about association of
one kind of thing with another within any boundary, including
the boundary of the other kind by being one of them, its
principles can be read as applying to the boundary at issue. For
instance the transitivity of implication can be read as applying
to attribution, provided the boundaries within which things are
associated are identically their own.
Such a procedure is not the most practicable, with attribution
viewed so distinctly as to be favorable stated separately. There
were seen other characteristics peculiar to it. Reference to only
some of things in statements of it, or a limited meaning of
negation, due to the same lasting attachment of many attri¬
butes. But what matters is that the two logics deal with any
association of one thing with another, with the negation of any
of these parts or wholes, and with what in like form follows or
does not from any of these parts or wholes.
Accordingly returning to the question whether all deduction
can be logical, it can be considered that most things deduced
from and deduced can be viewed as associations of one thing
with another, rather than as independent units. They are
customarily called premises and conclusions. As observed in
the introduction and afterward, in seeking knowledge one
wishes to find not only what can be known regarding things
known, what facts are connected with other facts, but what can
be known regarding any concept. In order that findings of facts
be made, as in induction or deduction, the concepts were seen
to require eduction, usually expressed by definitions. The
definitions, seen as fully statements in inferences, thus reflect
how almost everything is regarded not iindependently, but as an
association of one thing—here a thmg considered, perhaps by
a name—with another—here defining attributes, perhaps
recalled.
The explanation for these connections may be found in, for
the discussed sake of one’s pursuits, one’s interest in things true
in connection with things, even if only identifying attributes are
at issue. Subsequent connections may typically be that a thing
s
176
of given attributes exists, and that it is causally accompanied by
some other thing. But the connections, the accompaniments,
can be of limitless variety, and of present interest is that they all
fall under the equally unlimited accompaniments referred to in
logical principles, and which generalized are about whether or
not one thing is associated with another.
The associations occur, as expounded, within certain limits,
in accordance with which it might be revealingly said that the
time or place of one thing is a time or place of the other.
Inasmuch as the associations thus concern the same placement
of the things, possibly their selfsameness, logical principles
regarding them are about whether or not these placements
entail certain others.
Logical principles were seen for that matter, in the laws of
thought as represented by the law of double negation, not only
about what is entailed by placements of one thing with
another, but also about what is entailed by individual things as
wholes. The things can be viewed as above concepts of which
something is yet to be asserted, but they can also be any fact
as a unit, without regard to its parts. What is of account is,
however, that logical principles about them are likewise about
whether their placements entail others.
As observed regarding transposition, an entailment is itself
an association, by which the place of what entails is a place of
what is entailed, e.g. the place of A is the place of “A. When
the entailment is not by one thing but by combined things, of
presumably things likewise combined, then the difference
between those combinations and the entire one is likely that
the former are of things within narrower limits. Thus if trans¬
position is applied to causal laws, the entire principle is of
nature as place of the association stated, whereas the premise
and conclusion are of the, spatial or temporal, vicinity in which
the cause and effect, or their absence, occur, as place of these
associations. Of significance is that beside the placements with
one thing of another respectively represented by premises and
conclusions of deductions, when the premises or conclusions do
not set down undivided, but combinations of, things as of those
places then those combinations are likewise placements of one
thing with another.
Those combinations as well as the deductions are, as indi¬
cated, the stuff of statements, of concerned assertions of facts,
whereby one thing is connected with another, in contrast to
concepts as denoted by individual words. Accordingly all
deductions are about determinations of placements or their
absences, whether again of undivided things or of combined
things placed with each other. But such determinations were
seen the province of logic, as displayed graphically by the
177
diagrams. They disclose whether or not one thing is placed with
another, be the placement only the larger one, of the deduced
from with the deduced, or also the smaller one, of something
else in addition.
The pertinent factor in the last sentence is that logic informs
of, about what is posited, beside what follows, what does not.
By saying that it informs of the last it is meant, as indicated, that
logic reveals that what does not follow in accordance with it
does not follow at all. That logic indeed performs this function
may in accord with the preceding be further elucidated.
The disclosure by diagrams of spoken of placements were
seen to result from placements posited. When placements in
question are not disclosed, it is because the placements
posited, known, are not adequate. Thus from knowing only that
all A is 6 one cannot detect that all B is A, because the place¬
ments can vary as pictured. It may then be asked whether in
specific cases one cannot deduce placements in question
without knowing more placements than posited. The point is
that if one knows added placements required in logic then the
deduction can be logical, but that it cannot be if made without
that knowledge.
But the sufficiency of the logical information has the same
grounding whether it is about what follows from something or
what does not. As it can be ascertained through diagrams that
if certain placements are known then certain others follow
regardless of what the things so placed, so it can be ascer¬
tained that if only certain placements are known then certain
others do not follow regardless of what the things. The certainty
rests on the referred to impossibility to conceive the matter
otherwise, in one case than that an inference follows, and in
the other case than that it does not, whatever the things
concerned.
With knowledge of premises from which conclusions can be
logically deduced thus required, therefore, anything deducible
from something is so in accordance with logical principle.
The significance is that anything that follows from what is
known is by logical steps discoverable, the placements known
needing only to be appropriately assembled to perceive result¬
ing ones. The assembly of pertinent premises can nonetheless
be quite difficult and the desired conclusions not reached. The
task is simplified, however, by the explained reducibility of
logic to associations of one thing with another. For this reason
there is no need to construct further systems of logic, as is done
with regard to spoken of modality, relations, and more. All the
concerned particulars of things, their presence at issue in
researches, can be treated as just mentioned associations with
those things, deriving consequences as expounded.
s
178
The dispensability of additional logics does not mean that for
clarity a particular application cannot be formulated. This
happened when attribution was singled out as a case of impli¬
cation, it being unlikely that when attribution is meant it will
be understood from speaking implication. A distinction be¬
tween the two connections was noted also made in what are
known as propositional and predicate logics. Because much of
them is accepted and they suffer from fundamental weak¬
nesses indicated, added attention to them is warranted.
Beginning with the first of these and the concept of implica¬
tion central to it, it can again be noted that, known as material,
an implication, “A implies 6”, is held true if A is false or B true,
i.e. under all conditions other than if A is true and 6 false (p.l51
third and fourth pars.). The spoken of inconsistency of that
meaning with the general one is magnified by its usually
understood reference to the present as the time A or B is true or
false. Most implications have to do with connections holding at
all times, as would be expressed by ‘^When . . . then”, and they
are hence the less determined by a situation of the present,
related to their frequent designation by “If. . . then”.
As also alluded to, there has been dissatisfaction in logic with
that definition, a’nd another, for what is termed strict implica¬
tion, has been proposed. It states that “A implies B” holds not
if it is merely false that A is true and B false, but if it is impos¬
sible that A be true and B false. The definition appears to take
care of the above deficiency as to permanence, although the
meaning of impossibility and related terms is not made clear.
The sought after connection between A and B, however, was
found to still be lacking. If A is independently impossible or B
necessary then the definition applies.
Both definitions in the last two paragraphs also contain the
discrepancy that while by them either of “If A . . . then” or
“When A . . . then” are true if A is not or cannot be true, either
of the expressions assumes A as condition. In other words, the
purport is that supposing A is false, then supposing it is true,
such and such follows. Such contradictory suppositions were
seen illegitimate, leading to paradoxes and not representing
reality.
A paradoxical nature, though not in the same sense, is of
these definitions asserted in logic, because of their omission of
a tie normally expected between two things when one implies
the other. Attempts have been accordingly made for a further
definition of implication, one that, as suggested earlier, is
thought to require an inner, logical, tie in order to possess any.
But what the logical tie, called spoken of entailment and by
which one thing is deducible from another, is is itself debated,
and as to be seen in truth tables, by which deducibility is held
179
determined, the meaning employed belongs to none other
than those found unsatisfactory.
It is also advanced that any essayed definition of entailment,
or of other implication involving the tie looked for, faces
difficulties similar to the others. The argument in simple form
is that for such a definition of implication it is reasonable to
state that
1. A implies (A or Bj,
with 6 standing for anything, and
2. (A or implies (A implies Bj.
Taken is then for example the law of contradiction. Not both A
and A, which implies by statement 1 that (Not both A and A)
or B, and hence by statement 2 (Both A and A) implies B. The
last inferred signifies that something false would imply
anything, since any fact in place of the law of contradiction as
sample would be denied there. The error can be detected
without recourse to a sample.
The undesired result follows by transitivity from statements 1
and 2 directly, by which A implies (A implies 6). Statement 2
was in fact seen true by definition of ‘”or” as “Not to find one
is to find the other” {p.l43 last par.), and hence statement 1 can
be held to in effect say that A implies (A implies 6), giving a
very meaning of implication not intended.
The intention indeed when equating “A implies 6”, or “A is
6″, with ” A or 6″, and “Not both A and ‘6” was noted to be a
certain connection between A and 6, rather than a concern
with their independent status (ca. p.l49). The connection was
observed to be that 6 accompanies within certain bounds A
but not all things (e.g. p.152 first par.). Accordingly the
accompaniment cannot occur if A implies everything under A,
or, the equivalent, if everything implies 6 under 6. Should A
imply everything then there is no restriction to the boundary,
and all things would imply the same things; the like result is
explicit should everything imply 6. The gist of this reasoning is
that by contradiction the intended connection cannot become
true because A or 6 is true. Generally, if implication is defined
by a connection between things, the connection cannot contra¬
dictorily be deduced absent.
The attention in logic to material implication, meant decided
by an independent truth value of A or 6, is, alongside a
comparable meaning of “or” and “not both”, owing to its use
in truth tables, for believed validation of logical theorems. For
present illustration of truth tables, as well as for other purposes,
implication in general,
A implies 6,
can be symbolized by means of the earlier arrow (p.l63), as
i. A -> 6.
180
\
The truth table of Table 1 would accordingly be held to
demonstrate Theorem III.5 (p.l59).
A B
+ +
(A
+ +
+
+
i)
+
+
+
+
+
+
TABLE 1
CB -> ^A)
– + –
^- + +
– + +
+ – –
The variables, as indicated, are regarded as standing for prop¬
ositions though more accurately they would stand for their truth,
or falsity, since it is these which are in question. More generally
these affirmations or negations were seen extendable to
anything in a given domain, not only to propositions. It should
hence be fitting to in the tables symbolize them by plus and
minus signs for general presence or absence, instead of by
frequent ones for truth and falsity only.
In the arrangement then the theorem, as seen, is above the
horizontal line, to the right of the vertical line. To its left, A and
B are assigned beneath them all their combined truth values in
corresponding rows. Across them the corresponding truth value
is then first marked for each occurrence of A and 6 in the
theorem; for instance with the first assignment for A and 6, true
and true, the truth values under ‘6 and A are marked false.
After, the truth values of the next larger units of the theorem are
marked, namely of the first and last implications; these truth
values are what is determined in accordance with material
implication, true in all cases other than when the premise is
true and the conclusion false. Last, by the same criterion, the
truth values of the central implication, the theorem, are
marked; since true in each row, the theorem is concluded true.
It may be noted first that the truth tables avail themselves of
the laws of thought without proving them. They do so in the first
two columns, by assuming them exhaustive. Were it not for the
laws of thought, there could be combinations in which a propo¬
sition is both true and false or neither. The laws are more con¬
spicuously assumed under ‘B and A, by holding them false
when A and 6 are true, and true when they are false. The issue
has been tried to be bypassed by recourse to definition, as ob¬
served to happen elsewhere (e.g. p.l70third par.). “Not-A” has
been defined as true when A is false, and false when A is true.
It is not answered, however, whether, provided when A is false
it is designated “not-A”, not-A is in fact false when A is true.
It may also be repeated that since the premise of the
theorem assumes A, its later denials are inconsistent, and like¬
wise for the other assumed conditions in the theorem. But the
181
principal error lies in the conclusion that the th^orem is true
because it, the central implication, is true under each condition
listed.
To begin, although in the theorem the premise is maintained
to imply the conclusion in some sense other than merely
because the former is false or the latter true, those were seen
the reasons why the connection is considered to hold.
Relatedly, the notion that a logical principle be true if, by
whatever criterion, true under all truth conditions of its
variables like A and 6 is fallaciously substituted for its truth
regardless of what the variables stand for. This truth is the intent
of those principles, meant to be true in, as said, all possible
worlds.
The truth tables would even in compliance with their proce¬
dure not require the result of all truth combinations of their
variables. This because the theorems, in implication form, are
meant to assert that when the premises are true so are the
conclusions. In Table 1 it is hence only required that by its
arguments when A ^ 6 is true so is ‘B —>• A, regardless of the
truth of the last row. It happens that the theorems would be true
in the remaining rows, those of false premises, under which the
implications are likewise meant to hold.
To see what theorems so derived actually assert, however, it
will be advisable to reexamine the table. In it, as in other truth
tables, the theorem asserts through chosen symbols what is
asserted underneath it. There its variables and their negations
are either affirmed or denied in accordance with the first
assumed combinations and laws of thought. In other words it
is asserted by those marks that if A or 6 are true or false then
they are and their negations are not, in effect restating the
assumed or laws of thought. The rest of the marks assert
whether or not combinations of the affirmations and denials of
the variables are termed implications, as symbolized by the
theorem. In other words the theorem says no more than that
given certain truth combinations of A and 6, those or those due
to laws of thought hold, all nameable as in the theorem.
The same occurs in other truth tables, and it is consequently
not surprising that logical theorems, thought to be demon¬
strated by them, be called tautologies. But the demonstration
does not take place, even of, as remarked, laws of thought.
Whereas the combinations of truth or falsity may, by a defini¬
tion of symbols for them, resemble in table 1 the law of trans¬
position, they do not prove it as it is intended to be understood.
The actual law refers in its conclusion primarily to truth or falsity
of a time other than in the premise. Instead the table refers to
the same time only, the compass of the laws of thought. These
laws in fact are confined to the same place as well, so as not
182
\
to result in the conclusion of transposition even when of the
same time but different place, the least such difference com¬
prehended by that principle.
This is not to speak of the explicated revealing connections
to which the theorems are expected to apply. Should the central
implication be somehow revealing it would only be about the
material implications of its premise and conclusion, not about
others intended. Moreover while by the meaning of those
implications and laws of thought the conclusion, since the
required combinations for it result, does follow from the
premise, it does not by the criterion of the truth table. By it the
central implication is as noted true because of the independent
truth values of the premise or conclusion, a criterion insufficient
for deducibility of the latter from the former.
The theorems in the truth tables often resemble those valid
in the usual sense, since they can in that event be considered
offshoots of the others. Thus in the present theorem in Table 1
the material implications within the premise and conclusion
signify only that, for instance, not both A and ‘B can at the time
in question be true, the other truth alternatives diversified. But
this condition obtains also in normal implication applied to a
specific time, since “A implies B” was seen equivalent there to
^^Not both A and ‘B”. The revealing connection discussed is
applicable, because while A may be accompanied by 6 at all
times, other things may not, but the connection may be mere
coexistence. And by regular transposition, redundantly, if “‘Not
both A and ‘B” is true at the time, so is “Not both ‘B and A”. That
is to say that as a consequence the curtailed premise in the
truth table implies the like conclusion in it, and the same holds
correspondingly for other theorems.
However, whereas theorems true otherwise may be true in
accordance with truth tables, though because of expanded
senses unproved by them, theorems true in accordance with
truth tables can because of the same curtailment not be true
otherwise, hence being false, since interpreted in the usual
sense. Table 2 offers an example.
A B
+ +
A
+
– +
+ – +
+
+
+
(B A)
+ + +
– + –
-I- – –
– + +
TABLE 2
The theorem could as before be by the criteria employed
held to be true merely because when the premise is true the
conclusion is, needing only the truth of the first and last rows
183
beneath the theorem. But what would be a theorem should by
now be manifest as of a familiar definition.
It is a definition by which if something, A, is true then every¬
thing, 6, implies it, and which was indicated as objected to by
the users themselves of truth tables. Table 2 indeed, as will now
be clear, repeats, through its markings, that definitional state¬
ment. The conclusion is marked true whenever A in it, and in
the premise, is. So are of that definition the other rows.
The supposed theorem does accordingly obviously not apply
when the concluded is a customary implication, the connection
which can be taken to be of interest in logical principles. That
connection was expounded to require that not everything imply
the implied. The theorem can be read to say that when some¬
thing A is the case then everything 6 is accompanied by it, an
accompaniment not suited for implication, and the theorem
appears more absurd still if considering that most implications
are about timeless laws, while the A as condition in it is of any
one time. Perhaps most glaringly is the theorem false since
implication is commonly taken to hold between things some¬
how closely associated as discussed, to have in conclusion B
A of the presumed theorem nothing to do with a fact of A in the
premise.
That in truth tables one thing be held to imply another under
all conditions other than if the first thing is true and the other
false is defended by contending no difference from the other
implications in practice. In both cases, it is argued, if it is known
that A implies 6, then at a time in question it is certain that not
both A and ‘6, though any other condition may be true. That is
to say that should at the time A be true it ts held merely a trivial
convenience if hence by definition A implies 6, since the law
would not be in effect. The reasoning again commits, however,
the ordinary fallacy of affirmation of the consequent, long
cautioned about in logic itself. Because in either meaning if A
implies 6 then at a time it need merely be true that not both A
and ‘B, it does not follow that if at a time it is merely true that
not both A and ‘B then A implies 6 in either meaning. That is
the truth tables cannot, by deciding from merely not both A and
‘6 one kind of implication for which the theorems are to hold,
decide the same of an other kind.
The inappropriateness of truth tables deprives of an accepted
procedure for deciding whether a supposed logical principle is
valid. Such a procedure is understandably valued and, as
indicated earlier (p.111 fifth par.), thought to be often impos¬
sible. But it was herein actually supplied by diagrams, and even
when theorems were instead formally deduced, a like per¬
ception that they follow from others was seen utilized. It will be
seen furthermore that, because the required principles are few.
184
\
propositions stated in a symbolic language can be so connected
that by scanning them, without the aid of diagrams, it can be
decided what other propositions follow. The procedure as
suitable for logic as a whole will be displayed upon surveying,
next to preceding propositional logic, customary predicate
logic.
The symbolic language used in that procedure can be quite
simple, as it has as exemplified by form i (p.l79) been for such
as characteristic implication in propositional logic, but is not
also in predicate logic. In it characteristic attribution is bur¬
dened by a complex symbolism, indicative of yet more exces¬
siveness, attended likewise by inaccuracy.
Before considering the symbolism, it may be observed that
the excessiveness and inaccuracy of such as at once assuming
and denying the subject of a statement for supposed proofs in
truth tables also enters diagrammatic proofs accepted in
predicate logic.
As truth tables first listed all truth combinations of A and 6,
things connected in implication, so do these diagrams first
depict all combinations of attributes A and 6 and their
absences, things connected in attribution. How many the things
connected depends on the theorem tested by the method, and
the present diagram depicts the combinations when, in assum¬
ing in the theorem two connections before an inferred one,
three attributes are given. A, 6, and C. This occurs in discussed
transitivity and syllogisms, for the last of which the diagrams are
mainly used, by connecting A with 6 in one premise, 6 with C
^ in another, and A with C in a conclusion.
Correspondingly the circles in the diagram stand as marked
for A, 6, and C, with the area outside of either its negation as
usual. The compartments resulting where the circles overlap, as
well as the other areas when negations are included, then
stand for the combinations mentioned. For instance the area
outside the diagram stands for things neither A nor 6 nor C. An
initial defect lies in these negations, because not all were seen
to exhaust after affirmation all things. Of some things an
attribute can be neither affirmed nor denied.
The diagram would thus be used to verify the transitivity
principle (p.l64). By its first premise it is determined that
nothing is both A and not-B, and the area for that combination
is shaded out (horizontal lines). Likewise by the second premise
it is determined that nothing is both 6 and not-C, with the area
shaded out (vertical lines). It is then observed that every A not
shaded out is a C, to arrive at the conclusion of the principle.
Before more it can here, too, be noted that e.g. the laws of
thought are utilized without proof. Although ‘”Nothing is both
A and 6″, or ” ‘(Some A is 6)”, can be said to mean “All A is
185
‘(BJ” (p.l54 last par.), that the equivalence holds if 6 and ‘(B) are
interchanged is an inference through ‘(‘(B)).
To proceed it can be observed that the unshaded areas of the
circles depict the very premises as depicted in Figure III.7
(p.l63), those areas as standing for the lettered entities sufficing
for the conclusion in the first place. Therein can be found the
inconsistent affirmation and negation of a subject at the same
time. A, the subject of “A is B”, is first signified by a circle, only
to have a portion by shading repealed, and the same happens
to B, the subject of “B is C”. All of this is to say that the con¬
struction of all described combinations, and the subsequent
deletion of some of them, is superfluous, as was superfluous
the completion of all the rows in truth tables, were they of the
desired utility. This may be the more appreciated on con¬
sidering that when many circles or their substitutes are at issue
the diagrams become impracticable.
They were thought an improvement over Euler diagrams
(p.l40 third par.) because seemingly disposing of the several
believed required to disclose most conditions in theorems. The
conditions in the present one were seen to admit four diagrams
(p.l63) for all pertaining connections, held to be wanting
depiction. The now pictured diagram, however, is with that of
Figure III.7 equally without three of those connections, e.g. the
identity of A, 6 and C, and to supply them furthermore con¬
ceptually in the last depiction is, unlike in Figure III.7, ham¬
pered by the prearranged compartmentalization.
The same is accordingly the case in all such diagrams, which
also are not as supposed the answer to disclosing spoken of
existential import (e.g. p.l50). As was remarked, under ”some”
in contrast to “all” the subject is unjustifiably restricted in logic
to an existing one, since while “All A is 6” is taken to signify
that nothing is A and not-6, “Some A is 6” is taken to signify
that something is A and 6. To so construe the meaning of the
words has with other meanings of words been herein
repeatedly pointed out to be an arbitrary matter, in this case
particularly unsuitable for logical purposes, and in this case the
peculiarity is that the sort of existence is not made clear, being
applied to from a living one of a person to a mystical one of a
number. The existence as regards “some” is in any event
attempted to be worked into logical theorems, and in the
diagrams now discussed a symbol for it is usually an x where
some of one thing is premised to by overlapping be another.
In Figure 111.9 (p.l71) the overlap might thus be marked with
an X, and indeed there is no hindrance to marking the more
concise diagrams in this manner throughout if wished, for the
same conclusions as in the more elaborate ones. As a matter of
fact in the first diagrams the existence, if wished, could be
186
\
understood from merely the partial overlap of two circles, with
existence not admitted in full overlap of a circle, in “a\\” prop¬
ositions. This cannot be done in the later diagrams, which for
that matter would require regardless of existence a mark for
“some” to avoid misreadings. Taking for example the premises
“Some ‘A is 6” and “All 6 is ‘C”, chosen in the past for the very
explanation of the diagrams now looked at, they can by
diagrams herein used be pictured as at left. A conclusion is that
some A is ‘C. The other diagrams require the other depiction
here. The only shaded area is, by the second premise, for things
both 6 and C. As a result two partial overlaps of circles remain,
none of which assures that some of one of the circles is of the
other, existing or not, not to mention partial overlaps of circles
with negations, or of negations with negations, these two kinds
of overlaps figuring in the first premise and the conclusion.
There is in consequence no choice but to mark the assertion of
the first premise, that some A is 6, regardless of existence. The
conclusion sought, that some A is ‘C, is, additionally, far from
obvious. The first of these two diagrams, to be sure, has overlaps
likewise not signifying connections, e.g. of A and 6 or ‘6 and ‘C.
But these are exactly the connections understood as undecided
by the nature of the diagrams, which say no more than meant
by overlaps of lettered circles. Should those connections be
decided, appropriate marks can if need be be made, as were
in Figures III.2 (p.l44) and III.6 (p.l60).
The foregoing notwithstanding, existential import is endeav¬
ored to be incorporated into the symbolism of predicate logic.
In it
All A is B
would be symbolized by a less direct
ii. BxJ,
read “For all things x, if x is A then x is 6”, and
Some A is 6
similarly by
iii. (Ex)(Ax & Bx),
read “For something x, x is A and x is 6”.
The first of these—by speaking of things hypothetically A, not
of A directly—suggests that there may be no A, and the
second—by saying that something is A—declares that there is
an A.
The existence in the last case is also signified by the E-like
symbol, its parenthesized unit called existential quantifier and
sometimes read “There exists something x such that”. This
quantifier and the universal one, of the parenthesized x in the
other case, are meant to take the place of “some” and “all” but
they refer, as seen, to x as a new subject, with A and B regarded
as predicates, in after the quantifiers a reverse notation
187
corresponding to one of mathematical functions.
The needless addition of these subjects is also argued
required so as to distinguish them from those predicates,
attributes. But as expounded in the first chapter (ca. p.38
through rest of chapter), no such distinction need be made.
Things are indentified by attributes, and to say that a thing
which is A is a thing which is 6 is no more than to say that an
A is a 6. And the addition of, generally quantified, variables
like X for things as subjects is not an encumbrance alone. It is
held that quantification of predicates like A, which normally
are quantified subjects, belongs to logic of a higher order.
The ordering is related to the spoken of hierarchies of
languages and classes, in belief that predicates somehow
belong to a sphere beyond subjects, with a continuing order of
predicates of predicates. The quantifiers discussed, however,
are meant to already pertain to predicates, if A or 6 of the
examples be so named, (x) and (Ex) speak in effect of all and
some A respectively.
Their quantifying of things x instead results rather in un¬
wanted meanings. “AW” is not meant to be about singular
propositions, those of a single A. Yet since the reference is to all
things, hypothesizing some to be A, there may only be one A.
Similarly ”some” by referring to things in general, of which
some are A, does not mean that all A may not be held to be 6.
The hierarchical quantification of A or 6 as predicates, as do
other quantifications, restates for the matter of that the
understanding that the variables, as connoted by their name,
stand for all things in the situation on hand. The axiom of
mathematical induction (p.l38 fourth par.) has thus been given
in the form
iv. (A)(a)(b)((A0 & (Ao ^ A(a + ]))) ^ Ab),
read “For all properties A, numbers a, and numbers b, if zero
is A and if when a is A then a + Ms A, then all numbers b are
A”. By quantifying all properties A, predicates, it is meant that
the principle holds whatever the property may be. It is not
meant that concerned are, as would be by the usual quanti¬
fication, all things of a particular property, with A alongside
standing for all properties. That it when used does, as in the
preceding formula iv, is already understood by A as a variable.
The quantifier is hence redundant if not misleading, and so
are the more frequently used ones for numbers. In iv it is
equally understood by variables a and b that they concern any
number in their place. The same redundancy occurs further¬
more in the first of the two standard quantifications displayed
(facing p., ii). By the variable x is already meant everything in
the specified condition.
The second of those quantifications (facing p., iii) is cor-
s
188
respondingly awkward. Not only x but A and 6 are variables
which, as in theorems, stand for anything in whichever the
situation. The traditional ”all” and “some” does as indicated
above not quantify those variables, but the specific^thing they
may among many stand for. In “All A is B” A and 6 speak of all
kinds of things in that relation, and “all” speaks of all of a kind.
In “Some A is 6” A and B retain the universality of kinds, but
“some”, referring to those of a kind, does not. Accordingly, to
use the last quantifier, along with the first, instead for variables,
let alone ones more general than A and B, brings further
disarray.
The use of the existential quantifier is also supposed justified
by yet an additional workload imposed on the word “some”. It
has to do with relations other than that one thing is an attribute
of another, relations alluded to {p.l77 last par.) as basis of some
systems of logic, although all deduction, as observed, can be
achieved through the simple connections discussed.
One of the relations between two things would be symbol¬
ized by
V. (x)(Ey)(Rxy),
read “For all x and some y, x has relation R to y”. The relation
parallels the common universal proposition “All A is B”, in
which “some B” is tacit, which it can be with justice, since in the
fact that B, y, is connected with each A, x, is implicit that some
B, y, is, while no commitment is made as to whether all or
another exact amount of them is in some way involved.
Despite that parallel, the relation Rxy in form v (this p.) is
contrasted with those of Ax or Bx in ii and iii (p.l86) by holding
it to be dyadic, taking place between two kinds of things, and
the others monadic, concerning only one kind of thing. Subject
and predicate, represented by Ax or Bx, were observed,
however, to likewise characterize two kinds of things accordable
equal status, the two being of the relation that the first, or some
of it, is included among the second.
In this relation furthermore the quantifications, thought to
contrariwise serve the same function as in the relations now
considered, do not. They were observed to redundantly or
unsuitably quantify variables. In the present relations, however,
the quantifications, though seeming to be of variables, are
ones of all or some of a particular kind of thing represented by
them. As in “All A is B”, in “All x have relation R to y” all of a
kind rather than all kinds referred to by A or x are meant. The
appearance is false because x and y are used in place of A and
B, in belief that it is not attributes that are dealt with.
It is thus erroneous to think that the relations presently
considered expose an extension of the quantification used in
predications like ii and iii and numerical iv. The present quanti-
189
fications are rather used in holding that through them, partic¬
ularly those for “”some” further logical distinctions can be
expressed, as suggested on the last page. In scholastic logic it
was noted that a statement of the form “”All A have relation k
to some 6″ can be taken to say either that each A has the
relation to a different 6, or that each A has the relation to the
same 6. The attempt was accordingly in modern logic to
express the distinction by arrangement of quantifiers. It was felt
that the second preceding interpretation, the sameness of 6 for
each A, is clearer by speaking of A and 6 in reverse order, “”To
some 6 all A have relation R”.
In the symbolism,
vi. (Ey)(x)(Rxy),
“For some y and all x, x has relation R to y”.
But the ambiguity of words does not obligate logic to
symbolize every meaning. In the Middle Ages new logical
consequences were sought by studying the subtleties of
language, but as in other comparisons of words with fact,
diversities in language do not signify diversities in logic. It
would be observed about relations v and vi that the first follows
from the second. But like other deduction this is true by the
investigated standard logical principles. Since any y is by the
meaning of “”some” identical with some y, what is true of any
y is true of some y. To wit if it is true that all x have relation
R to any y, e.g. the same one, then they have the relation to
some y.
In addition the arrangements of the quantifiers did not
achieve the intended statements of sameness. The relations
with the remaining arrangements are
vii. (x)(y)(Rxy)
and
viii. (Ey)(Ex)(Rxy).
Exchanging somewhere x and y is not required. If exchanged
either in the quantifiers or next to R, the relation will be pas¬
sive, e.g. in v to all of one thing some of the other will have the
relation; but the nature of the relation is beside the point, the
interest being that any hold between things of quantifications
as given. And if x and y are exchanged throughout, the result
is merely a renaming.
The passivity of the second variable after R indicates that by
vii is not meant mutuality of the relation, as in “A is identical
with 6″; such a relation would be given by adding an exchange
of each x and y, a new statement since not independent, but
not required here since of the same structure. Statement vii
rather says that each x has the relation to each y, in a usual
listing of all combinations of the quantifiers, as in previous
combinations not necessarily required for the purpose. For the
s
190
purpose of signifying or not by the arrangement that the same
things of a kind are of a relation to things of another kind vii is
not needed, for when all are of a relation to all, the same of
each is always part, whatever the arrangement.
In contrast the notation employed is not adequate for the
same purpose when some are of a relation to some, as in viii,
which parallels the other common proposition, the particular
”Some A is 6″. In viii, first quantifying y is to mean that the x
have the relation to the same y. When the y may not be the
same, they would accordingly be quantified after the x. Then
the X, coming first, should be the same with respect to the y. As
can be confirmed by the diagram, however, when the y are the
same for the x, the x are for the y, and conversely of course, and
hence it is never signified by the symbolism that the y, or the x,
may not be the same, as they may.
This result can as elsewhere be reached by customary
deduction. By saying that the same thing is of a relation to a
number of some other things is meant the same as merely
saying that the thing, without speaking of its sameness, is of the
relation to each of those things. Correspondingly, to say that
some X have a relation to the same y means that each of those
X has the relation to the same of each of those y, or by the last
sentence simply to each of those y. By the same sentence
therefore each of those y are of the relation to the same of each
of those X, or simply to the same x. And since if one is the same
for the other then the other is the same for the first, by trans¬
position if one of these situations is not the case then neither
is the other.
The additional overburdening of the term “some” by having
it express beside spoken of existence sameness becomes thus
a further barrier to its ordinary use, of quantifying a subject
as not necessarily of “all”, without regard to the sameness or
existence.
Existence was seen (e.g. p.l49 last par.) expressed as a matter
of course independently of quantification, and plain significa¬
tion of it is in fact available through customary symbolism. In
propositional logic, in which variables stand for propositions,
their truth is expressed simply by stating the variables. Variables
were further, in theorems about them, seen applicable to any
thing, its truth replaced by more general existence. And since
the same variables were seen usable in predicate logic,
existence in it can be expressed in the same simple way, by a
variable alone. Quantification will be seen equally irrelevant
to above sameness, as well as to distinctness, whose knowl¬
edge may be as desirable. Quantification can for that matter be
largely done without altogether.
It was already observed how quantification of variables by
191
”all” is redundant, and how “some” is implicit in considering
things, possibly negations. The latter quantification is usually
made explicit only when the former is not necessarily the case
but may otherwise be understood. It would be said “Some coal
is browm” because “Coal is brown” would be understood to
refer to all of it. This understanding suggests that not only
variables but all universal, all words referring to more than one
thing, could do without universal quantification, especially in
a standardized logical language, where ambiguity is avoided.
In such a language the universals, names for particular kinds
of things, were seen not even used but represented by vari¬
ables, “All A is B” nonetheless not speaking of all things of the
variable but all of the particular thing represented. There is
obviously no problem in comprehending this, and neither
should there be if the quantification is omitted, as done in
implication. In it “A implies 6” is understood to speak of all
things of a kind as leading to some others, the variables in
addition denoting all kinds to which the sentence may apply.
Universal attribution can similarly be expressed by
A is 6,
as was done on occasion herein before.
There is no need to worry about so-called indefinite proposi¬
tions, like “Man is bearded”, in which it may not be certain if
all or some are at issue. It is the uncertainty by which things are
qualified as some, which can for logical purposes be done by
edict. If the speaker knows of the universality, the qualification
can be omitted. The unqualifiedness of the above displayed
expression also makes it useable for not only universals but
individuals, as was implication in a broad sense.
As evidenced in this section, logical principles are often valid
for multiple and single subjects alike, and the sole distinction
will be disclosed to be that of the last section, that the com¬
plementariness of 6 and ‘(B) holds only for a single A. It does
because an attribute denied of a single thing, as observed (e.g.
p.l43 second par.), is meant to be so fully, but not so with
universal propositions. As in them, in their internally negative
opposites the interest is in inclusive principles, not following
from negation of internally affirmative ones, whereas the
internally negative of a singular proposition is to merely state
that the subject is other than stated by the internally affirmative,
in equivalence to its negation. The equivalence was seen
furthermore not to follow for the ordinary negative attribute,
herein symbolized by ‘6 as compared to ‘(B). Similarly to the
case of the universal A, the law that a single A is either 6 or ‘B
does not hold unqualifiedly.
Most notable, since where the difference occurs it is of
negated propositions, it is of little importance in customary
s
192
deductions. For these deductions affirmative propositions, ones
not negated and to which negated ones were in the last section
seen convertible, will be found the ones of use.
These propositions were mentioned at the start of this section
as of more interest, and they are the ones spoken of (p.l83 last
par.) as connectable in symbolic language, so as to enable easy
inference. A suggestion of such inference is provided by the
pursuits in general of man or animal. An affirmative proposi¬
tion signifies knowledge that one thing may lead to another,
which in turn may be known to lead to a further one and so
forth, for inference of connections between remoter things.
Deductions of this nature are made abundantly in all walks of
life, including consciously deductive endeavors like geometry,
without necessarily awareness of their conformance with
logical principles. To avail oneself of the few basic principles
can thus be regarded as preliminary to the predominance of
other deduction, and it is primarily for that deduction that a
simplified symbolism is of utility. Basic principles were observed
to require conceptual substantiation instead.
Should it be insisted that they, too, be amenable to the sym¬
bolism, i.e. that the difference between universal and singular
propositions be given by it, the difference can be additionally
symbolized where of account. That is in an unquantified is
6″ A can normally be of any quantity, adding suitable sym¬
bolism if the quantity is wished specified. The difference for an
individual A a logical exception, it is especially pointless to
mark it where not needed.
It may be remarked that the difference between the singular
and universal is thought to go farther. The belief was mentioned
(p.l61 fourth par.) that of something predicated of a subject
there cannot only be one. Accordingly while it is traditionally
acknowledged that “”All A is 6′” implies “”Some 6 is A”‘ the
inferred, predicating A, would be thought not to hold if there
is only one. The belief was at that mention noted faulty, and an
added reason is that when a single A is a 6 then some 6 is in
fact the A, belief notwithstanding. The mistake is grounded on
the discussed notion that subject and attribute are distinct, with
the subject held an individual and the attribute a universal,
although there can be many of the subject, or one of the
attribute. The distinction between the individual and universal
is further grounded on the discussed disputes about universals
as independent of the individuals partaking in them. But
universals, the words concerned, were observed meant to refer
to the very individuals.
Omitting in accordance with the foregoing the universal
quantifier to state “”A is 6″”, a symbolization of the same,
analogous to that of implication (p.l79, i), can be
193
ix. A — 6,
It has apart from its extreme simplicity the advantage of
providing for the mutuality of the relation, for identity, in its
accustomed double line,
X. A = B.
Mutual implication, equivalence, may similarly by signified
by the likewise known double arrow,
xi. A B.
Turning to particular propositions, those represented by
”Some A is 6″, it will be recalled that they can be broadly taken
to say
A can be 6.
The linguistic utility lies in a corresponding symbolization,
which can be
xii. A/—/B,
obviating quantifiers as prefixes, to facilitate the mentioned
(e.g. facing p. second par.) connecting of premises to derive
conclusions.
The strokes indicate qualification of the relation, as well as
its reciprocity by their duality again.
In implication “some” was noted to be usually expressed
only by negation of an opposite principle, but if need be the
statement “A can imply 6” can be considered, symbolized as
A/^/B.
The concern in the preceding regarding these symbolisms
was the quantification of the subject, historically the one
attended to, no doubt because of the interest in laws or their
absence. They were observed to be about whether all of one
thing are connected with another, the other mostly of interest
without specific quantity. This may be held a reason why
“some” is left tacit for it. Understandings of quantifications not
made express extend to “all” as well, as indicated of variables
and other universal, and they include moreover much more
than would seem from the above.
There it is not explicit whether the displayed propositions
symbolized are to stand alone or be part of others, such as
logical principles. In the forms symbolized they indeed would
be parts of others. For example in formula x it is not stated that
all things are identical with all things, but as part of a longer
statement the reference may be to any two identical things. The
formula can also be viewed as representing any complete,
likewise symbolized, deductive statement of identity, e.g. of a
mathematical equality. The point was to assert disposability of
the quantifications at issue in either case.
It is the first of these cases which is usually considered,
without the thought that a more complex condition as a subject
about which something is deductively true might also be
s
194
quantified. Yet it is the last case that truly pertains to all
instances of something. For illustration can be taken the law of
transposition as symbolized in Table 1 (p.l80),
xiii. (A B; ^ (‘B A).
Were it not the custom, as an instance of what is understood,
not to quantify the A in implications, all A and all ‘B would be
specified to be concerned, as done in attributions. It would not
be specified, however, that all implications are referred to by
the first one in the law, as likewise not done when laws are
attributions, e.g. equations. But in present xiii it is only that first
implication which unexceptionally applies to all instances. A
and ‘B as about a kind are fully of all of it only when about
timeless principles, while the law also applies to limited
occasions regarding a kind. Still less unexceptional are the
variables, noted in various usage also needlessly quantified.
They on first use in xiii are only of things in the relation given,
and on second use only of negations of these.
This does not take away entirely from the spoken of inclusive¬
ness of variables and other universals. They were earlier spoken
of as about all of the named things in the situations given, by
which they are thus restricted. But it is still all that are so mainly
understood, though only some might. It is understood that
above xiii, unlike Theorem III.9 (p.l71), refers to all of certain
things under A when in the relation, not to some. Much more
is understood, not merely in the symbolism, whose meaning
can be codified for easy grasp, but in correspondihg standard
language, lacking the uniformity.
A statement in that language of the above law, more in
keeping with xiii than its form in Theorem III.5 (p.l59) and
without even variables, may be ”\i one thing implies another
then the negation of the second implies the negation of the
first”. In concord with the foregoing it is understood that the “if”
applies unqualifiedly to all cases of the condition it speaks of,
that, further, the “one thing” and the “negation of the second”
apply to all cases of them at the given occasion, and that
“thing” and “another” apply to all kinds of things in that
relation to each other. In general by every universal, every word
denoting more than one thing, as well as by every combination
of them describing something universal, are in usage meant all
of those things within the given restrictions, of which there may
be none. If all are not meant, they can be expected to be
quantified. Variegated language has exceptions as said, with
indefinite propositions unclear about inclusiveness. Cor¬
respondingly inclusiveness, that all of a thing within perhaps
a limit are meant, is often specified. But there is no need for the
exceptions in an unambiguous symbolic language.
That is why it is redundant to speak in it of all, that meaning
195
already contained in it where not otherwise quantified. The
redundancy is particularly notable in mathematics, since in it
quantification is in practice most frequent. An example was
furnished by iv (p.l87), and another, more like the preceding
xiii in variables that are restricted, can be presented. It is of the
discussed commutativity of multiplication, the principle quan¬
tified as
xiv. (a)(b)(a X b = b X a).
There is no need for the quantification, since by the meaning
of the variables the statement already speaks of all numbers a
and b multiplied, and to be remarked is again that while the
universally quantified variables are in fact restricted to those
multiplied, the only fully universal part, referring to all in¬
stances without restriction, is the first multiplication as a whole,
though not likewise quantified.
Variables can of course be fully universal too. As appears
evident from what was said, those things are generally meant
fully universal that are subjects of statements, possibly of ones
part of other statements. This may again be attributed to interest
in principles, as was original quantification (p.l93 seventh par.),
and in deductive principles that meaning again can be con¬
stant, dispensing with the quantification. Its omission was in¬
deed seen inadvertent in principles like xiii and xiv, where the
subjects are composite. But when the subjects are variables
alone the practice can be forgotten. The same does not happen
when the statement is an implication, as suggested previously.
In the supposed theorem of the other table (p.l82),
XV. A ^ (B A),
under A as subject all things, or all propositions, without restric¬
tion are understood, as are under 6. But variables as subjects
in mathematics would like other variables be quantified. This
is more or less concealed in iv (p.l87), where quantified sub¬
jects are a and b. Regarding it as well as xiii (facing p.) and xv
(this page) it may be observed why subjects of statements part
of other statements can be more inclusive in conclusions than
in premises.
In xiii A concerns all of a kind, whereas in xv 6, not being a
repeat, concerns all things altogether. The reason is again
found in what is meant. Principles are meant to assert some¬
thing about each of the subject individually. Thus in xiii each
instance of the premise implies individually the conclusion.
Hence the subject within the premise itself is only about all of
a kind as an instance named by A. However, in xv, since a
conclusion in entirety applies to each instance of the premise,
6 names at once all kinds of things, namely all things since all
of each kind are at issue.
It may also be noted that while in xiii A, though unquanti-
s
196
fied, names each time all of a kind, in iv a, though universally
quantified, names each time a single number. A number can
of course have many instances. A person has two eyes, two ears,
etc. But as clear now, quantified are not all those Instances, but
all numbers. There is hence as suggested (p.l87 fifth par.)
additional inconsistency in quantification, which was designed
for, in purport at least, not variables, but particular things they
stand for. It might in fact with equal justification be required
that particular numbers in similar use, too, be in mathematics
universally quantified, due to their many instances. But as
explicated, there is no need for such quantification in the sym¬
bolic language, including superfluously where in fact all things
represented by a variable are referred to, as in the case of b
in iv.
There the variable stands for the subject of the inferred. In an
example of a mathematical variable for the subject of a whole
statement, similar to last xv, that every number equals the
product of two numbers, perhaps of itself and 1, would be
quantified as in
xvi. (a)(Eb)(Ec)(a = b x cj.
The quantifications are not only superfluous but in accord¬
ance with the foregoing wrong with regard to ‘”some”, the
existential quantifier. “Some” is viewed as not necessarily “all”
and “all” is applied, to repeat, also within restrictions. In
present xvi b and c are restricted by the product that equals a,
but although to accordingly quantify them would no doubt
confuse they represent all numbers of that product, not merely
some. It is rather the product that, at least regarding the number
it equals, does not vary, and of which it might be said that
some, not any, equals any number.
Even of that product or any other predicate, however, can it
be said that, redundantly, all pertains to the subject that
pertains to it, being restricted by it alone. In contrast as sug¬
gested (p.l94 third par.), whereas what amount is in question
regarding the predicate and other things not a subject is
dependent on something else in the statements and hence
cannot be less than determined by it, a subject is in this regard
independent and hence can refer to less than necessarily all,
it can refer to only some. Accordingly quantification by “some”
in the sense of allowing less than “all” is suited again for
subjects only.
The appropriatness of that quantification is thereby virtually
eliminated in mathematics. Mathematics is not apt to state
something about an unclear some of a subject. Even probability
or improbability alone are weighted toward one side or the
other, and normally the degree is specified. The unsuited use
of the quantifier has due to its mistaken association with
197
existence furthermore led to uncalled-for suppositions of
existence. Many mathematical principles are accordingly called
existence theorems, although merely stating that with certain
things certain others are connected, the ones quantified
‘^some”, not as existing, but because of vagueness of amount.
The statements as instanced by above xvi can to boot be viewed
as universal ones, speaking ‘”all” in the subject, which in
contrast to “some” regarding it is not held to imply whichever
existence.
“Some” was observed to in addition be called upon to try to
denote sameness (e.g. p.l89 first par.). The function was seen
deficient, and as may be gathered from the preceding, the
relations at issue can if needed be expressed without the
quantification. Supposing they are of mathematics, they are
formulated below in a not unfamiliar way. The relation of v
(p.l88), in which the sameness of the related to is undeter¬
mined, can be
xvii. a R b.
It can be read “All numbers a have relation R to a number b”.
The relation of vi (p.189), in which the same is related to by
all, can be
xviii. o R b, (c R dj (a R dj,
read “All numbers have the relation to a number, and if a
number c has it to a number d then all numbers do”. If by vi be
meant, as not made clear by the quantification, that the x may
also have the relation to different y, then the strokes for “can”
(p.l93) can be used for
xix. a R b , (c R d)/-^/(a R d).
“All numbers have the relation to a number, and if c has it to d
then can all numbers”.
Interposed can be a state of v not covered by the quantifica¬
tion and perhaps more likely than the last two, that all numbers
have the relation to a distinct number, among them earlier
spoken successors if in question (pp.137-138).
XX. a R b, (c R dj ‘(‘c/R/dJ,
to read “All numbers have the relation to a number, and if c has
it to d then it is not true that a number not c can”. The negation
sign for logic as well as the strokes are made here use of for
further purposes. A usual notation, used for successors and
slightly lengthier, would be o R b, (c R e)(d R e) (c = d).
Corresponding to vii (p.189) is
xxi. a (b R a),
“All numbers a imply that all numbers b have relation R to
them”.
As to viii (p.189), seen to fail in distinguishing the sameness
from its absence, it can first be given in an indeterminate sense,
in likeness to xvii above.
s
198
xxii. o/R/b,
“A number a can have relation R to a number b”. With, similarly
to xviii, the relation to the same number, viii \A/ould be
xxiii. a/R/b, (c R d) ^'(‘c/R/d),
“A number can have the relation to a number, and if c has it to
d then it is not true that ‘c can have it to d.” If similarly to xix
those a may also have the relation to different b, viii is
xxiv. a/R/b, (c R d) (‘c/R/d),
‘A number can have the relation to a number, and if c has it to
d then ‘c can” And if similar to xx the relation in viii be to
distinct numbers,
XXV. a/R/b, (c R d) -^'(‘c/R/d),
“A number can have the relation to a number, and if c has it to
d then it is not true that ‘c can”.
The point in the last pages was to illustrate the needlessness
of quantification where used in logic and related fields, and to
generally establish a simple logical notation by which deduc¬
tion can be performed as mentioned. For that deduction logical
principles are likewise simple, and as evidence a few
remaining ones are hereupon added to the former, together
with further comments.
The entities of which logical principles were herein observed
composed comprise basically no more than things viewed as
undivided and represented by variables like A, and composites
of things as connected by shared placement represented by
symbols like the arrow for implication, with possible
negation of either whole units or things connected in them.
Symbols were added for these negations, which can be held to
include the concept “some”, or “can”, as negation of an
opposite principle and symbolized on its own. Also symbols like
parentheses were added to delimit units, and of importance
now is that with the entities thus simply understood they can be
easily listed, in order to determine from which follow which.
The list of undivided entities can be small, corresponding to
those of the laws of thought as A and A, and perhaps ‘(A), “A,
‘(A) and ‘(‘(A), more of the same leading to needless repeti¬
tion. It is not difficult to see, as by diagram, that from any of
these follows only what the laws of thought prescribe. No other
undivided entity 6, or composite A -> 6, or negation regarding
these can be inferred.
A word about the laws of exportation and importation,
which came into account with modus tollens (p.l62). They
respectively state that (A & B) C implies A (B C), and
the converse. As a result modus tollens could be stated as ‘6
(fA B) A) or, exchanging variables and negations, A
((‘B -> A) -* B). There appears hence a consequence of A not
part of the above. Instead, however, 6 was viewable as a
199
consequence of A combined with ‘B —* A, and the last unit is
an additional assumption in both cases. Therefore for sim¬
plicity’s sake only certain such arrangements will be specified.
Since a statement was seen to mean the same with parts im¬
ported or exported, if a new letter, e.g. 6, appears in the con¬
clusion it should, with any unit it belongs to, be imported to the
premise. The procedure is possible, since the new letter, with
the unit, was indicated not to follow from a premise, but must
be posited in a conclusion too, as above. The old form of modus
tollens, ((A B) & ‘B) —* A, would correspondingly be restored.
By the same token ‘6 of the premise in this form can be
exported to the conclusion, making for, as discussed, trans¬
position, (A B) (‘B -* A). The object is to state as clearly
what follows from units of two letters as from those of one.
The number of two-letter units used was seen more extensive
than the other, in particular with respect to attribution rather
than implication. It included in this respect not only the
universal form, A — 6, and negations of the whole or part of
it as well as reverse sequences, but also the particular form,
A/—/B, and negations and reverses regarding it. For the
deductive purposes mentioned, however, much of the
negations and reverses can be neglected. External negations
especially were for these purposes said (p.l92 first two pars.) to
be convertible into their affirmative equivalents, by the laws of
complements, an opposite converting not here required. And
internal negations and reverses need not be considered in the
units as independent premises, since the variables apply there
to negations as well and with indifference to sequence. Con¬
sequently the main interest here is what conclusions not ex¬
ternal negations may be drawn from the two in this paragraph
symbolized two-letter units alone.
Such inferences were furnished in the first and last theorems
of this section (pp.l59, 171). In both cases the inferred was a
reverse connection, the second found derivable through the
first, and in accordance with the second of those principles
another such conclusion, mentioned before (p.l92 fourth par.),
can be drawn from the premise of the first, mainly if of attribu¬
tion. It is that
When A is B then some B is A,
because to speak of, all, A is by definition to speak of some A.
This principle can as easily be confirmed by a diagram of
A — 6, notwithstanding that it has been denied of not merely,
as noted with its first mention, a singular A, but also a universal
one, in conformity with the contention that whereas “all” may
not signify the existence of the subject, “some” does. It need
not be repeated that the distinction is uncalled-for, and due to
it it is evidently also denied that When A is B then some A is B.
200
This added principle may also be considered, although of less
account than the displayed preceding one, inasmuch as know¬
ing specifically that, all, A is 6 makes the vaguer inference from
it of little interest.
In consequence of transposition, that when A is 6 then ‘(B)
is ‘(A), follows on the same grounds also that when A is 6 then
some ‘(B) is ‘(A) and some ‘(A) is ‘(B). Put very simply, from “A
is B”, as to be seen, follows no more than, excluding external
negations, ” ‘(B) is (A)” and corresponding forms of “”some” for
both connections in both directions.
The last of these forms.quoted, “”Some ‘(A) is ‘(B)” or “‘(A)
can be ‘(B)”, is noteworthy as a consequence, because again
implying that, revealingly as earlier expounded, not everything
is 6. If everything were, then not only all A but all ‘(A) would be
B, with that consequence false.
That there is no other consequence of “A is 6” can first be
observed for any in question not a connection between A and
6 and their negations. As in the case of A alone and negations
regarding it, it is evident, as by the diagram, that no other entity
C or unit containing it can be inferred. Only units with A or 6
follow. Moreover it cannot be a unit with one of them alone,
which would signify that when “A is 6” is true then there is such
a unit since, as in other implications, from the truth, the
existence, of the premised follows the existence of the
concluded, either perhaps a negation. But the connection
between A and 6 was explicated not to require the existence of
either, which therefore cannot be inferred. And a connection
between units of the same letter, viz what follows from any one
of them, was observed (p.l98 next to last par.) derived only in
accordance with laws of thought.
Only units used with A and 6 connected remaining, they
beside the above are A — ‘B, B — ‘A, ‘A — B, ‘B — A, A/— /’B,
‘B/—/A, A — ‘B, B — A, ‘A/—/B and B/—/’A. Keeping in
mind that the present demonstrations apply to both attribution
and implication, the negation signs are somewhat liberal.
A should strictly be ‘(A) in attribution. But since the issue is
what does not follow from “A is 6″”, or “A implies 6″‘ if some¬
thing does not follow for ‘(A) it follows the less, by what was
said, for A. Thus among the few more things determined above
to follow, those regarding ‘(A) do not for ‘A, unless transposition
applies. In that event as in others of limited negation it is not too
difficult to confirm that no more follows than with negation
unlimited.
With that understanding of the preceding list of ten sym¬
bolized connections, the first six do not only not follow from
A — 6, but are impossible, as can be confirmed, except for the
third perhaps, by the last diagram. The third connection once
201
more expresses that not all is meant to be B, and its denial can
be deduced from above following A/— /’B by complement law,
by which the rest of the six are from the known likewise denied.
The remaining four do not follow in the more usual sense, by
which both they and their negations are possible. The nega¬
tions of the first two are by complement law equivalent to the
other two, and conversely, and hence if all four are possible
none follows, since any that would would contradictorily make
impossible another. The diagrams at right, complying with the
assumed A — B, disclose each possibility. Specifically, the
upper depiction is of B/— /A, the lower of its negation in 6 —
A, and A/— /B and A — ‘B follow respectively also by standard
deduction.
Coming to ”Some A is B”, or “A can be 6”, as a source of
inference, less follows still. Only the converse by Theorem 111.9
(p.l71) does. On the same grounds as in the previous case, only
the connections with A and 6 in each need as consequences
be considered, despite the faulty position that the existence
of A and 6 individually is implied. The units to be viewed are
A – ‘B, B – A, A – B, ‘B – A, A – B, B – A, A – ‘B, ‘B – A,
A/-/’B, ‘B/-/A, A/-/’B, B/-/A, A/-/B and ‘B/-/A.
The first two are under A/—/B impossible, as seen by its
diagram, or is deducible from the known by complement law.
Of the remaining twelve the negations of the first six are by the
same laws equivalent to the other six, and conversely, and
hence again if all are possible none follows. The diagrams at
right, complying with the assumed A/—/B, disclose each
possibility. Specifically, the upper depiction is of A — B, the
lower requires its negation in A — ‘B and ‘B — A, and all other
of the twelve units follow also by standard deduction.
With respect to either of the two units, “A is B” and ‘A can be
B”, of which the consequences have in the foregoing been
explored it may also be asked whether if combined with other
units with A or B something follows that does not from the units
separately. That something can was in fact seen regarding “A
implies B” and ‘B, in modus tollens. But this combination and
resulting A were seen transformable into transposition, in
which the first unit alone implies a like connection between the
other two. The same transformations were seen possible with
any two units implying something, and when one of them is
thus an above tie between A and B and the other of an
independent letter, what is implied by the last unit transported
was already determined. Should there be more than one such
independent unit, the same transportation and result apply. The
question then is what follows from only combined units of the
ties regarding A and B. By testing again through diagrams it can
be established that whatever the number of the units what
202
\
follows is still in accordance with the preceding. The testing is
fairly brief, since the units with A and 6 are soon exhausted.
Consequences different from a combination of units than
from them separately were also seen regarding complements.
Thus from “”Not all A is 6″” and “”Not both some A is 6 and some
A is “(B/” together but not separately follows “”All A is ‘(BJ”, and
likewise if B and ‘(B) are interchanged. The same was found not
to hold when there is one A. Then the denial of “A is B” is
equivalent to “A is ‘(BJ”, and conversely. This was the only
logical difference between one and more A noted, and it may
be confirmed as in the preceding that this is the only difference,
with the externally unnegated connections here considered for
deduction usable without regard to how many the A. More
specifically, with only one A the same can be seen to follow
and not follow from both “A is B” and “A can be B”. By the latter
it may seem that e.g. more is impossible, as that some A be ‘B,
but since by not stating that A is B A”s singleness, or its “”all””, is
not known, “”some”” not stated otherwise, the same possibilities
remain.
In this survey of conclusions that follow or not from premises
of two letters the letters, perhaps negated, were not treated as
separate units, and it can further be asked what follows if they
are. What follows from each was determined in the laws of
thought, and no more can be seen deducible. Connections of
premise with conclusion as well as connections within either
were observed to consist in placement of one thing with
another. And when in a premise two letters or their negations
are not thus known to be connected, no conclusion of such
placement can be, by diagram if wished, found reached other
than for the separate letters by laws of thought. The different
letters obviously could in absence of more knowledge be
connected in any of the above ways.
The same inconclusiveness occurs when assumed are three
or more letters of such independent units, simply because since
no connection between them is known, any is possible. And
when only two out of three letters are connected as described,
a similar situation takes place. From A ^ B and C together
follows no more than found to follow from them independently,
with no connection between one and the other. Accordingly, to
find what new conclusions can be drawn from three-letter
premises, all three letters must be connected.
The by now most familiar such connection is the universal
A — B coupled with B — C, and one using the particular
A/—/B coupled with likewise B — C was in certain forms
mentioned also (p.l64 sixth par., p.l86 first par.).
The inference from the first of these of the transitivity, A — C,
was furnished in Theorem III.7 (p.l63) dr the like principle for
203
attribution on the page after, not to mention like visual
confirmations elsewhere (pp.l40, 184). A similar transitive
inference from the other combination was indicated with its
early mentions, namely that
When some A is B, and B is C, then some A is C.
This inference was noted to in fact be in accustomed manner
deducible through the other. By the other any considered thing,
for convenience named A, that is 6 is C, and hence since in the
second combination a certain thing considered, some of any
other A, is 6, it is C.
There is a further transitivity implied by the first combination,
apart from like implications by both of them on “‘some”. Both
imply “Some A is C” and hence “Some C is A”. But the first also
implies ” ‘C is A” not only because A is C, but because ‘C is ‘6
and ‘B is A, i.e. by transitive law reversed. The significance is
that, knowing of transposition, in a sequence of implications
from say A to Z it is as facile to infer by transitivity that ‘Z iniplies
A as that A implies Z.
Put simply again, a consequence of (A — BJ(B — CJ is transi¬
tive A — C, the other results following by mainly transposition
or conversion as before, or by transitivity itself. By that
transitivity was also seen A/—/C deducible from (A/—/B)(B —
C), and inasmuch as it may not be immediately clear what
combinations result in transitivity, a simple rule can be applied.
If each of a thing, 6, of which there may be one, is of another,
C, then anything. A, which is of 6 is of C. The deduction is in
ordinary situations exemplified by the several times mentioned
act of inferring an instance from a principle, B — C being the
principle and A — C the instance, without normally bothering
to state that A is a 6.
To verify that no more follows from the preceding configura¬
tions of three letters than the spoken of, it can again be noted
that neither a unit of one letter, perhaps repeated within the
unit, nor any with a yet unused letter does, as happens however
many letters are assumed connected. The first does not follow
because, as pointed out (p.ZOOfourth par.), the existence or not
of a thing A does not follow from the connections, and
connections pertaining to it alone are decided by laws of
thought. And the second does not follow because, nothing
being known about the unused letter, anything about it is
possible. For a similar reason, with connections of A with B and
B with C no other connection can be concluded of A with B or
B with C as a result of the combined premises. If assuming e.g.
A — B first then, nothing being known about C, any connection
of it with B is possible. This means that whatever the connection
of A with B there can be any connection of B with C, and of
course the opposite. But this means that whatever one con-
204
nection, or a series of them, none is implied of another, the
other still of any enabled by its own nature.
In those configurations hence only connections that follow
regarding A and C need be looked for, others having been
determined.
Of these connections the ones not found implied or
precluded by (A — B)(B — Q, through transitive A — C, are, as
with A — 6 previously, A — ‘C, C — A, A/—/C and C/—/A.
Since one half contradicts the other, if all are possible none
again follows, and the first two diagrams here, complying with
the assumed, disclose that result.
The connections not found implied or precluded by
(A/— /B)(B — C), through transitive A/—/C, are the also
familiar A – C, ‘C – A, A – C, C – A, A – ‘C, ‘C – A,
A/-/’C ‘C/-/A, A/-/’C C/-/A, A/-/Cand ‘C/-/A. That
these contradictories are possible and hence do not follow is
disclosed by the further two diagrams, which comply with the
assumed.
Considering the connections of ‘”all” and “some” between
A and 6 and between 6 and C without change of direction
within the connections, the two configurations remaining are
(A — B)(B/—/C) and (A/— /B)(B/— /Q), both pictured next.
The question is whether any connection regarding A and C
is implied by them.
The complete list of the connections is A — C, C — A,
A – ‘C, ‘C – A, A – ‘C, C – A, A – C, ‘C – A,
A/-/’C, C/-/A, A/-/C, ‘C/-/A, A/-/C, C/-/A,
A/—/’C, and ‘C/— /A, and since one half contradicts the other,
if all are possible none follows. The further two diagrams
comply with the assumed in both configurations and disclose
all those possibilities. Therefore none of them follows.
If configurations of other directions are considered, the re¬
maining two are (B — AJ(B — CJ and (A — BJ(C — B), because,
due to the indifference of letters or convertibility of “some”,
other changes in direction result in forms already given. For
example (A/—/B)(C — B) results in (C — B)(B/— /A), of a pre¬
ceding configuration.
Of the two added forms, pictured next, the first one is seen
to imply A/— /C. The same can be deduced in standard manner
by the last two principles displayed (pp.l99, 203), and the
transitivity thus corresponds to the others in exemplifying an
instance. A/—/C, of a principle, 6 — C. No other unit with A
and C other than those found affirmed or denied through
A/—/C follows, as confirmable in accordance with those re¬
maining (this page fourth pan).
All of them are disclosed in the next two diagrams, which
comply with the assumed (B — A)(B — Q. Hence none of
205
them, as before, follows. As to the lost form, (A — B)(C — BJ,
it reveals itself to be transformable into the preceding one.
By transposition it becomes (‘B — A)(‘B — ‘C), and hence the
some follows from it os from the other but for interchanged
affirmation and denial of letters. To wit, o consequence is
A/-/’C.
Denial may also be combined with affirmation, to create
further of these forms. But these ore likewise transformable into
former ones, or no connection regarding A and C follows. Only
6 has ground to be denied, while affirmed in its other mention,
since negation of A or C does not alter the form. But for
instance (A — B)(‘B — C) is equivalent to (‘B — A)(‘B — C), on
above form. The results ore similar by like transpositions in the
other forms except in (A/— /B)(‘B/— /C), where not performoble. But no connection for A and C follows from it, os con in
the previous manner be confirmed by the diagrams here, which
comply with the assumed.
It may be further asked whether results not in accordance
with the few listed principles do not follow if any of the above
configurations are combined with other connections of A with
6 or 6 with C, as wondered about two letters only. That no more
follows was for the most part answered (p.203 last par.), it
remaining whether a connection of A with C does, other than
which follows from one inferred of them before. From it alone
follows the same that does from such connection of any two
letters, since the letters are indifferent, and the question hence
is whether a connection of A and C follows in a novel way due
to 6.
It can by diagram with preceding ease, however, be deter¬
mined that no connection in question of one thing. A, with
another, C, results from a third thing, 6, if each 6 is not con¬
nected, as in discussed form of a principle, with C, one of the
other two things, of which some of the second. A, is also con¬
nected with 6. It is the transitivity of acquaintance.
For these reasons no further principle enters also if an as yet
not assumed connection of A with C is added, a connection
more likely as a conclusion than additional premise. If for
example to A — 6 and 6 — C is added C — A, the identity of
A, 6 and C results by transitivities. From 6 — C and C — A
follows 6 — A and hence due to A — 6 the identity of A and 6;
from C — A and A — B follows similarly the identity of 6 and
C; and from these identities the identity of A and C. But no
connection of either thing with another does not result from an
exampled principle, since if deriving from a third thing, it does
by transitivity, and if not, it does by the other laws.
The same reasoning by now evidently applies to configura¬
tions of four or more letters. Any connection between one and
s
206
another follows either, through o further letter, by transitivity, or,
through its own letters, by the other lows. And on unconnected
letter, negations also considered, follows in accordance with
laws of thought.
Unconnected letters, or their negations, when observed to
follow in modus tollens otherwise (e.g. p.201 last par.), were
seen connectable, to comply with the preceding. The reason is
that these single-letter units do as noted (e.g. p.2(X) fourth par.)
not follow from connected ones alone, and hence an uncon¬
nected one is a premise, transportable into the conclusion. Thus
when a conclusion will not be such a resulting connection, it
will as a one-letter unit follow by laws of thought from a like
one in the premise. Confusion may though arise, because units
of single letters—or by the foregoing (p.l90 fifth par.) more
generally the truth or not of propositions, or still more generally
the existence or not of things—are often derived by way of logic
other than laws of thought. Characteristically if known that A
implies 6, and A is found, 6 is inferred, casting doubt on the
herein observed redundancy of modus ponens (p.l62 last four
pars.). In this case if A is transported the redundancy was seen
explicit, and it should be added that since with another letter
C instead in the one-letter premise no conclusion follows from
both combined (p.202 fourth par.), modus tollens and modus
ponens are the only possibilities. And the deceptiveness of
language once again intrudes.
When a state is asserted in language without qualification of
time, it is usually the present which is meant. As when saying
that there is a solar eclipse. But when saying that when the
moon comes between the sun and the earth then there is a
solar eclipse, the meaning is timeless. The timeless statement
is an example of ‘A implies B”, and when added to its sup¬
position is in modus ponens “‘and A then B”, the meaning is
equally timeless, for the principle is to as such apply to all
times. What happens when a present state is, independently,
asserted is in general rather a matter of an instance.
Because of the unqualifiedness of the statement, when A is
stated respecting an occurrence, the same seems referred to as
by that letter in “A implies B”. But since the latter as a rule
speaks of all times of A as times of 6, and the former of this time
of it, it is an instance of A and might be symbolized by another
letter. It is not so symbolized, or a corresponding statement
qualified, because the content is distinguished only by time,
but it nonetheless plays such a logical part. The subsequent
inference is by transitivity, which can take the form “If a time,
A, is a time of 6, and times of 6 are times of C, then that time,
A, is a time of C”, the time, as formerly, possibly a place.
Accordingly if the present, otherwise expressed by A, is a time
207
of B, it is a time of C, expressed by C, as presently existing, os
would be done with 6, the two alphabetically the former A and
6, when on knowing the connection between them and that the
present is a time of A, the present as a time of 6 is inferred. The
concealment by language extends farther.
Through it an unqualified assertion speaks covertly not only
of the present, but also of present knowledge of the asserted,
in concord with the expounded relation of knowledge to reality.
This should be clear from the very nature of inference, in which
upon knowing one thing another becomes known. Accordingly
when saying that A implies 6, the purport is that when A is
known, knowledge of 6 is ensured. This is especially notable
when an instance of a principle is put as “\i A then B”. Meaning
the present again, by ^^if” is meant, unlike by some of ^’when”,
that A is not known, by which to know 6. That is to say it is
meant, more pointedly than with the principle, that should A
at any time be known, knowledge of 6 is ensured. In this case
if it is also stated that A, namely that A is known, A is an
instance in that sense. The present knowledge of A is an
instance of knowledge of A, and hence knowledge of 6 is
ensured. More can be said.
An instance may not be what might be called proper. As in
former numerical cases, an instance of something may truly be
identical with it. It can be that in “^if A then B” at issue is not
only a single connection but a single occasion on which upon
making certain that A, one infers by the connection that 6. The
connection could be that if when a certain food cooking comes
to a boil the heat is reduced then the food will be as desired.
Then upon seeing that at the correct time the heat is reduced,
by which one infers the desired result, the actual making
certain of the needed condition is identical with the certainty
assumed. The certainty first had the attribute of being assumed
and then of being actual, i.e. it was like other identities char¬
acterized in different ways, and the attributes might again be
symbolized by different letters, with e.g. A identical with 6. The
first connection would thus be “\i B then C”, but although the
actuality A by transitivity ensures the actuality of C, it could not
be stated as would be customary that if A = B and 6 C then
A -> C.
Such a statement is not groundlessly called a conditional,
with the ”if” clause the condition for the “then” clause, but not
necessarily true. The statement says that the conclusion is true
provided the two premises are, without affirming either of the
two. But the first of them was meant to assert the actuality of,
affirm, 6. The point is that the actuality cannot be put into a
principle, which consists in a conditional. It should be first taken
into account that the above 6 C, or “If 6 then C”, is also a
s
208
conditional, viewable as an example of premise A 6 in
modus ponens, as might in it above premise A = B and
conclusion A ->• C be viewed as respectively premise A and
conclusion 6, Then by A 6 is not affirmed A and hence B,
which rather are the subsequent intentions based on the
conditional. Should the actualities be meant, then to restate in
modus ponens that A and hence 6 would again be redundant.
But A -> 6 is meant to only assume, and should in such a
principle subsequent A dnd 6 state actuality, that A and hence
6 are known, then they cannot at the same time be merely
assumed in A 6, but are known in it, for equal redundancy.
Hence above mentioned present knowledge as a proper
instance of one of the same thing does for transitive inference
strictly not apply. Only knowledge of a particular time other
than the present can be the instance of the knowledge
assumed. And for the same reason in the last discussed non¬
proper instance, that of the identity of the assumed and the
actual knowledge, the knowledge, of some particular time,
cannot by a principle be stated at once as both assumed and
actual. If stated as actual, being thus of the present, it cannot
also be merely assumed, and if stated as assumed, being of
another time, it cannot also be actual.
For the actuality to in regard to the conditional A ^ B be
stated of A and hence 6, as noted to be often done, the letters,
as well as their connection, would stand as independent
assertions, in what is called an argument form but is as a whole
mistakenly likewise treated as conditional. The form may be “A
implies B, and A; therefore B”. There seems a return to modus
ponens, but now stated is not a principle but a succession of
things ascertained, not all of them at once, the symbolism
serving to represent, not to prescribe.
Of interest is that such a deduction, when not of the proper
instance in transitivity, cannot be made a principle, and modus
ponens is hence of the same futility as were principles that,
though not equally redundant, prescribe case inferences from
principles (e.g. p.l41 second par.). To through a connection infer
one fact from another thus requires conceptual perception of
the entailment in each case, and pertinent is that it takes place
in accordance with listed logical law, by which the perception
is of one thing coinciding with another, here by transitivity. To
generalize this transitivity put in the discussed language of
placement—actuality is the place of, the assumed. A; the place
of, the assumed, A is a place of B; therefore actuality is a place
of B.
The transitivity by which from the identity of an actual A with
an assumed one, and from the implication by A of B, B is
inferred can go as far as to concern actual and assumed exist-
209
ing knowledge. At issue is as usual not knowledge by someone
else but by oneself or mankind collectively, since the interest
is in that very possession of knowledge. And an A connected
with 6 may not only be hypothetical but known, but not as so
connected. A like situation occurred with an equality like
3 + 1=4 (p.l37). Both sides of the equation were in a
sense known, but although it may accordingly be known that
2 + 2 = 3 + 1, it may be known only by inference that in
consequence 2 + 2 = 4. It was observed that in the process
while something may be known, e.g. the second equation
here, the knowledge may not be in consciousness, e.g. when
affirming the first equation, so as to by it actually affirm the
third one. Likewise when affirming a connection between A
and 6, knowledge of A, of its existence, may though possessed
not be in consciousness, wherefrom to infer 6.
Despite the subtle difference regarding A between its knowl¬
edge that is not and is in consciousness, when stating A and
consequently 6 something else is meant by them than when
stating their connection, though the language be the same.
This is clear from the fact of inference alone, the inference of
course being that 6 exists, inasmuch as A does. By virtue of that
difference the inference was seen of transitivity, to like all
deduction be made in accordance with not more than some of
the basic logical principles given.
Whether something follov\/s from something else can accord¬
ingly be decided in accordance with a rather few and simple
principles, despite the contention that many logical and math¬
ematical principles cannot be proven. The contention largely
stems, as discussed (pp.111-113), from the mistaken belief that
the statement “This statement is unprovable” is true and cor¬
respondingly unprovable, and it reached a field as extensive as
mathematics in peculiar ways.
One of them is built on a system by which linguistic forms
used regarding that statement are assigned numbers, where¬
upon on supposing to prove the truth of that statement, the
proof is inferred to hold for those numbers. The numbers and
especially mathematics are, however, irrelevant here. The
supposed proof does not concern those numbers but things to
which they are assigned. If a cause be numbered 1, and the
effect 2, the consequence is not that the number 1 causes the
number 2.
Irrelevant for another reason is the ascribing of unprovability
due to that supposition to truths about all of an infinity of things,
e.g. numbers. In support of the supposition that “This statement
is unprovable”” is true it is argued that every means of proof of
the statement can be proven to fail, since its proof would con¬
tradict it. It is alongside argued that not all means of proof can
s
210
be proven to foil, since the statement is otherwise proven
unprovable and hence by its content proven, contradicting it
again. But the meanings of ‘”every” and “all” are the same,
while contradictory things are said with regard to them in the
last two sentences. Should there be doubt about the sameness,
it is enough to exchange those words in the sentences, to see
the arguments retained.
The second of these sentences actually reveals the earlier
observed paradox unrecognized. It discloses that by stating the
first one, a form of the customary proof alleged of “This
statement is unprovable”, the proof constitutes a contradiction
of it. In lieu of that recognition it is maintained that there
somehow is a difference between stating that something holds
for all of a kind and that something holds for the first of them,
the second, etc., as might be held said by speaking of every.
Since unprovability is in the arguments, further, spoken of in
terms of all attempted proofs that would fail, indefinite though
their number may be, they would be viewed as infinite in
number. It is correspondingly held that, as noted, truths about
all of an infinity of things are not provable. In the preceding,
however, the issue is the number, regarding a particular thing
yet, of any proofs attempted, not the number of things to be
proved.
There is more irrelevance. Inasmuch as it is thought that for
the statement in question contradictorily all means of proof
cannot be proven to fail but not the first, second, and the rest,
it is believed, though the enumeration is but a way of speaking
of all, that truths holding for an infinity of things can only be
proven for particular ones at a time. This procedure is further¬
more confounded with the one of proving something of a thing
one step, rather, at a time, as conclusions are usually reached,
by drawing from initial premises inferences that serve as further
premises, and so on.
It is indeed possible through regular inference to find at one
time that something is true of an infinity of things. If from an
attribute certain things have in common, as by definition,
another is inferred, it holds for an infinity of the things, since it
does for anything of the first attribute.
In the disputing of provability regarding infinities the famous
last theorem of Fermat, the 17th century mathematician, is
often cited. It has to do with equations like those of the men¬
tioned Pythagorean theorem (pp.86, 87), by which a^ +
equals c^ when a, b and c are the sides of a right-angled
triangle with the last side the longest. By Fermat’s theorem the
equation never holds if 2 is replaced by a higher natural num¬
ber, with a, b and c likewise natural numbers. The provability
of the theorem is questioned, because the numbers above 2
211
continue into infinity. It is acknowledged at the same time that
with some numbers in place of 2, for instance 3 and 4, the
theorem has been proven. But this is contrary to the generally
proposed unprovability. A proof that the equation does not hold
for 3 or 4 applies whatever the allied numbers a, b and c, and
these continue likewise into infinity.
In point of fact, all logical principles, not only mathematical
ones, apply to infinities. By applying to anything, they do to
things of an infinity like numbers.
Utilizing then the not many basic logical principles, together
with likewise uncluttered notation, there is a brief scanning
procedure for deciding whether certain conclusions follow from
given premises. The arrangement of symbols enables, as a
matter of fact, a procedure more positive. Beside finding
whether conclusions in question are entailed by the premises,
the arrangement can be used for finding whatever is entailed
by them.
This can be accomplished by combining them in symbolic
form in a linkage resembling customary chains of equations
and remindful of a sorites, a series of syllogisms stating all
premises and omitting intermediate conclusions. The li-nkage
differs from a sorites by, apart from employing only the sym¬
bolism, only once listing an attribute, a variable in general. The
last indicates that the linkage also differs by its application to
additional material, i.e. implication, by which it differs from
chains of equations as well. It differs from both of those forms,
further, by extending also in multiple directions, for multiple
connections of a variable. It is illustrated by
A
B->C-^DE-^F
G
With attribution marked in contrast to the arrow of implica¬
tion plainly by a line, the general rule can be that, as in other
reading, connections be read from left to right or from top to
bottom, or, for more connections, inclined between those,
namely from upper left to lower right.
A single train of reasoning is not apt to be as multidirectional
as suggested. Illustrated is nonetheless beside such as separate
implications by A and 6 of the same thing, C, a combining of
two directions, implications, by C, should the implied, D and E,
combined imply something else, E, with E alone implying G.
The individual implications, or other connections, can thus be
united in various ways, of importance being that they be
appropriately linked, to facilitate deductions.
It will be obvious now that the deductions rely foremost on
transitivity, the justification for linking one connection with
another. It will be equally clear from what was said that
transitivity between letters of any distance takes place wherever
there is a continuity of implications. It can accordingly be from
the depicted deduced for example that A implies F, or that B
inplies G, true without need of D.
More can be deduced by also making use of transposition.
Accordingly for example ‘F implies ‘B, or ‘G implies A, true
without D. Still more deduction is possible in view of
convertibility of “”some”, of “”can”‘ as used mostly for attribution.
Here the results are mainly that, taking into account that from
“all” follows “some” as so-called subaltern, some F can be said
to imply, be accompanied by, G, and the like is the case with
‘A and ‘6, as well as D and E since implied by C.
The four principles in the last two paragraphs referred to—
transitivity, transposition, and perhaps conversion and subalter¬
nation—are virtually all required for standard deduction, as
performable through the linkages. The laws of complements,
seen the only other fundamental ones, were also seen to
involve indirect, or external, negation, basically about what is
known to be false rather than true, the laws translating one into
the other. The known to be true was noted the interest in the
deductions, and the complement laws, applied in life at least
as much as a matter of course as are the others, thus serve the
subsidiary role by which on reaching a denial one makes for
those deductions an opposite affirmation, and on reaching an
affirmation one does not lapse into its denial.
Other principles, even parts of the preceding, were seen
consequences of the basic ones, with for example identity and
equivalence signifying ties between A and 6 in both directions.
This suggests further possibilities of the linkages. Beside using
the same letter to end one statement and start another, it can
be used for, beside a connection in the opposite direction,
reverses in letter sequences that include occasions when
negation is involved, as well as to state the denoted entity’s
discussed (pp.206-209) existence or not, the knowledge of the
same.
In mentioned identity and equivalence, or in A/—/B, the
sequence can be reversed for any thus suitable linkage with
another connection, as in B/— /A — C, and in the case of two
statements like A 6 and C ‘B they similarly can by
transposition of one of them be united, as in A 6 ‘C. When
existence of an entity, or of its negation, is known, a simple
mark like an underscore will do, as in
As explained, the knowledge or existence at the time is a form
213
of an instance of knowledge or existence meant regarding the
entity, A, to imply the same of what is connected with it, 6.
Consequently knowledge of the existence of the implied is by
transitivity likewise ensured, as commonly understood, and the
same is true of course of each succeeding linked entity, C or
more. A simple mark for deletion will similarly serve when
known is an entity’s or its negation’s nonexistence, as in
In accordance with transposition then is ensured the knowledge
of nonexistence of each preceding linked entity. These matters
on existence and nonexistence apply of course as much to
attribution as to implication, the first being part of the second.
Apart from availability of the linkages for both implication
and attribution, the two can using the same variables be com¬
bined, as indicated before and given by
A-^B-C.
The difference between the two connections was observed to
be substantially that in implication one thing is accompanied
by another within a distance if any that depends on the con¬
nection, and in attribution that distance is nil, the one thing
being the other. In consequence, with attribution hence a form
of implication, when the two are combined, transitivity of
implication is in effect. This consequence is another understood
as a matter of course, it being quite clear that if A implies 6,
and 6 is C, then A implies C.
It may be added that the transitivity also follows from an
observed transitivity of attribution, by which if one thing is the
same as another then what is true of either is true of the other
(ca. p.l66 last par.). This may be more apparent if the last
displayed connections are interchanged, for
A-B-^C.
Since A is the same as a 6, and the 6 implies C, so does A. Not
much harder is it to see that in the precedingly displayed since
6 is the same as a C, and 6 is implied by A, so is the C. The
important is that however attribution and implication be
combined, the implication is transitive.
While many known facts can for deduction be in linkages
thus combined, it is not requisite that all reasons for that
knowledge, itself often gained deductively, be stated in the
same manner. Since the linkages are constructed for viewing
how one thing is connected with another through transitivity, if
a fact is not determined through it, there is no need to similarly
depict the derivation.
Some statements in a linkage were, as in reversing a
sequence, already seen derivable directly through other logical
laws, which were few and simple enough to, along with the
applied to, use unstated. Some of these, e.g. transposition and
214
conversion, os well os transitivity itself ore so used in fact for
deductions from linkages themselves. Statements in a linkage
may also be derived by transitivity, justifying perhaps also a
linkage disclosing it. But more likely than not the deduction is
a case inference, too brief to need depiction. As iterated, such
deduction, from a principle, is an everyday process, often per¬
formed without stating either the likely familiar principle, “B
implies C”, or especially the specific connection, “A is a B”,
merely stating the instance of the principle, ‘A implies C”.
Nondeductive determinations were treated elsewhere, some
of them of definition, others of observation, and they are apt to
become the facts combined in a deductive linkage, or be the
basis from which those facts themselves are in preceding ways
deduced. The last can take place principally when the deter¬
minations are quite general, the linkages serving mainly to
combine particular facts.
They need hardly be used for additional logical principles,
since things to be decided by means of logic were found to
require no more than the principles listed. And if things subjects
of logic are equally of certain generality, they may likewise
require no linkage. Mathematics offers an example.
In its chains of equations, an application of linkage, parts,
often numerical, are equated as a result of more general
equalities, ones likely using algebraic variables. The particular
equalities are, as above referred to, case inferences derived
without stating the pertinent connections, here the general “B
equals C” and specific ‘A is a 6″ to state the instance ‘A equals
C”. It is of interest that, although the inference is made without
ado, it contains a complication similar to a preceding transitivity
(last p. fourth par.). There implication by A of C is logically an
attribute, resulting from the sameness of A and a 6, or of 6 and
a C. At present an attribute is the equality of 6 to C, because of
which A, same as a 6, is likewise equal to the corresponding
C For this inference thus equality is not about 6 and C as shared
attributes identifying things (p.l65 Theorem III.8), but is jointly
with its object, C, an attribute of the subject, 6. It is also
noteworthy that, similarly to application of transitive or other
logical law (p.14l third par.), the C inferred as equal to A is not
any C, but one applying in quantity to A. From a X b = b X
a one does not conclude that 2 x 3, a case of the premise a X
b, equals b X a, but 3×2.
Despite the complexity, such a result is derived with ease
mentally, to set it down for a chain of equations. An inference
is usually simpler still if of a case of such as a causal law or
definition. The transitivity is likely of standard implication or
attribution, and the implied or attributed may not be as
specific. And as in mathematics, it is particular cases that
215
mostly ask to be linked together.
In the large, general facts were observed in the last chapter
to furnish the basis for dependence on things as realities,
because it is through those facts that the things are known to be
effectual. But it is eventually not the general facts that are the
concern, but the particular instances, which when combined
may lead to ultimate realizations. Having investigated those
facts, laws governing reality, it is appropriate to as a result of
them thus combine any specific determinations, and find what
can be inferred.
The linking of the determinations in the symbolic form
suggested may help to serve this purpose, and in the main has
it been intended in this chapter to provide a stable foundation
for deduction and the inquiries herein making use of it. Among
these and relying also on ascertainments of the preceding
chapters are the farther explorations following.
216
Chapter IV
THE TRANSCENDENT
REGIONS
As indicated in the introduction, much of philosophy has
increasingly denied the possibility of demonstrating any
metaphysical facts, any what are thought of as unexperienced
realities, let alone any that would transcend the material world
as known to man.
The views were noted bolstered by the successes of observa¬
tional sciences, and advances by science in general appeared
to increase antipathy toward what were largely speculations.
Deductive science itself, however, informs of the fallacy of the
denial of the antecedent. IfA implies 6, it does not follow that
not-A implies nof-B. If realities held to be of experience,
specifically ones of the natural world, are demonstrable, it does
not follow that others are not. As noted also, all inferred, on
which the preponderance of knowledge depends, is of the
unexperienced.
The opinion that instead deductive truths, comprising the
large part of inference, do not speak of reality can as indicated
be attributed to the generality of logical or mathematical
principles, which do not refer to particular realities that may be
sought. It is one thing, however, to consider those principles in
isolation, and another to confuse them with their application
to actual problems. The significance of those principles lies in
those applications, and they are indeed applied abundantly,
whether by conscious reference to them or by compliance with
them unwittingly. The first is done in profuse mathematics, and
the second is indicator of the obviousness of logical funda¬
mentals, whose instances are without knowledge of the prin¬
ciples perceived in particulars in question, unlike in induction,
where the instance cannot be known without the principle.
How pervasive the use of mathematics is for findings of realities
would seem needless to say were it not for above contentions
to the contrary. It need only be kept in awareness that quantities
regarding things, such as distances through trigonometry, have
217
been computed for millennia. How common in getting to know
realities is utilization of logic was previously remarked regard¬
ing several theorems, beside the repeated observation of in¬
ferring by everyone of instances from principles by transitivity.
Animals, too, avail themselves of that transitivity, in learning
from repeated experience how to go about life, and they know
likewise that in connected causations if, for instance, pushing
at a tree will make it shake, and this will make it drop its fruit,
then the first event will result in the last.
The knowledge of causation brings to the fore that man and
animal make also other inferences of realities than deductive
ones, to wit the inductive ones having to do with laws of nature.
The point is that the acquaintance with nature as a reality, as
consequential as expounded in the pertaining chapter, rests
fundamentally itself on inference, on certainty of things not
experienced.
There is in ordinary life thus continual and legitimate ascer¬
tainment of unobserved realities, as explicated in the last two
chapters. Many of those realities were for that matter seen
presupposed by the average man and scientist alike in their
pursuits, to carry hence greater certainty than scientific findings.
It is reliance on these findings that is primarily behind the
emphasis on experience. The findings were moreover observed
to be to a large measure of the conjectural nature of theories,
which suppose at their foundation unperceivable realities.
Whereas scientific assertions looked to are correspondingly
subject to lack of substantiation, fundamental things presup¬
posed by them were found to be to the contrary, among them
ones called metaphysical.
Belonging to these were free will and a similar purposive
independence in live organisms from physical and chemical
laws. These as susceptible of demonstration about nature have
been matters of most dispute. Matters that appear more remote
than what is known as nature are furthermore equally suscep¬
tible of demonstration. The inquiry can accordingly proceed
into realities that may be regarded as transcending nature.
Section 1
THE EXISTENCE OF GOD
It was observed in the first chapter that if something not
linguistic is to be about an entity ascertained, it is necessary to
have a concept of it, it is necessary to know what the meaning
of the name applied to it is. In corresponding demonstration
s
218
through discourse the meaning must therefore somehow be by
language conveyed.
It is sometimes proposed that some concepts, as might be
that of God, are ineffable, that they cannot be expressed or
described in words. All concepts can be expressed in words,
however, since any word can by choice be made to name a
concept for that purpose. It is sometimes said that proper names
do not have a meaning, and some dictionaries do not list them
for that reason. But all a word needs for a meaning is that it
designate something. Should the question of the ineffable be
whether a concept can be described—that is, defined by usu¬
ally a number of words to as presently of interest communicate
the meaning—it was already expounded that all concepts ad¬
mit of appropriate definition.
This is the more feasible if the concepts are of general
acquaintance, as are those concerned in this disquisition. Yet
it is contended that in particular the present concept emerging,
that of God, is unknowable, because of his presumed other¬
ness from all that may in the here and now be known by man.
As explicated likewise, however, the use alone of a word, lest
it be without meaning, presupposes knowledge of a concept
behind it.
Knowledge of a concept of God is in accordance with the
beginning of this section in addition requisite for attempting
proofs regarding him, specifically his existence. It has instead
inconsistently been put forward that it is possible to know that,
but not what, God is. Essayed proofs of his existence supplied
informal definitions nonetheless, and a further issue is whether
a given definition is appropriate.
In conformity with previous remarks, definitions are some¬
times adapted so as to enable an inference desired, which as
a result does not pertain to what it was expected to. The defi¬
nitions may not be of the expected for some other reason, and
what is of account is that in each case the words defined are
used equivocally, the subsequent reasoning misapplied.
With respect to the definition of God there is especial like¬
lihood of misguidance, because the concept is variously
infused with particulars of divergent beliefs. The definition can
accordingly lose sight of the common concerns regarding God,
concerns which it is in this writing the object to address, and
which thinkers in general are presumed to have in mind. They
are nevertheless neglected in mentioned proof attempts, which
will in that light and otherwise now be reviewed.
FORMER ARGUMENTS
It may be noted that the definitions given with the argu-
219
merits, sometimes by the same author, are disparate. They
would accordingly not pertain to the same entity, they could in
particular not all pertain to the common conception of God. On
one occasion thus St. Thomas Aquinas, an above questioner of
the knowability, gives nonetheless, in his believed five ways of
proving God’s existence, five respective definitions. Several of
past arguments need correspondingly not be closely examined,
because what they would prove would not be in accord with a
satisfactory concept of God.
Such arguments are those called cosmological. Among them
may be considered to be the first three of the referred to ones
of Aquinas, and they maintain the existence of a primal, an
uncaused, cause of all things. But the general concept of God
can be held to require some power additional to that of first
causation, a causation that may be as blind as most others
appear to be, whether they themselves be caused or not. The
proposal that a primal cause would be of greatest perfection,
as advanced by Aquinas in his fourth way though not holding
such attribute decisive below, is derived neither through natural
nor logical law. The same is true of merely the contention that
there is an uncaused cause, to briefly remark on the lack of
soundness of the cosmological arugments besides.
In conjuction with them arguments called ontological and
teleological form a triad of what appear the most debated
regarding the existence of God.
Of these unduly the most seriously taken is the ontological
one, associated with St. Anselm. Interestingly Aquinas ques¬
tioned the definition commencing it, although it may be
regarded as suited by connoting God to personify men’s highest
yearnings. The argument is
l.i. God is that than which nothing greater can be
thought.
But, the argument proceeds,
l.ii. It is greater to exist in reality than in thought.
Hence, it is inferred,
l.iii. If God exists only in thought then he is not that
than which nothing greater can be thought.
It is therefrom and from l.i concluded that
l.iv. God exists not only in thought but in reality.
The argument though invalid has been falsely held refuted
by Immanuel Kant. As did Hume before, he alleged that exis¬
tence is not a predicate or attribute, that accordingly nothing is
added to an object of thought by positing it to exist in reality.
The understanding is that if the distinction in premise l.ii above
is thus unwarranted, what is inferred from it, and l.i, does not
follow. The distinction is irrelevant to the reasoning, however,
not leading to the conclusion regardless.
s
220
The distinction can indeed be made, whether or not it be
held namable as about an attribute. As noted more than once,
whether or not existence be called an attribute is a matter of
definition. And between a thing as a conception and as a
reality there is more than a slight difference. The question as to
fact or fancy is a principal one in human endeavors (p.25 last
par.), with the greater importance accorded to things that exist
in reality, not only in thought.
In consequence assertion l.iii, which does follow from the
two preceding it, is acceptable. As a matter of fact a similar as¬
sertion can be made with respect to most any definition of God.
As was observed about real entities like chairs (e.g. p.38 fifth
par.), a thought of them is not one of them. It was observed also
(p.39 fourth par.) that a thought of a reality conjectured, e.g. a
quark, is likewise not identical with it, although the reality may
not be certain either. Conjectured realities are many, all the
assumptions, again, the truth of which is sought after in man’s
endeavors. Such an assumption is the existence of God. That is
to say by God is envisioned a reality, not a thought, a thought
of him not identical with him. Hence whatever the definition
l.i, the existence referred to in l.ii is the assumption, and as in
l.iii accordingly if God exists only in thought then he is not the
being assumed.
But as in the case of other assumed realities, the assumption
does not imply the reality. The ontological argument faultily
elicits conclusion l.iv, by in accordance with the premises
pitting existence in thought against existence in reality,
overlooking that both remain conceptions, and equivocating
the second of them with the actual.
In response to objection that something else of perfection,
such as some island, may be imagined to exist though it does
not, Anselm replied that whereas things other than God can be
conceived not to exist, God cannot. The impossibility was
further explained to mean existence that is necessary, in the
sense that it is eternal. However, other assumed realities,
whether assumed permanent or perishable, can likewise not
be conceived not to exist, since contradicting, as above, the
assumption of existents. To be precise, in questions of realities
it is conceived that assumed existents may not exist, but in that
case their concept is in practice deprived of one of the things
by which it is formed, their existence.
The ontological argument has been thought recently revived
on the basis of the just mentioned necessary existence as part
of the concept of God. By that necessity if he ever exists then
he always does, since an existence not eternal would by the
definition not be of God, and if he ever does not exist then he
never does, since any such existence not eternal would again
221
by the definition not be of God. By this conception then,
speaking of necessity and impossibility in place of “^always”
and “‘never”,
2.i. It is either necessary or impossible that God exist.
But, it is argued,
2.ii. It is not known to be impossible that God exist.
Hence, it is inferred,
2.iii. It is not impossible that God exist.
In consequence and by 2.i. it is concluded that
2.iv. It is necessary that God exist.
The falsity of the argument is hinted at if the second premise
is replaced by “It is not known to be necessary that God exist”
leading to the opposite conclusion. The error lies in equivocat¬
ing the two uses of “possible”, or of “not necessary” referred
to in the first chapter (p.37 sixth par.). The progression from 2. ii
to 2. iii is fallacious, because while something may be possible
in the light of man’s knowledge, it may be impossible in fact,
to never occur as in question.
Of the three mentioned types of argument for the existence
of God the teleological may be of widest appeal. It is repre¬
sented by the fifth above way of Aquinas, and it reasons from
design or purpose exhibited in nature to a knowing designer
beyond. It reasons by analogy, comparing the purposiveness of
organisms to the similar purposiveness of a watch or of an
arrow in flight. Both are produced by, in the persons of a
watchmaker and an archer, beings of intelligence, of knowl¬
edge, whence it is surmised that the purposiveness of organ¬
isms has a comparable source.
At times the argument proceeds from considering a general
order in the universe, ascribing a designing intelligence as
cause. The concepts of order are sometimes opposing ones,
however. In one sense what appears referred to is a uniformity
as exemplified by laws of nature, and in another sense it is a
utility of arrangement in contradistinction to what is looked
upon as chaos due to the same forces of nature left to them¬
selves. It is clearer consequently to consider in the argument
the analogy with regard to purpose alone.
Analogy is inadmissible, however, as a form of inference. It
constitutes what is known as the fallacy of undistributed middle,
by which, in no more than unwarranted generalization, if some
6 are known to be C, it does not follow that anything A that is
6 is C. That and other lack of implication was treated more fully
in the last chapter (p.204 fifth diagram for here), and here it
would mean that because some things of purpose have a
knowing author, not everything else of purpose needs to. The
traditional teleological argument thus takes a leap from
premises to conclusion without adequate connection. Yet, its
major components ore well-founded for on earnest inquiry into
the existence of God.
Among the foregoing definitions of God, that of o power with
knowledge in likeness to conscious man seems closest to the
universal conception of a supreme being. And unlike the
grounds adduced for his existence in other arguments, purpose
in organisms, which pertains also to men and their preservation
or interests, seems a reality of most concern as a possible
manifestation of the workings of God. Both, knowledge and
purpose, were explored earlier herein, and that the two con¬
cepts can be deductively rather simply connected so as to yield
the searched-for conclusion will be found presently.
PROOF
In view of the observation that the universal concept of God
can be taken to encompass the attribute of knowledge, it can
be appreciated that he is accorded that attribute in major
religions. But as indicated, a religion tends to embellish the
concept with additional specific beliefs, not necessarily com¬
monly shared. The disadvantage is that determination that a
being of certain fundamental powers of concern exists is incum¬
bered by specifying further attributes, of for instance how those
powers are used. The search will rather be more advantageous
if directed into the existence of a supreme being less narrowly
defined, while the concern. The common concern indeed can
be considered to be existence of a being of the fundamental
powers nameable as sovereignty over man.
To be more explicit, the general conception of God can be
considered to be of a being of the powers of man to a superior
degree, such that he sees man’s condition, has altogether
knowledge of it and the conditions connected with it, and
possesses the power to act accordingly. The attributes consisting
in that knowledge alongside the power to bring it to bear, these
traditionally expressed by the appellations of the Omniscient
and Omnipotent, may in result be regarded as defining God.
Proof of the existence of God so defined is then undertaken
here. While in doing so both attributes may be termed powers,
whence the title of Almighty may be applied, it will be fitting
to speak of the power of action as power, and of the power of
knowledge as knowledge. Because there should be little
question, further, as to the requisite power, with any of nature’s
events by which the requisite knowledge would be revealed
presupposing that power by their presence, the inquiry will
center on that knowledge.
Knowledge, in man, was earlier defined as awareness of
something real (p.46 last par.), and the real as what is con-
223
sequential in man’s purposes {p.44 last par.), the two meanings
suggesting, as also considered elsewhere (p.96 third par.), that
the knowledge serves as a means toward fulfillment of
the purposes. Viewing knowledge thus as a means, it admits
of a more penetrating definition, one resting on seeing human
knowledge and other of man’s faculties in terms of their
function, in keeping with the purposiveness of organisms in
general.
What can in them be excluded as having function are entities
which are the basis for it, namely the self and those ingredients
of consciousness which are the ultimate purposes, objects of
pursuit. But other parts of consciousness were seen os having
functions in those pursuits. Passive perception supplies infor¬
mation on which the pursuits ore based, and active will brings
the pursuits into being. The processes may be intermediate,
based on previous activity, and supplying further information.
The awareness as to which of the information one may be
guided by, which of it represents reality, constitutes knowledge,
whose function is correspondingly to be the guide in one’s pur¬
suits, to be the means, alongside one’s power to act, toward
fulfillment of one’s purposes. Knowledge con in consequence
be accordingly defined, inasmuch os it is its function which is
of concern, not things extraneous to it, os may be acquisition
of the knowledge through o material body.
In this larger sense, that goes beyond human attribute alone,
knowledge is with power thus understood os the means to pur¬
posive activity, to purposive events, events that ore, proximate
or eventual, objects of purpose. And since o means is some¬
thing by which something else is brought about, it con be
designated o cause or its component. Knowledge can thus be
defined as that which is with power the cause of purposive
events.
There may arise questions as to the manner in which knowl¬
edge with power is a cause, since dissimilar from accustomed
causation. But as observed regarding causation, such as
causation of bodily motions by thoughts, its manner is irrelevant
(e.g. p.62 second par.). Of account is that one event is always
accompanied by another. In the case of causation by knowl¬
edge and power, the knowledge can be likened to a tool or
material by the use of which something is effected, and the
power, in being applied, can be likened to any such use. It
suffices that these two factors be understood as that which is
instrumental in producing purposive events.
The simplicity of the proof searched for should now reveal
itself. It can be formally framed as follows, combining the
foregoing definitions with previous observations, customarily a
matter of course. It should be prefixed that when referring in
224
the proof to the scope of the powers and of events caused by
them, meant is a correspondence between these magnitudes,
in concord with earlier observations on force as measured by
result.
1.1. God equals knowledge with power of concerned
scope.
1.2. Knowledge with power equals the cause of
purposive events.
Hence, by 1.1 and 1.2,
1.3. God equals the cause of purposive events of
concerned scope.
1.4. Purposive events of concerned scope exist in
nature.
1.5. Events that exist in nature imply the existence of
a cause of them.
Hence, by 1.4 and 1.5,
1.6. A cause of purposive events of concerned scope
exists.
Therefore, by 1.3 and 1.6,
1.7. God exists.
The proof may be derived by means of a symbolic linkage
as discussed (ca. p.211 through end of chapter).
P^C= K=G,
with definitions
P = the purposive events
C = their cause
K = the knowledge with power
G = God.
Because of the importance of the proof, some logical detail
will be reviewed.
As discussed regarding linkages, which mainly connect par¬
ticular facts, it is cumbrous that they be accompanied by listing
of general facts that are bases. In the present case, while
premise 1.1 above is in the linkage repeated by the terminal
equality, in the first equality there premise 1.2, a general
definition, is represented by the particular instance; and while
premise 1.4 is in the linkage repeated by the underscore of the
first letter, in the implication there premise 1.5, a principle of
nature, is represented by the instance for purpose. Conclusion
1.7 will thus be reached by seeing existence implied in the links
on through the last letter, without intermediate conclusions 1.3
and 1.6, not dispensable where used if to ensure clarity.
Both forms of proof also combine implication with attribu¬
tion, the last as equality. The equality occurs in premises 1.1 and
1.2 and conclusion 1.3, the whole approximating a traditional
categorical syllogism. Instead premise 1.5 states alone an
implication, an existence of what it connects stated in turn by
225
premise 1.4 and conclusion 1.6, the whole approximating o
syllogism called mixed hypothetical. And conclusion 1.7 is
derived by use of ports of each, 1.3 and 1.6, with existences
connected through equality. The symbolic linkage similarly
connects existences through oil other connections.
More strictly by particular previous logical observations, in
the first form of the proof the first of the three inferences is of
o nature noted with respect to mathematical equations (p.214
fourth par.); in 1.2 the equality is the attribute of the subject, of
which 1.1 states o specific instance, 1.3 stating the specific
equality applying. The second and third of the inferences ore
of o nature observed to be mistaken for modus ponens (co.
pp. 206-208); in 1.5 the implication is by o general condition, of
which 1.4 states o specific instance, 1.6 stating the specific result;
this statement of 1.6 con similarly be viewed os about on actual
existent, identical with the assumed one of the second of the
equated in 1.3, to ensure the actuality of 1.7. The like occurs
in the symbolic linkage, viewable os simply connecting
implications.
As suggested by that linkage the reasoning, despite the
apparent intricacies, provides little difficulty in practice, resting
on common transitivity, by which one progresses from one of
connected facts to another. Fact, existence, though in the lost
chapter assigned primarily to one only of the logics, is, os
indicated before by finding the other logic port of the some, o
question at the bottom of inquiries in general, so that even
when the overt determination is only that something is of o
certain nature, beneath is on interest in corresponding exis¬
tence. In accordingly reasoning by successively connecting one
thing with another, be the connections implications or attribu¬
tions, it is natural to consider that the reality of one thing leads
to the reality of another.
Yet an above formal procedure may needlessly complicate
the deduction for a listener unaccustomed to it if of some
length. If instead truisms are largely left tacit, as in mentioned
enthymemes, it may be found easier to proceed from one
statement to the next. The truisms are understood regardless,
similarly to inadvertent application of logical principles in the
preceding, and to interpose their statement can obstruct
apprehension.
In the proofs above, as a truism could be regarded the
definition of God, in 1.1 and the ending equality in the symbolic
linkage. But beforehand understood would be especially
premises 1.4 and 1.5, and counterpart underscore and
implication in the symbolized proof, as well as intermediate
conclusion 1.6. All these comprise in fact the basis for the
popular teleological argument (p.221 third par. through p.222
N
226
second par.). It is remaining premise 1.2, or the first equality in
the linkage, which is the missing link in that argument and not
as evident as the mentioned, with 1.3 seen an intermediate
conclusion corresponding.
In this light the worded proof can be informally abbreviated.
1. God is knowledge with power.
2. Knowledge with power, as in man, is that by which
purposive doing is brought about.
Therefore
3. The purposive doing in living things is brought about
by knowledge with power, namely God.
The first premise, a form of preceding 1.1, was observed
viewable as a truism, but it is given to affirm a meaning seen
much obscured by diverse beliefs, and to state the attributes to
be proven.
The second premise is a form of definition 1.2, indicated as
not necessarily evident. That the definition is appropriate is in
the present premise briefly clarified between the commas.
The conclusion has, for continued brevity, two easily appre¬
hended parts, one before the comma, and the other after.
It will be noted that for simplicity’s sake other matters in this
proof remain tacit beside 1.3, 1.4, 1.5 and 1.6 of the preceding,
these four implicit in the conclusion now, including the
concerned magnitudes. In the present first premise the
concerned magnitude of the attributes is likewise understood,
as is their identity with God. In the second premise an identity
is more explicit, but less perhaps is that those attributes are
causes, that may in the conclusion be inferred in conformance
with nature’s law. And in the conclusion implicit is, apart from
the premises referred to, that the knowledge with power and
consequently, as stated in likewise obviated 1.7, God exists. All
this can be held comprehended by a hearer of the demon¬
stration, as were observed comprehended beside truisms
complex meanings behind ordinary words.
It may yet be argued that assertions contained in the demon¬
strations, whether unspoken or express, may not be as accept¬
able as they appear. The spoken of scope of the attributes, even
if great in totality, might be argued not to belong to the same
power, but be divided among units as small as minutest parts
of organisms, parts referred to in a previous chapter as thought
to have purpose built into them (p.95 last par.). The extent of
knowledge and power that could be then said to be present in
them would be negligible, adequate only to their limited
function. In similar confinement, even if a supreme being is a
cause of things, he might be proposed not to exist today but
only in the past, as held in some beliefs. The dependence on
him as ministrant to human cares, in hope of which he is also
227
called the One and the Everlasting, would by these hypotheses
be ill-founded.
That the powers in question are of the concerned scope, how¬
ever, is determined by observations preceding those regarding
minute parts of organisms. They are the observations of the
coordination of purpose in the organism’s parts, directed toward
the preservation of its whole, as well as the species. This phe¬
nomenon of far-reaching coordination presupposes its own
cause, to wit the discussed powers appropriate to the purpose.
It is this large-scale purposiveness of organisms which is seen
to exceed, and have as part of its object, the conscious
purposiveness of individual men or animals, which conscious
purposiveness has as its part the motivating cares or feelings of
the individuals possessing it. The knowledge and power at
issue extend to all of these, as well as the surrounding world,
utilized in behalf of the purposes.
The coordination spoken of betokens a single force or power,
the power which thus coordinates and accordingly resides in
the described capacities of knowledge and action. What con¬
stitutes a single entity was brought up (p.51 last par.) during the
discussion of the self, noting that it is a matter of what is desired
to be regarded as a unit. And in similarity to the oneness of the
self as determined by the selfsameness of primary mental
purpose, the oneness of God can be held determined by the
selfsameness of projected purpose toward which different parts
of the organisms and different powers of God as coordinator are
directed.
The singleness and powers of God appear by the preceding
determined with respect to only the purposes of particular
organisms and their species. Considering man as an instance
of these, however, the determinations are of the pertinence
required, just as it was sufficient to ascribe reality to things by
their consequentiality to man alone {p.46 second par.). As
explicated, the concern regarding the nature of a supreme
being is at the basis his sovereignty over man, even if not
extending elsewhere. That power was found exhibited by
jurisdiction over man’s purposes, including elements of
consciousness, and there is no other supreme power of purpose
exhibited with respect to man. To elucidate, the only other
purposive, not mechanical, forces known to affect men are
other creatures or organisms. When the purpose is by the
consciousness of the animal, a fellow sentient being, it is not
of a supreme being in question. And when it is of the function
of the organism, when a supreme being behind can be
inferred, the purpose is known not to be directed at the interests
of man, but of the organism.
Upon thus ascertaining a single supreme being over man, a
s
228
being that acts in accordance with man’s interests, it may still
be considered what the full extent of the deity’s powers may be.
For as signified by the common reference to him as all-knowing
and all-powerful, it is of prevailing interest whether his powers
may be unlimited, extending to the capacity to meet any of
those interests, man’s purposes.
Considering limitless power, the power of action, it has been
interpreted as of actual predestination of all that ever occurs.
The free will of man or animal would accordingly be pre¬
cluded, as its earlier observed meaning as unimposed by any
force would necessitate. But the unlimited power does not
require its unlimited exercise. The will was indeed earlier
herein found to be free, the freedom in point of fact seeable as
bestowed upon man in virtue of God’s power, in vi^w of such
as organs of sense perception and action, supplied in service
of that will. Wondering how much farther the same God’s
power reaches than the control over man’s organism, it may be
taken into account that beside the ancestral connections of a
species, or even the connections between all species proposed
by evolutionary theory, organisms, not excluding plants, are so
constituted as to be largely of dependence one on another.
One of them can be or can provide the sustenance for another,
setting the moral issue for the meanwhile aside. That their
interests may be in conflict does not confute a common
supreme being. The interests of members of the same family
can be in conflict as well. Inorganic matter, furthermore, is
likewise an object of dependence with regard to organisms.
Not only in the life-sustaining sense applying to the elements
of old, to water, fire, air, and earth, of which four the universe
was thought to be composed. Also in the comprehensive sense
by which all that has reality does because consequential to
man, as set forth in this treatise.
This consequentiality of nature, in making significant the
governance of all of it by the same laws, is further seen as
established by the very purposiveness of man’s organism. The
organs of action and sense perception are made to affect
nature and to perceive the results, as in frequent informing
through pain or pleasure of influence on one’s body. Acquain¬
tance with the world at large, and consequently its presence as
expounded, is to man for the matter of that given by way of the
body’s purposiveness. For the purpose of sleep or other re¬
quirements the world is removed from consciousness, possibly
replaced by dreams or other images. Reality was indeed noted
held to when unperceived be present because of its possible
perception by man. The world and everything that occurs in it
is together with other perceptions, all finding its realization in
man’s consciousness, in consequence the product of God’s
229
power. And his power can be said to extend beyond, toward
capacity to meet any of men’s purposes, in accordance with a
plain process of induction. That the world with its laws of
nature, including the living form’s activities toward fulfillment
of its purposes, extends beyond the present, whatever direction
events should take, is already an ordinary induction, by which
men’s actions were observed to be guided. This is to say that
under God’s power fall, in concord with those purposive
activities continued, all events that would be in man’s interest.
As regards unlimited knowledge, it has been likewise ques¬
tioned in the light of man’s free will. It would be contradictory
that God have foreknowledge of all future events, which would
be predetermined accordingly, while man be given a choice in
their determination. As implicit in that argument, however, the
supposition that if man has a free will, the future accordingly
undetermined, then God does not know all is equally con¬
tradictory. The recognition that if future events are known then
they are determined discloses the understanding that knowl¬
edge is meant to be of the determined as reality or fact, in
accord with knowledge as considered before in this writing. To
affirm man’s freedom to decide a future not yet determined and
to also require that that future be the object of knowledge is
hence inconsistent with that meaning of knowledge as about
the determined. The question regarding God’s knowledge is
indeed that no fact, not what is not, be hidden from him. To
these facts belong the intents of a free will itself, everything but
the part of the future left undetermined, the determination of
which, perhaps depending on man’s use of free will, is in view
of God’s unlimited power nonetheless at his disposal. And that
his knowledge as well is unlimited revolves around the same
factors that play part in that power. That power was observed
displayed by the purposive bringing to man’s consciousness all
reality, which finds its presence in that consciousness. In it was
in fact seen to find its realization all that to man is. The real¬
ization was found to be either voluntary or involuntary, the first,
unlike the second, of purpose which belongs to man’s will
rather than of that of the body. And by the preceding God
possesses the power to fulfill either of those purposes, and
he possesses that power, manifested in the organism’s pur¬
posiveness, purposively. But by possessing the power to purposively fulfill all that may be purposed, that is the involuntary
objectives in consciousness as well as the objects of man’s
volition, he possesses by the deeper meaning of knowledge, as
means to fulfillment of purpose, all those means, to wit all
knowledge.
Turning to the other question regarding the plenitude of what
is demonstrated in the displayed foregoing proofs, to whether
230
God’s existence as cause of purposive events is unceasing, the
answer is supplied by the continuance of those events, taking
again free will into account. Because of the free will of sentient
beings, the course of things outside it, particularly of bodies of
those beings themselves, is, by being subject to those wills,
equally undetermined. Accordingly the response of those
bodies or other organisms to the changes wrought is contingent
upon those wills, to mean that only after their exercise is the
specific purposive behavior of the organisms decided. As a
result the existence of God as author of that behavior is a con¬
tinuing one.
Remarkably, the fundamental disclosure of God’s existence,
underlying the substantially alike demonstrations of it so far
presented, does not depend on the definition of God given,
although those attributes are requisite. Nor does it corre¬
spondingly depend on the definition of knowledge, with power.
To demonstrate the existence of the God who has specifically
the attribute of knowledge is universally comforting, because
it connotes immediately that he is aware of man’s concerns and
may act accordingly. But because the desire is at heart for his
power thus to act in behalf of man’s purposes, the appropriate
power of purpose, also, will define God.
The proof of God’s existence based on such a definition
retains the resemblance to the teleological argument. In that
argument occasionally only a supreme designer is spoken of,
without further particulars. His existence is alleged to follow
from evidences of design. The notion of design, as suggested,
is there ambiguous, having to do either with purposeful
function, or order of some other kind. Sometimes the word is
moreover meant to designate the product of a designer, to in
circular argument make positing of design in nature assume a
designer before inferred. The concept of designer is regarding
his product comprehended in addition in a passive sense, as of
one who fashions things, not who activates them. For these
reasons and in compliance with present terminology it will be
more fitting to speak of purposer and purpose than designer
and design.
As an ingredient of the proof purpose, as an occurrence of
appropriate scope in nature, was already considered in the
previous demonstrations. And that instead of defining God by
knowledge and power he may in the same active sense be
defined as purposer, of likewise appropriate scope, was
indicated in the preceding. To continue, similarly to the defini¬
tion of knowledge and power but without the same need for
preliminaries, purposer should without much dispute be
definable as producer or cause of purposive events. Possession
of this active power was explained as the interest in regarding
231
a being as a purposer.
The present proof then proceeds along the same line as did
the first one (p.224).
II. 1. God equals a purposer of concerned scope.
11.2. A purposer equals the cause of purposive events.
Hence, by II. 1. and 11.2,
11.3. God equals the cause of purposive events of
concerned scope.
11.4. Purposive events of concerned scope exist in
nature.
11.5. Events that exist in nature imply the existence of
a cause of them.
Hence, by 11.4 and 11.5,
11.6. A cause of purposive events of concerned scope
exists.
Therefore, by 11.3 and 11.6,
11.7. God exists.
Not even the designation of God as purposer is required,
furthermore, rendering parts of the proofs omissible. As
remarked before the beginning of the first proof, force or power
is determined in accordance with result, and presently it is
indeed a power determined by purposive events of certain
scope as effect that was observed the interest. The proof need
consequently only begin by taking as starting definition from
either the first or the most recent proof their first inference, 1.3
and 11.3.
111.1. God equals the cause of purposive events of
concerned scope.
111.2. Purposive events of concerned scope exist in
nature.
111.3. Events that exist in nature imply the existence of
a cause of them.
Hence, by III.2 and III.3,
111.4. A cause of purposive events of concerned scope
exists.
Therefore, by Ill.l and III.4,
111.5. God exists.
Further reflection may go farther still in simplification. It is
customary to query whether there must not be some purpose
to nature, with the implicit understanding of a responsible
power. Although such purpose is patently before the eyes of
men, in the form of life as propounded, it manages to escape
notice that that very phenomenon is one that is sought after.
While the significance of ordinary objects of knowledge,
specifically nature’s purposive events, can thus become
obscured to the adult mind, it can be wondered whether a
natural association of comparable events with a supreme being
is not made by men, and animals, in their early stages of life.
Sentient beings do in fact become aware in their incipient
experiences that something acts with regard to their interests.
The source of that action may for the most part be the mother,
but the young, learning very soon of causation, may not
identify the originator immediately with her. Of importance is
that there is a display of some thing’s activities for the purpose
of affecting the young’s interests, and its concept of what
performs these actions can be equated with that of a supreme
being, although not yet, and in the case of an animal presum¬
ably never, known by a name. The question can be then what
happens in the time between these primal perceptions and
those of the grown individual.
It is in conflicting view proposed that the concept of a su¬
preme being is a transference of the image of one’s father. But
as indicated in the preceding paragraph a transference can be
seen to rather take place in an opposite direction. And apart
from learning of something purposeful directed at itself and not
yet known as a parent, the very young perceives as likewise
purposeful other things as well. Not the least of these is its own
body. In known curiosity about it the infant soon discovers that
the body wants to, like other creatures in entirety, be treated
well, at the cost of pain. This preservational purpose of one’s
organism thus becomes a virtual surrogate for one’s own pur¬
poses, in animals thought to be of an instinct of self-preserva¬
tion. And the world at large can be supposed to at first appear
as equally purposeful. The playing of a kitten with objects as
if they were animate is conspicuous in this regard.
But in time the inanimate character of a large part of nature
is recognized, and to animals not even life in a plant may be
evident, although they may retain a sense of purpose regarding
their surroundings. With reference to one’s own body and those
of other individuals, they come to be identified with their
owners, because of the close connection between the two. It is
consequently not surprising that during growth one turns
attention to parents and others of control as beings of purpose
that affect one’s life, while losing sight of the containing bodies
and other organisms as purveyors of their own and, by sup¬
plying the powers of sentient beings themselves, overruling
purpose.
Because of that association of action directed, well-inten¬
tioned or not, at one’s interests with familiar persons or crea¬
tures, it is equally no wonder that a supreme being be expected
to be revealed in anthropomorphic form. It may justifiably be
assumed that God does resemble man in a sublime manner, if
for no other reason than the thereby present embodiment of
man’s highest longings, as suggested previously. But as also
233
expounded, the association of the relevant manifestations with
their appearance as manlike actions is unnecessary for their
establishment. The return to and uncovering of the fundamental
attributes at issue is the foundation of the ascertainment of the
existence of God, as represented by the foregoing proofs,
progressing from connotation of more particular human
qualities to the principal ones sought, and found in the pur¬
posiveness of life.
It may still be argued, in persisting skepticism regarding so
important a proof, that life may yet be artificially created.
Whatever possibilities exist, however, the furnished demon¬
strations could be invalidated only if the laws of nature failed
to prevail, not to speak of the laws of logic. Should artificial life
be possible, all power would not be placed into man’s hands.
Man does not create nature’s laws, including those of live
organisms, but is merely enabled knowledge of conditions
in which they take effect. Creation of artificial life would
accordingly only be creation of conditions under which life will
enter, just as conditions can be maintained under which life is
preserved. What causal laws characterize life and what is
implied by them will not be demolished but constitutes reality
on which everyone depends.
Section 2
FURTHER INQUIRIES
THE GOODNESS OF GOD AND MAN
So far these laws were determined to imply the reality of God
as possessor of appropriate powers. It is naturally of concern
how these powers are used—specifically, whether they are
used in behalf of man’s interests, whether God can be deter¬
mined to be good.
The questions regarding goodness, or the good in general,
have been major subjects of philosophical investigations,
known under the names of ethics and aesthetics. As in the case
of other subjects, the questions are on one hand about the
nature of the thing discussed, that is to say what sort of conduct
is ethical, or moral, and what an aesthetic experience, often
termed the perception of beauty, consists in; the questions
are on the other hand about the means by which those
implicitly desired goods as ends can be achieved, although
the means are sometimes identified with the subjects dealt
with, with the ends.
s
234
It can at once be observed that what the nature of ethical
conduct and an aesthetic experience is is again a matter of
definition. Though both are designated as good, of that conduct
only its final object, a well-being of the recipient, would be
likened to that experience, generally meant to be satisfaction
for its own sake, rather than gratification over something
because it leads to something else. That satisfaction thus per¬
tains to the purpose observed to characterize man and animal
(e.g. p.50 second par.). But there need as with other definitions
be no dispute about which entity called good is at issue, it
depending on choice.
In aesthetics the entity of interest does differ from that in
ethics foremost be needing to be only about the satisfaction, or
a perceived thing causing it, not about action that has someone
else’s similar feelings in consideration. It differs in addition by
having to do usually with, if not works of art, then other objects
of satisfaction that are supplemental, not with critical ones,
which are primary aims of morality. These satisfactions, con¬
centrating on works of art, are correspondingly associated
principally with those things that please through sight or sound,
that have been termed the beautiful.
The designation of beauty in this connection has fallen into
disfavor. It has long been acknowledged that works of art can
deal with ugly subject matter, and it therefore seems inappro¬
priate to regard them as beautiful. There is a difference, how¬
ever, between a work of art and the content it represents, and
it is the appeal of the work, not the content, which is referred
to by the work’s aesthetic value. What works of art are able to
accomplish is to be pleasurable while the same may not be
independently true of their subject matter. There is as a result
no cause to abandon the term beauty, be it only a word. It is
sometimes contended that neither beauty nor aesthetic appeal,
nay any appeal at all, is required as a mark of art. But in that
event a different meaning of a word, of art, is again adopted.
If desired, anything else may be named art, but it would not be
the subject contemplated.
This observation should not be mistaken for a frequent one
opposite in purport, which expects of art that it depict a content.
Content may not even be expressed, the requirement of art
meant to be that it, among man’s creations, please alone, in the
above aesthetic sense. It may of course do so alongside some
other function, insofar as that is something additional. Architec¬
ture can serve as example, and also illustrate absence of such
as pictorial content. Neither is art for art’s sake, if understood
as emphasis on formal features independent of the perceiver,
what is meant, with delight of the perceiver the test instead.
It could be maintained that in view of the profound truths art
235
is often said to convey, at least some of it should be evaluated
on the basis of content. But works of artistic merit often portray
matters of discredited value, and the most serious facts can be
presented in artless fashion, without it being decided to
accordingly change the meaning of art. This is not to mean that
what is appended to art is immaterial. If not intended to be
self-sufficient then it certainly counts what else it performs, and
it may be also appraised from that standpoint. As a matter of
fact beauty can be an expression of the fittingness of that which
is beautiful, as can for instance in nature appeal be a sign of
benefit.
Nevertheless whether concerning art or more broadly the
aesthetic, the meaning of the good in mind is of the satisfaction
for itself, or of the perceived quality in a thing that directly
elicits that feeling, not that has a less direct import instead.
This meaning has as indicated been taken issue with not only
by denying that some sort of satisfaction need result from the
thing perceived or that the thing need not be associated with
some other matter, but also by denying that any perceiver
response is of account, the value of the aesthetic object held to
reside elsewhere. The contention is supported by pointing to the
subjectivity of that response, which may differ from person to
person, whereas an aesthetic object is viewed as being one
objectively. It need only be repeated, however, that what is
called aesthetic is a matter of choice, and of interest at present
are things that cause certain satisfaction. If they do to no more
than some, then so be it, but there is good reason to believe
that things regarded as beautiful have at least potentially a
general appeal, although particular tastes in such as foods may
vary. The many differences encountered in responses can be to
a large extent attributed to the disrelated indirectness of some
appeal. A thing is often held beautiful because associated with
some other.
Questioned is also the directness of the appeal with regard
to literature. It is argued that being dependent on language as
conveyor of meaning, its beauty is to be found in that meaning,
frankly allowing that its aesthetic status may be in doubt. But
similarly to a painting, the most beautiful subject matter can be
described in poorest language, and a realization in literature
is that language, as other things, can have its own beauty. The
cognition of that is sufficiently displayed by rhymes in poetry.
This formal attention belongs in fact to that also given to things
of beauty in order to define them. Their formal aspect rather
than the perceiver’s favorable response would thus become the
criterion of their aesthetic character, as would something else
associated with objects of aesthetic appeal if accordingly
defined. Such definitions of the beautiful can be seen unsuited
if only because they by finding the objects to be of the appeal
intimate it as the criterion after all. Moreover rather than stating
what beauty is meant to be, they hypothesize the conditions
under which it is brought forth. In accordance with a previous
mention the beautiful as a sought after end is consequently
confounded with the means to it.
The irrelevancy to it, to its perception, of many factors sepa¬
rate from it was noted above. And the search for what in the
beautiful object itself, in its form in which perceived, accounts
for its appeal is comparably not germane.
The elusiveness of any rules regarding such form is notorious,
with formulas prescribing it known too well not to secure good
works of art. Where rules have been attempted, they often have
been lacking logically.
Cited in many ages as principal requirement for a work of
beauty is unity, yet the word is as indefinite as noted of order
(p.221 fourth par.), spoken of sometimes in place of it. The unity
is explained by qualifying it as organic, through inter¬
dependence of all the parts of the work. It can accordingly be
asked in what manner the parts are dependent on one another,
and the answer can be found in the very meaning of the
beautiful. As that which satisfies as explained, the beautiful
must please by means of whatever formal parts it is composed
of. If one part clashes with another, the import is that the work
does not please in that manner, which is to say that one part is
dependent on its compatibility with the other.
Into account is here taken the frequent reference to discord
in art, particularly in a musical work. The matter may be lik¬
ened to disquieting contents alluded to. By whatever standard
sounds may be held discordant, they please together in the
composition, for that is the criterion. Dissonance also finds its
similitude in contrast, regarded in unison with repetition like¬
wise an aesthetic requirement. The combination may also be
known as theme and variation or simply sameness and dif¬
ference. But the two need not come together for a thing to be
beautiful. The beauty of geometric uniformity, exemplified by
a square, has long been recognized, as has the appeal of
nature’s many irregular shapes, exemplified by a cloud. To say
that the aesthetic must hence have either uniformity or irregu¬
larity would exhaust all things, since either attribute is the
negation of the other, and all things are certainly not regarded
as beautiful.
Requirements that beautiful things possess certain formal
attributes usually furthermore engage in dual fallacy observed.
One is an inference that because some things of those attri¬
butes are beautiful all are (the fallacy of undistributed middle),
and the other an inference that because all things of those
237
attributes are beautiful all beautiful things are of those
attributes (the fallacy of affirmation of the consequent).
More apropos is that a protracted study of what formal attri¬
bute in things thus satisfies is beside the point, because what
does is decided by direct perception of them. It would not be
endeavored to instruct what in the taste of foods makes them
pleasurable, placing a restriction on what they may taste like.
It is to be understood that in question is not what in things may
be desired by people for whom a product would by someone
else correspondingly be prepared. In that case differing pref¬
erence or need would be considered. As earlier regarding
knowledge of meaning of words of one’s own usage, or any
knowledge possessed by an inquirer, the question is rather
what in aesthetic objects appeals to their producers themselves.
More generally the question as to what in beautiful things
makes them appealing has to do with their appeal to oneself,
and as one knows one’s meaning of words without hypothesis,
or has access to any of one’s own knowledge, one need not
investigate by given rules of form whether a thing has aesthetic
value, for one can ascertain the same directly. The matter has
bearing chiefly on the artist, the aesthetic form of whose work
is accordingly not guided by rules, but by satisfying perception,
granting the skills required.
In ethics the issues are of a similar nature. The original
meaning of the ethical is oftentimes lost in efforts to formulate
rules by which moral conduct should abide. That meaning was
indicated to be and is herein considered as conduct in behalf
of a well-being of other individuals, a well-being generally
called happiness and to be sought for all mankind, as well as
animals, within possible extent. As in other cases when deter¬
minations about a subject are confounded with it, however,
through consideration of ways by which that happiness would
be provided those of them that were found of a generality were
adopted as definitions of morality in place of its fundamental
meaning.
The results can be more disastrous than mere redefinition.
When the means that may lead to human benefit thus replace
it as ends, then on supposing them to be of higher value,
human lives are often sacrificed for the sake of them. Such
resulting actions then become in a turnabout the means to the
misconceived ends, and in consequence further misconcep¬
tions can ensue. Seeing those actions as unconscionable, the
oft quoted rule may be embraced that ends do not justify
means. The use of the rule, however, can likewise bring about
the opposite of the intended. If the end is the happiness spoken
of and the means contrary to a referred to code, the results of
prohibiting the means to the end can again be deplorable.
238
Whether it is the means or the ends that are important depends
on what they are, in an example of how morality can be
foresaken if dependent on rules.
The primarily important is that in accordance with the con¬
cerned meaning of morality equitable happiness be always the
object. This purpose can itself be considered the decisive rule,
as was in matters of fact the rule that an actuality be perceived
appropriately to what is meant to make it one. Inasmuch as that
purpose can be regarded as obligatory it can also be spoken of
as duty, although that word has unfortunately been attached to
the extraneous rules discussed. They are referred to as about
duty for duty’s sake, connoting a behavior which is aimless.
And the point is that the behavior to be regarded as a duty is
one aimed at other individuals’ happiness.
The characterization of the aim of ethical conduct as that
happiness is similarly to beauty criticized as platitudinous, as
not designating the desires of everyone, or as forgetting that not
everyone may find happiness in the same things.
That happiness be a platitude is of no relevance. Any con¬
fidence placed in what appear to be newly established facts is
self-defeating if it rejects the existence of constants, on which
all knowledge is founded.
That everybody’s desire may not be happiness depends
again on the meaning assigned to it. That meaning can herein
broadly be taken to be the very fulfillment of one’s desires.
Happiness is by some equated with pleasure, which in turn is
often assigned to sensory satisfactions. It is then argued that
there exist other satisfactions or desirable things, such as
knowledge. Concerned as said are things desirable for their
own sake, and it can be questioned whether knowledge or
other suggested goods belong to them. Anything so desirable,
what is the ultimate object of desire by anyone, can regardless,
however, be understood as meant by the satisfaction or happi¬
ness. It is the meaning by which desire is customarily connected
with satisfaction, the object of the former being the latter. The
two may not be in a sequence of time, desirability meant to be
merely satisfactoriness itself, perhaps of a present state. Nor
should the desirable be distinguished from the desired, under
the notion that some do not know what makes them happy,
desiring things undesirable. At issue is what makes individuals
happy knowingly, not what someone else thinks their lot should
be. In the same strain are of no concern allegations of uncon¬
scious desires, however discovered in absence of meanings
outside consciousness, with conscious happiness the object.
That all may not be made happy by the same things seems
the same question as the preceding on all wanting happiness.
But now it is granted that all want happiness, satisfaction, and
239
the query is whether what is satisfying to some is to others. As
with beauty, however, the matter is not relevant to whether the
happiness of fellow beings is the object of moral behavior. If a
different thing brings happiness to each, the things sharing no
other attribute, it is still the benefit intended, though its
realization would be quite difficult.
But as it can be presumed that the beautiful is to a large de¬
gree the same for all men, so it can be presumed that with all
men the immediate conditions for happiness are likewise to a
large degree the same. Similarly to differences by regarding
things as beautiful for reasons other than direct appeal, dif¬
ferent desires can be held to substantially be for conditions
expected to lead to those more fundamental in conducing to
happiness. And because these fundamental conditions for
happiness can be presumed to be considerably the same with
mankind in general, they make feasible the discernment of
means that may generally apply in seeking moral ends. Such
a means is the universally sounded golden rule, that one
should do onto others as one would have them do onto oneself.
By having one’s own desires as express criteria the rule can be
seen as manifestly built on the assumption that the circum¬
stances for happiness are the same for all. Less manifestly this
assumption can be viewed as the foundation of other, more
specific, rules as well, such as contained in the ten command¬
ments, in accordance with which one would wish to be treated.
As indicated, however, ethical rules as means may not be
always appropriate for ensuring human happiness as end.
Even if the golden rule, expressing the basis for others, is
interpreted to apply to only ultimate sources of satisfaction,
these may not be uniform from person to person. As men can
differ in their taste for a food, they can differ in their vision of
an ideal existence. In this regard it may be recalled that the
aims of morality are predominantly the satisfaction of more
fundamental requirements than the sensuous ones that may be
represented by preferences of the palate. The concept of
happiness as the aim correspondingly warrants a yet more
profound characterization in addition to the foregoing one. It is
not the indulgence of momentary appetites which is of
concern. Instead it is a comprehensive mental contentment, in
which the senses, as signs of physical needs specifically, have
a part, but which includes such intangibles as receiving love,
or dispensing it, in reciprocal morality. This deeper meaning of
happiness, of contentment, is easily perceived as a general
one, inasmuch as grosser pleasures would be admitted not to
bring happiness, contentment.
Recognizing the encompassing nature of happiness as
compared to particular pleasures, the conditions for happiness
s
240
may nevertheless likewise not be identical and the golden rule
not apply unequivocally. The golden rule is nonetheless of
exceptional value, especially with regard to tenderest feelings.
Men regrettably often think themselves and perhaps those near
to them to be of greater sensitivity than others, if not imagining
others to be devoid of feeling altogether.
While rules in ethics can perform amiss, especially the more
particular the conditions they would prescribe, there may be
more general conditions for happiness that are commonly
required, and to which a rule may accordingly apply. Aside
from fulfilling others’ more direct desires, one can provide
means by which they can be fulfilled by their possessors.
Among these means, next to physical ones, was indicated to be
knowledge, acquired, in all learning of reality, through
dependence on information received. Consequently in order
that information one imparts be dependable. In accord with
commonly sought trustworthiness, one should always be
truthful, there are occasions of course when someone’s
knowledge of a truth is expected not to serve a good end, but
in that case witholding the truth rather than lying is in order,
since to allow for exceptions destroys credibility. But it should
also not be thought that by observing a satisfactory rule one’s
obligations are always discharged.
The widely acknowledged exhortation that one should
always be truthful, which includes keeping a promise if within
one’s powers and if any other required conditions are met, is
one defectively justified in conjunction with the by many en¬
dorsed categorical imperative. In accordance with it one should
act only on maxims one can will to be universal laws. It is
argued that if, in a rule opposite to the above, it were a uni¬
versal law that everyone in difficulty can make a false promise,
then no promises could exist, since no one would believe them.
It is concluded that one should will truthfulness a universal law.
Objectionable about this and conjoined arguments is that
any purpose for the commended behavior is disavowed, in
particular any benefit that may accrue to someone. Thus
explanation is wanting why it is bad if a listener must disbelieve
a liar, or if as a result promises cannot exist, not to mention that
avoidance of disbelief through truthfulness is already a
purpose.
Further the inference that no promise could exist as a result
of disbelief has no basis, and it signalized a shift of attention
regarding ethics from concern with benevolence to forms of
non-contradiction. The suggestion is that false promises could
not exist because contradictory. But they would merely be
untrue, as is any statement inconsistent with fact. More
pertinent is that the proposal that a contradiction or other
241
logical falsity is immoral is out of keeping with the intended
meaning of morality.
Furthermore even provided the reasons offered for not
making false promises or lying are acceptable, the conclusion
about truthfulness does not follow. If it is wrong that one should
always lie, as was also posited, or that everyone in difficulty
can make a false promise, it does not follow that one should
never lie, or that a false promise cannot be made when in
certain difficulty only or for some other reason, or that any of
these cannot be done by only some.
On this ground it would seem that though it be found unac¬
ceptable that one should for arbitrary reasons lie or make false
promises, there may be times when doing so could be justified.
This view is in fact adopted by many, maintaining that for
instance at one’s deathbed one can be told a pleasant lie, since
not verified. The requiring of that latitude, however, itself
implants doubt about someone’s truthfulness, to defeat its
purpose.
More troubling than the arguments on particulars like truth¬
fulness in prescribing the categorical imperative is the prescript
in general. That it mean the same as the golden rule is with it
expressly repudiated, and one can be at a loss as to what
action be willed universal law, for one can so will most any¬
thing, including what is to one’s own advantage. The indication
is that the action be moral or, although the categoricality is to
preclude a purpose, benevolent. But if the action is asked to be
moral then the rule is circular, describing morality via morality,
i.e. failing to be more illuminating so modified. And if the
action is asked to be benevolent then it should by the rule be
only taken if it can be willed a universal law. Therein lies the
rule’s injuriousness. It replaces action guided by humaneness
with action according to prescribed procedure, the baleful
consequences of which doctrinaire ways are legion.
The past approaches to ethics, as well as aesthetics, as
an elaborate field on the order of a science can in fact be held
ill-considered, similarly to before noted treatment of the human
mind as if an unconscious mechanism, determining behavior
in accordance with to be studied rules, discounting free will.
The question of the concerned moral conduct is, as in the case
of the beautiful, largely confused with its meaning, which not
only informs of its sought after nature and is a matter of defi¬
nition, but is understood by everyone, including children, as
kindness. It is because the nature of moral conduct is thus
common knowledge that people are held responsible for, sup¬
posing them ascertainable, acts of ill will but not acts of igno¬
rance. As to particular acts that can be regarded as moral, it
was already observed that certain fundamental conditions for
242
\
happiness can be viewed as applying quite generally, as can
the perception of certain things as beautiful. Corresponding to
this view is again the golden rule, an expression of a natural
comprehension that to be kind is to treat others as one likes
to be treated oneself. Morality by consisting in the fostering of
the happiness of others can in point of fact be viewed as
founded on the conception that others, in their need for hap¬
piness, possess sensitivities like one’s own. The golden rule
correspondingly does not set up a procedure one must follow,
merely speaking of like happiness to be striven for for all, just
as a work of art may want to be of beauty by a shared meaning
of the word, without requirements regarding the work’s form.
Beyond thus meeting the desires of everyone, a rule in a nar¬
rower sense would accordingly apply to everyone in every
instance of the thing at issue. A rule that would apply to every
aesthetic object beyond its appeal, or to each of a kind, was
noted hardly formulable. When expanding appeal to the
sphere of ethics, however, certain consistencies, extending to
indirect connections, are observable in things beyond their
production of well-being. Unlike the concerned results of cer¬
tain kinds of pictorial or musical works, the tastes or smells of
certain kinds of things can fairly regularly be determined to be
pleasant or unpleasant. More broadly there are kinds of things,
notably those vital to the preservation of one’s body, that are
known to generally produce pleasure, as do nutrients, or pain,
as do injuries. For such reasons provision* of sustenance and
shelter are regarded as foremost moral obligations. And of
consequence is that these connections between certain con¬
ditions and resulting welfare have to do with sciences external
to ethics. What conditions favorably or unfavorably affect the
body is the province of medicine and by extension biology,
physics, and so on.
Whereas it is rules of science, or rather laws of nature under¬
lying it, that thus determine how moral objectives be attained,
one can of course act morally without looking to scientific
verification for each of one’s moves. As expounded previously,
scientific knowledge presupposes an ordinary knowledge of
worldly reality, and within that knowledge falls acquaintance
with man’s and the animals’s general needs such as mentioned
sustenance and shelter. Furthermore in personal relations one
is likely to know more about the other individual’s situation
than would a scientific thesis that speaks about everybody. This
does obviously not mean that one has the ability to achieve
every moral purpose, as does not science alone, both it and
knowledge of individual need being correlates of each other.
What matters is that one is equipped to act morally, in particular
by means utilized for oneself in like situations, on the preced-
243
ing understanding that certain basic requirements are the same
for other beings.
As regards moreover not the intermediate requirements of
such as the body, but the desired ends of the individuals
themselves, their happiness, sciences are still less of
dependency. One simple way of learning what other persons
desire is to ask them, for the concern as noted is what makes
others happy in their own consciousness, not in the opinion of
someone else. Even animals manage to communicate whether
something pleases or displeases them, and men’s tool of
communication, language, was observed to be one of the
common acquisitions through induction, and one whose
understanding by science is presupposed in its ascertaining of
human desires, if not presumptuously again telling others what
their desires are. Another such common induction was
observed to be, with the aid of language, that others share,
beside one’s bodily functions, the mental ones, along with
corresponding experiences. To these experiences belong
feelings, the satisfaction or dissatisfaction with circumstances,
the objects of moral concern. Hence when particular commu¬
nication is hindered one can use as guide, in likeness to the
creating of art, one’s own response to a contemplated condi¬
tion, in moral pursuit of it for another being. Viz insofar as the
conditions for happiness are the same one has to for their
knowledge only rely on one’s own feelings, without injection
of scientific principle, and insofar as the conditions differ a
principle is of no avail.
A person thus has a comprehension of what consists in moral
action, which includes, alongside seeking of ultimate con¬
ditions for happiness of others, the affording of means by which
they may themselves attain that end. Among these was seen
to be knowledge imparted through truthfulness, and most
moral action is in fact of furnishing such implements, as in pro¬
viding circumstances for physical welfare, the receiver enabled
then to make use of them. That action be so divided is in justice
when considering that if one person be regardful of another’s
interests, the other would share in responsibility, each doing
with regard to each as capacitated.
This mutuality suggests that men can also so arrange their
lives as to benefit one another as best they can due to their
developed competencies, such arrangements, larger than of
the natural family, known as political units, chiefly states.
Political theories abound, and by the foregoing important is
that beside the fundamental goal of attainable happiness for
all there ought to be a refraining from inflexible rules, in par¬
ticular from imposing by government conditions not in accord
with the people’s desire. That desire once again, not someone’s
conception external to it, was noted the criterion of the cir¬
cumstances for happiness, its political expression usually called
democracy. Emphasis on this desire can be in conflict with
both what are termed the political left and right. In one, the
endeavor is for instance to abolish private ownership of prop¬
erty; yet it is doubtful that there are many who do not wish to
treasure things as their own, with even animals resenting
intrusion. In the other, individuals would be left to their own
resources in fulfilling their basic needs, such as care for their
health; but the very organizing by society begs for securing by
it of everyone’s essentials one cannot obtain alone. The focus
is on bringing happiness in concord with shared wishes, rather
than on implementing a conceived political system, the appro¬
priate morality requiring by the aforesaid no further govern¬
ance than the knowledge supplied by the particular expertise
of those chosen to articulate and carry out those wishes.
The matter of morality being a commonly comprehended
one, it appears mainly requisite to in informing of its attributes
make known what the subject discussed, with largely avoid¬
ance of rules, unlike in searches for further attributes by prin¬
ciple of objects known. The searches regarding morality were
indeed seen to revolve to a large extent around its definition.
The question posed was noted part of that concerning the good,
which has been held indefinable, because upon proposing a
meaning, other things were found to be named good as well.
But as.reiterated, not only can a word be defined, but meaning
is a matter of choice, and the choices can be several, resulting
in different senses. Accordingly as indicated at the beginning
of this section, there are diverse kinds of things referred to as
good, accounting for the discrepancy about the word. As a
good was noted to be designated an aesthetic experience or its
object, the beautiful, as well as presently discussed ethical
conduct and its objective, happiness. Good are called also the
immediate circumstances for happiness, and anything that may
by its workability lead to it. And good is the person who seeks
these for others, who acts morally.
The last designation suggests that of interest is not merely
that to seek the happiness of others defines moral conduct, but
that it is a precept by which man should abide.
In keeping with the tenor of this book, or with expectations
in general of supporting of statements made, it could be asked
how the moral precept is justified. A longstanding contention
is as a matter of fact that ascriptions of value in ethics or
aesthetics cannot be substantiated, that they even lack truth or
falsity, as mentioned in the introduction. As said there the
.contention is related to the questioning of facts not verifiable
through the senses, and it is accordingly argued that moral
245
injunctions are not statements of fact, the truth or falsity of
which statements alone it is possible to establish. It is specif¬
ically argued that because moral injunctions say what ought to
be, they cannot be inferred from what is.
The argument is again falsely dependent on language,
however. To say that something ought to be done is similar to
saying that something is necessary, both expressions sometimes
substituted by phrases containing ^’must”. And in both cases it
is as before simpler to say ‘This must be done” than the what
would be held true or false “If this is not done then the desired
result will not be obtained.” That result can for particular moral
codes be considered to be the happiness spoken of. To hold
then that these codes need no justification is equivalent to the
mentioned hurtful proposal that they be observed for them¬
selves, not as means to beneficial ends.
But at the base of the supposed unprovability of moral codes
can be held to lie the thought that nature imposes no moral
laws, unlike causal ones and perhaps those of logic. Thus it
may seem that while the acknowledged laws cannot be
ignored without risk, moral ones can since appearing arbitrary.
But even if arbitrary, laws imposed by men can evidently not be
ignored either. Children know that if not behaving they will be
punished, and the like applies to laws of governments. Moral
law is, however, not so arbitrary. Man-made laws or rules,
including some called moral, can be contrivances of tyrants. But
many result from the very same want prompting one to obey
laws of nature. It is the want to be satisfied, to be happy. It is not
only the inanimate that affects one’s happiness, but animals
and particularly other men as well, to give rise to assented to
laws or rules aiming at happiness for all. Underlying is what in
affinity with past usage may be termed a natural law of
morality, of seeking the others’ happiness. For one should if for
no other reason seek it because as one cannot with impunity
go counter to laws of nature, one cannot with impunity mis¬
treat others.
One prefers to think that moral behavior is also appropriate
under all circumstances, however, not only when recompense
by others is at issue, and some may ask what the justification
may be. Religion maintains retribution by God, similar to the
responses of men. These are perceived as actions coming from
outside oneself, but as suggested by the preference of moral
conduct independent of circumstance, the motivation can also
be found within one’s own feelings.
The feelings are best known by the name of love. The word
is like others used in various senses, and it is certainly not its
meaning as much enjoyment of its object such as food that is
here considered. In this sense the meaning can become the
246
opposite of the intended, connoting selfishness instead of
selflessness. In its pertinent use the word is likewise ambiguous.
It sometimes and with no less importance does not refer to
feeling but to action, as implied in the entreaty to love one’s
neighbor. The present meaning of love has to do with the
feeling behind that action. It is love as a feeling of benevo¬
lence, of wanting the happiness of another. The feeling is
sometimes called sympathy, referring to the sharing of another’s
gladness or distress, the meaning implicit in preceding love, by
which happiness of another is desired by oneself as well as by
the other. This feeling can accordingly be taken into account as
motivation for morality, motivation that reaches farther than
that of external requital, so as to be always present.
That sympathy be the ground for moral action has been
contested as inappropriate because not everybody may feel
sympathy or do so in equal degrees. Other grounds adduced
for moral action, however, are likewise not certain to fit every¬
one. Whichever the rewards or punishments upheld to induce
men to morality, their equal effect on all can similarly be
questioned. It is true that men will often respond to coercion
rather than performing the same action of their free will. But
supposing the action a moral one and not some other, the
implication is not that the person is void of sympathy. Instead
it can be presumed that the person does not let such feelings
interpose, under some notion of advantage derived. Coercion
may prevent the assumed advantages, but being heedless of
one’s feelings may have a different result. Mankind is quite
conscious of the many cases in which one’s self-centeredness
will subdue one’s feelings of compassion. Of equal awareness
is the affection in man’s and the animal’s care for their young.
The last indicates that love, one’s feelings for the interests of
others, is as fundamental a function shared by sentient beings
as are the other functions previously cited.
On finding one’s own happiness the justification for acting
toward the happiness of others, whether the reward is the
internal one of love or the external one of requital, some might
ask what the justification is for wanting to be happy. In the vein
of the arguments that reasons cannot be given for morality, it
is advanced that for some moral rules more general ones can
be given as reason, and for those perhaps others, but that to
escape infinite regress, a basic rule cannot be justified. On
further thought it is admitted that a like issue can be raised in
regard to all knowledge by asking for a reason why any
claimed basis for it be accepted, and it is maintained that in
real knowledge basic facts can somehow be agreed upon,
without any more proof. But the throughout this volume found
evasiveness of basic facts belies this belief. Objectivity versus
247
subjectivity is moreover irrelevant. As also seen in what went
before, knowledge does not depend on consensus, but on
correct criteria. It will be well in this respect to examine
justification.
Expressed as reason for something, it was earlier already
found confused with cause. The cause of an event is sometimes
called the reason for it, but so is its purpose if it has one. Fur¬
thermore the criterion on which a determination is based is
called its reason likewise. For examples, as a reason for the
finding of circulation of the blood may be given in one sense
its movement due to the pumping of the heart, in another sense
the function of distributing oxygen in the body, and in a third
and incongruous sense scientific observation. A commcJn
ground can be detected. In determinations their criteria as
reasons provide certainty, also present when knowing the
function as reason of organic occurrences, and what events as
reasons cause what others. The certainties were further seen to
serve purposes of man, the purposes reasons, once more, for
one’s actions.
Among these kinds of reasons it is only material causation
found to regress indefinitely, in conformity with the principle
that all physical states have a cause. Even this infinity, however,
is qualified. It is only required as long as the cause is physical.
The free will as cause was observed to be uncaused. In the area
of logic it was likewise observed that principles are not always
based on broader ones, but can on their own be perceptually
ascertained. Such ascertainment was in fact explicated to
underlie all knowledge, with certain perceptions the criteria of
truth, with no further confirmation. The adequacy was in places
explained by equating description with explanation. No further
reason after a decisive one also concerns its sense as a pur¬
pose, as in general indicated in a former chapter (e.g. p.50). In
an organism the purpose of preservation could be held
ultimate. But in man the ultimate purpose is consciously
contentment, happiness, the eventual reason for intermediate
purposes. It is sought for no other reason but itself, as observed
in this section also with respect to its meaning, and hence, lest
engaging in contradiction, requires no further reason. The
obviousness may be made clear by stating redundantly that
one is quite happy to be happy.
The possible attainment of satisfaction with no other end but
itself was observed to (e.g. p.234 first par.) also characterize the
good regarding beauty, as well as, when the satisfaction
concerns the sought after happiness of others, the good
regarding the person who by that endeavor acts morally.
Corresponding to this meaning of goodness applied to a person
is its meaning applied to God. By the goodness of God is meant
248
that his purpose is the happiness of man and other creatures.
In searching accordingly for whether God is good it can be
taken into account that it was through purpose itself as
exhibited in organisms that he was found to be revealed. The
remainder of nature does not behave purposively, whereby it
might display either benevolent or malevolent intent. And the
encompassing purpose of organisms was seen to be their
preservation. This purpose was observed to in sentient beings
be provided for also by, in cooperation with components of the
body, those of the mind, including feelings of pleasure and
pain, the feelings moreover extending in sympathy to the inter¬
ests of fellow beings. Therefore insofar as preservation is the
interest as road to happiness of man or animal, God is good.
The preceding may be viewed as a reply to what is known as
the problem of evil. It is asked why if God is all-powerful and
good does he permit pain. But in view of its function as a
warning about bodily harm, pain is not synonymous with evil,
on holding evil to connote ill intention. Yet it may be remon¬
strated that while the organism thus functions toward selfpreservation, it cannot be said to, in the economy of a world in
which one organism is sustenance of another, function toward
preservation of others.
In answer it may serve to be remindful that it is in the end
consciousness which determines the nature of reality, and
whose contentment, even in the eyes of other individuals’
sympathy, is of interest beyond a physical universe. The fortunes
of this world are not necessarily the ultimate ones. It should be
considered in this respect that conditions inordinate in anguish
become removed from consciousness in forms like faint. But it
may be persisted that if bodily existence is not the only one why
should one suffer at all in connection with it, why should one,
it may be asked, suffer altogether, and not be happy always.
The conditions for happiness may in this regard be again
examined.
It was noted that by happiriess is meant a comprehensive
mental contentment, and the contentment may contain par¬
ticulars which in themselves are grievous. Happiness can thus
be compared to the mentioned work of art in which depicted
particulars displease apart, but please as part of the whole.
Indeed as may be also the case in a drama, what dissatisfies
independently may become satisfying because of the antici¬
pated outcome. This is in accord with the experience that not
all that is painful is unhealthy, or that not all that is pleasurable
is healthy. Man learns to renounce momentary satisfactions for
more lasting ones. And that in the process the preliminary
conditions become likewise satisfying is evidenced by the
overwhelming majority of things that please as a means to
249
something else, rather than for their spoken of own sake.
Knowledge or money, prized in opposing ways and having not
even direct physical functions, are prominent examples.
Furthermore even failures and misfortunes represent and are
valued as instructive experience. Any of these attitudes is
expressed by the adage that all is well that ends well.
Still it may be protested why there should be troubling
particulars, why all should not be trouble-free. Again those
particulars can be compared to darker incidents in a literary or
other composition. Beside the value of the incidents in the light
of the larger work, the work cannot be fully appreciated
without the incidents. In a broader context it is said that one
cannot appreciate the good without the bad. In truth fulfillment
unearned, attained without hardship, possesses for man no
lustre. And the more lasting the ultimate fulfillment the more
in imbalance is a sparsity of merit.
It may continue to be argued that although the worthiness
appeals to the human sense of justice, things might be
otherwise. Man might be happy under other conditions. But
such argument is no longer relevant. At issue is what in fact is
basis for man’s happiness, and whether that basis is put at
man’s disposal, as signification of God’s benevolence.
As disclosed in the previously said, man is furnished the very
attributes that in confrontation with difficulties enable one to
seek fulfillment of one’s purposes, in consonance with the
described preparatory conditions for man’s happiness, to by
the expounded imply God’s goodness.
HEAVEN AND IMMORTALITY
The preceding observations call forth the conceptions of
an afterlife and heaven, the circumstance of consummate
fulfillment or happiness.
Because heavenly ideals may differ, because they should
coincide with individual yearnings and be envisioned
accordingly, little more need be said on the nature of heaven.
It may only be noted again that desires can often be held to be
alike in certain fundamental aspects, and therefore the same
can be assumed in the present regard.
As complete realization of one’s longings heaven is not
thought to occur in this world. Neither a state of eternal bliss
nor a temporal dispensation of justice is a culmination awaited
in an individual’s worldly existence. Consequently it is hoped
that there is an afterlife, that human beings, and others,
moreover are immortal.
The possibility of immortality, of survival of the self after
bodily death, was indicated (p.l2 third par.) to have been
250
denied on the proposition that the self consists at least in part
of one’s body. The indication was given in connection with
definition, which can be made to so characterize the self as to
preclude the inference of immortality. As likewise observed
(e.g. p.50), however, the self universally meant to survive the
body is of course a disembodied one, often referred to as soul.
This self was earlier described more specifically as an element
of consciousness distinguished by a sensibility characterized by
desire for happiness, attained perhaps in a future life and by
the self newly embodied.
Granting that the self resides in consciousness, argument is
also advanced that consciousness is in some manner identical
with part of the body, the impossibility of which sameness, too,
was explicated before (p.56 third and fourth pars.). Having
to do with logical, intrinsic, rather than natural, extrinsic,
connection is also the proposal made that mental entities
devoid of a body are unintelligible. But one need only abstain
from observing one’s body to find that one’s consciousness does
not vanish, one’s body indeed seen as only part of many objects
of one’s perception.
It remains to find that the existence of consciousness is not
inexorably tied to the existence of the body without being part
of it, that the mind, the self, indeed may always survive. It
should at first be considered that the death of the body does not
signify its disappearance, but its disintegration. This happens
in conformity with conservation law, discussed in a previous
chapter (e.g. p.74 last par.). Hence should the death of the
body result in the death of the mind or self, the implication
is likewise not that the mind or self disappears, but that it
somehow separates into pieces. Indeed conservation law is as
was expounded a fundamental inductive experience regarding
the permanence of the world in its elementary material, as
compared to ephemerals such as objects of dreams. Hence
whatever in sentient beings the basic material of consciousness
or the self, it, too, can be considered permanent. Unlike the
case in organisms or other material entities, further, conscious¬
ness or the self are not meant to be of a particular arrangement
of constituents. That is to say, they exist if their constituents,
which need not be multiple, exist. In consequence, with the
constituent, the material, of the self continuing to exist, so does
the self.
Argument may continue by saying that conservation law
applies to material things, but that it may not apply to the self,
because no self has been observed to remain after death. In
this regard it can be remarked that if the self does not remain,
neither does reality, found dependent on one’s possible
perception of it (e.g. p.44 last par.). It could be replied that
251
reality may not remain for the deceased, but it may for others.
A rejoinder can be that since reality is as an ingredient of
perception separately determined by each individual, it would
cease to be each time an individual would, without the
continuity ascribed to it. More of pertinence to answering the
question whether worldly conservation applies to matter but
not to mind can be found in the very continuity of worldly lives,
of species.
As did induction of other minds depend not on constant
observation of them, but on the observed constancy of other
basic features in accordingly alike beings (ca. p.lOO last par.
through p.l02 third par.), so does the induction that selves
continue to exist not depend on their observation, but on the
constancy of continuance of other like basic phenomena. The
self when occupant of a living body was seen as a force which
is the purposeful cause of certain of the body’s activities,
simultaneous with its other purposive activities, those of the
organism. These signify the organism’s life, and correspond¬
ingly when the organism dies, it is not only the self that ceases
its activities in it, but the body’s activities end as well. Yet the
force which is the cause of the last mentioned activities does
not cease to exist, manifesting itself in the offspring or other
members of the species. It is the force herein identified with
God. Viewing ther rest of nature equally by its activities, foun.d
to make it significant, the conservation of worldly entities, their
continuation, is thus learned to in all known instances extend
both, to those instrumental in events that are mechanical, to
entities that may be called material, and to those instrumental
in events that are purposive, to entities that may be called
mental. Inasmuch as all laws regarding nature are inferred
from constancy in all known instances, it can be concluded that
the selves of man and animal continue to be.
Immortality alone may not be thought to provide happiness,
nor, should it for some, may it be held to be the wish of all.
Nevertheless it can as in the case of other fundamental feelings
be presumed that contentful immortality is of deepest shared
desire. Whether it be immortality or other objects of ultimate
happiness that are pondered it is the goodwill of God that in
accordance with the aforesaid ensures that the destination
of the course of the trials of everyone be heavenly existence.
252
\
INDEX
abstract entities 10, 36, 39-40, 48
acceleration 79-81, 89, 92-93
active mind 62, 223
aesthetics 2, 233-237, 241, 244
affirmation 16, 63-66
analogy 221
analytic connections 4, 11, 31, 36
analytic philosophy 12
analyzability 28, 33
angle 124-133, 171
definition of 124
degree of 125-126, 130
reflex 125
right 124, 128-133
definition of 128
straight 128, 130, 133
animate, the 96
Anselm, St. 219-220
a posteriori 13, 63
apperception 51
a priori 13, 63
Aquinas, St. Thomas 219, 221
argument 5, 8, 13-14, 22, 30
argument form 208
Aristotle 48, 114, 140
art 234-237, 248-249
assertion 16, 18, 23, 176
atom 47
attention 65-66
attribute 17-18, 21-22, 26-27, 29,
33, 38-42, 48, 55, 69, 101-102,
105, 114, 135, 137-138, 144-
146, 148, 150, 157-158, 161,
164-167, 170, 175-176, 184,
187-188, 192, 210-211, 219-220
definition of 38
attribution 145, 151-152, 161, 170,
172, 175, 178, 184, 191, 194,
199-200, 203, 213, 225
attributive logic 135, 141, 144-146,
148, 160-161, 171
awareness 9-11, 13, 18-20, 23-24,
27, 29, 46-47, 52, 65-66, 139
axiom 1, 105-106, 112, 137-138
beauty 233-237
behavior 18-19, 24, 57, 61, 241
behavioral science 5, 60-61
behaviorism 23-24, 56-57
belief 13
Berkeley, George 46
body 12, 50-53, 55, 62, 94-96,
99-100, 223, 228-230, 232,
242-243, 248-251
brain 56
Carroll, Lewis 139
categorical imperative, espousal
of 240-241
causality 36, 58-59, 61-62, 70-103,
152, 172-173, 176, 217, 223,
251
definition of 71
final 71, 95-100, 102, 251
mechanical 95-100, 102, 251
certainty 2-3, 5-6, 13, 34-35, 106,
112, 134, 138, 177, 217, 247
chance 34-35, 37, 59, 74, 98
circle 125-129, 131, 133, 174
definition of 125
great (equator, meridian) 126,
133
classes 20-22, 115-117
classification 41
commutativity 169-170, 174, 195
complements, la\A/s of 37, 135-158,
173, 199, 202, 212
complex entities 29, 52
concept 3, 6-7, 12, 19-30, 34,
42-43, 48, no, 114, 140, 175-
176, 217-218, 220
definition of 19
concrete entities 10, 36, 39-40, 48
conclusion 3, 5-6, 8, 17, 31-32, 58,
77-78, 82, 93, 107, 134, 136-
137, 139, 141, 150, 158-159,
163, 165, 172, 174-177, 181-
186, 193, 195, 199-200, 202,
205-206, 210-211
253
conditional 207-208
conditioning, supposition of 61
congruence 130-131
conjunction 170, 174
connotation 15-16, 39
consciousness 2, 10-12, 16-17, 20,
22-26, 29-30, 45-47, 49-50,
52-54, 56-57, 60, 62, 64-67,
73, 101-102, 137-139, 143-145,
152, 209, 223, 228-229, 238,
248, 250
conservation law 74-76, 250-251
consistency 133-134
contradiction, law of 3, 13, 27,
135, 141
contraposition 159, 162
conversion in logic 171, 203, 212,
214
countability 120-121
cylinder 128-129
Darwin 96
declarative sentence 15-16, 18,
108
deduction 2-7, 13, 26-27, 29-31,
33-35, 41, 48, 63, 65-66, 72,
85, 103-217
of instances 105-106, 139-141,
158-159, 163-165, 203-204,
208, 214-215, 217
definition 2, 6, 11-15, 17-20, 26,
28-43, 47, 51-52, 64, 69, 82,
105-106, 133-134, 136-138,
143, 146, 168, 170, 174-175,
180, 218
ostensive 102, 142
demonstration 8, 13-14, 16-17, 28,
30, 72, 82, 140, 159, 181,
216-217
denotation 15-16, 39
Descartes 55
description 17, 75, 79, 218, 247
determinism 34-36, 57, 59, 114-115
diagram 140, 153, 159, 177, 183-
186, 198, 201-202, 205
diagrams 6, 41, 79, 86, 99, 109,
125, 127-133, 140, 142, 144,
147, 152-153, 159-161, 163,
165, 168, 170-171, 177, 184,
186, 190, 199, 201-202, 204-
205
dichotomy 55
disjunction 170, 174
distance 124-126, 128-129
distinct entities 23-25, 40, 43, 68
double negation, law of 32, 36,
144, 176
dualism 55
eduction 6-7, 12, 25-28, 41-43, 48,
175
ego 50
Einstein, Albert 76, 80-81
Eleatic philosophy 122
empiricism 23, 57, 63, 140
British (classical) 23, 57
energy 74-76, 91
kinetic (of motion) 75-76, 91
potential 75
entailment 152, 167-168, 176,
178-179
enthymeme 17, 30, 225
entropy 97
epistemology 6, 48, 57
equality 130-131, 136-137, 146, 161,
165-168, 171, 193
equilibrium 88, 90
equivalence 144, 166, 171, 193
equivocation 27, 218
essence 12, 34, 38-41
ethics 2, 233-234, 237-247
Euclid 1, 8, 109, 123-125, 128,
130-134, 167-168, 174
common notions of, 1st, 2nd
and 3rd 167; 4th and 5th
168
Elements of 8
postulates of, 1st, 2nd and 3rd
128-129; 4th 129-130; 5th
123, 131-133
propositions of, 1st 174; 2nd
130-131; 4th 109, 131; 6th
109
Euler diagram 140, 185
euphemism 28
evil 28, 248
evolution, theory of 96, 228
excluded middle (or third), law of
4, 135, 141, 146
exclusion, laws of 142-146, 148-
149, 156-157
existence 1, 3, 6, 10, 18, 24-26, 38-
39, 61, 93-94, 106, 115, 135,
142-143, 145, 149-152, 161,
254
172-173, 185-186, 190, 197,
199-201, 203, 206-209, 212-
213, 219-220, 225
existence theorems 197
existential import 150, 172, 185-
186
existential logic 135, 141-144, 151-
152, 157, 159-160, 171
experience 13, 37, 57, 68-70, 73-
74, 85, 101, 112, 122, 216-217,
243
explanation 75, 79, 81, 89, 247
exportation in logic 198-199
extension (matter) 67-70
extension (meaning) 15
extensionality, axiom of 41, 165
external reality 67-103
extrinsic connection 4-5, 63, 73-
74, 127
fact 2-3, 5, 11, 13, 16-17, 26, 34-41,
45, 85, 95, 98, 107-108, 123,
134-135, 150, 158, 173,
175-176, 220-221, 225
fallacies 1, 5, 27, 77-78, 134, 174,
183, 216, 221, 236-237
affirmation of the consequent
77-78, 134, 183, 237
denial of the antecedent 216
undistributed middle 221, 236
falsity 2, 11, 18, 25, 108-110,
113-115, 134-135, 151, 158,
172-173, 180-183, 244-245
Fermat 210
Figures, A 6; I 41; II 99; III 140;
lll.l 142; III.2 144; III.3 147;
III.4 152; III.5 159; III.6 160;
III.7 163; III.8 165; III.9 171
finitude 122-123, 132
force 75-80, 89-93, 95-96, 98, 100,
224, 231, 251
centrifugal 89-92
motive 90-93, 98
frame of reference 81-88, 92
inertial 81
free will 34-35, 45, 57-66, 95, 217,
228-230, 241
function 96, 101, 223, 246
geometry 1-2, 81, 109, 123-134,
167-168, 192
God 4, 217-233, 247-249, 251
definition of 222, 224, 226,
230-231
existence of 4, 217-233
cosmological arguments 219
ontological arguments 219-
221
proof of 222-233
teleological arguments 219,
221-222, 225-226, 230
goodness of 233, 247-249, 251
golden rule 239-242
good 28, 233-249, 251
government 9-10, 243-244
gravitation 77, 79, 92-93, 98, 126
Newton’s law of 77
gravity 75, 77, 79, 81, 90, 92-93
happiness 50, 237-251
definition of 238-239
heaven 249, 251
hierarchy in logic, supposition of
no, 116-117, 187
homonym 27-28, 39
Hume, David 56, 72, 219
hypothetico-deductive method 5
idea 20, 23
idealism 23, 48, 55
ideal perception 2, 22, 84-85, 88,
125, 127, 130
identity 17, 33, 42, 51-53, 56, 69-
70, 84-85, 141, 146, 152, 157,
165, 170-171, 193
identity, law of 135-136, 151
idiom 42
immortality 12, 249-251
implication 3, 49, 136, 143-145,
151-152, 157-158, 160, 163-166,
168, 170, 172, 175, 178-184,
191-195, 198-200, 211-213, 225
material 178-183
strict 178
importation in logic 198-199
impossibility 34, 36-40, 104, 173,
178, 220-221
inanimate, the 96
inclusion, laws of 142-143, 145-
149, 156
indemonstrability, supposition of
1, 4, 105-106, 135, 216-217
indeterminability (insolubility),
supposition of 1, 3-4, 112
255
individual 39, 191-192
individual sensibility 50-53, 56,
69, 250
induction 4-7, 11-12, 27, 30, 35, 41,
48, 61, 63, 74, 85, 100-102,
112, 139, 175, 216-217, 229,
250-251
ineffable 218
inertia 76-77, 90
inference 1, 4-7, 10-12, 18, 27-29,
31- 32, 35, 47, 65-66, 73, 84,
100-101, 104-106, 134, 136-137,
139, 141, 162-165, 168, 174-
175, 177, 185, 192, 207, 216-
218, 221
rule of 104, 163
infinitesimals 122-123, 127, 132
infinity 115, 119-123, 126-127, 132-
133, 209-211
informative statements 17, 136
informing statements 113-114
instruments 67, 85, 98-99
intension 15
interactionism 62
intrinsic connection 4-5, 31-34, 63-
73, 104-215
intuition 112
justification 13, 247
Kant, Immanuel 117, 219
knowledge 3, 6, 10-13, 17, 19, 22,
24-27, 34, 37, 46-49, 53-54,
57-58, 63-70, 72, 93, 96, 99-
101, 105-107, 175, 177, 192,
207-209, 216-217, 221-224,
226, 229, 237-238, 240
definition of 13, 223-224, 226
innate, supposition of 3
language 6-7, 9-21, 23-26, 28-30,
32- 33, 39-40, 100, 102, 106-
108, 110-111, 114, 117-118, 121,
123, 133-134, 138, 170, 189,
191, 194, 206-207, 217-218
interpretation of 106, 134, 138,
163, 174
laws of nature 27, 36, 41, 49, 58,
61, 72, 74, 85, 93-94, 96, 98,
127, 140, 151, 217, 221, 229,
233, 242
laws of thought 135-158, 160, 176,
180-182, 184, 198, 200, 202-
203, 206
Leibniz’s law 31-33, 166
life 9-10, 50, 70, 94-102, 231-233,
251
light 9-10, 81-88, 126
path of 126
speed of 81-88
limit 85, 123-124, 133
line 123-133, 174
definition of 124
straight 124-133, 174
definition of 126
Locke, John 23, 29
logic 1-5, 7, 13, 16, 26-27, 31-34,
36, 41, 48, 58, 63, 65, 72, 85,
104-217, 224-225, 233
Lorentz transformation 86
love 239, 245-246
mass 74, 76-77, 79-80, 92
gravitational 76-77
inertial 76-77
materialism 50, 55-56
mathematical induction, axiom
of 138, 187
mathematics 1-5, 7, 16, 27, 65, 81,
104-217
matter 1, 40, 47, 51, 55-56, 61-62,
65, 67-70, 74-77, 92-94, 124,
151, 250-251
meaning 2, 4, 9-20, 23, 25-26, 28-
29, 33-34, 39, 81, 102, 106-
107, 133-134, 136-138, 143,
172-173, 185, 217-218, 237
knowledge of 10-13, 19-20, 237
meaningless statements, supposi¬
tion of 1-2, 11, 18, 106, 114
measurement 81-88, 169-170
memory 11, 22, 27, 46, 52-54, 63-
65
metalanguage 110-112
metaphor 28
metaphysics 1-3, 44, 48, 106, 216-
217
mind 1, 49-69, 99, 102, 241, 250-
251
involuntary 57-58, 62-67, 72, 99,
102
voluntary 57-66, 99-100, 102
modality 34, 37, 177
modus ponens 162-163, 206-208
256
modus tollens 114, 159, 162, 198-
199, 201, 206
morality 233-234, 237-247
motion 76-93, 122, 125, 169-170
Newton’s laws of 78-81, 85, 90-
92
relativity of, supposition of 77-
78, 80-93, 169-170
name 16-22, 25-26, 33, 40, 42,
136-137, 139, 154, 175, 217-218
proper 21, 218
natural law 245
natural science 2, 5, 18, 60
natural selection, theory of 96-98
nature 4, 48, 72, 74, 93, 96, 100,
102-103, 125-127, 130, 142,
151-152, 217, 221-222, 228,
231, 251
necessity 31-41, 45, 72, 104, 154,
172-173, 178, 220-221
negation 104, 142, 144-146, 148,
152, 157-158, 160, 173, 175,
184, 191-192, 198-200, 212
external 158, 199-200, 212
Newton, Isaac 76-81, 85, 89-90
Newtonian (classical) mechanics
76, 89
nondistinct entities 24-26, 28-30,
40, 43, 47, 110-111
non-Euclidean geometry 133, 138
number 118-121, 123, 137-138, 169,
174
natural 118-120, 137-138
definition of 138
rational 120-121
real 118-121
objectivity 68, 84, 87, 235, 246
one-to-one correspondence 169
order 221, 236
other minds 100-102, 251
ought 245
paradox 1, 3, 27, 107-134, 158, 178,
209-211
Berry’s 118
Burali-Forti’s 121
Cantor’s 121
Grelling’s 117
of Achilles 122-123, 132
of the barber 115-117
of the husband 113-114
of the king of France 113-114
of the liar 107-111, 113, 158
of the race course 122
of the unprovable 111-113, 209-
211
Richard’s 118-120
Russell’s 115-117
part 122, 168-169
particle 47, 67, 75, 77, 92, 98-99
passive mind 62, 223
perception 10, 23-27, 29-30, 46-
48, 50-51, 53-54, 56, 63-70,
72-73, 84-85, 99-105, 127,
129, 139-143, 145, 152-153,
159, 174, 177, 183, 208, 216-
217, 223, 228, 247, 250-251
subject of 51
perceptual attribute 25-27, 29-30,
123, 127
perceptual element 24, 28-30
person 50-53
phenomena 24, 52, 54, 69
phenomenalism 24
phenomenology 12
pi 118, 121
place 129, 142-143, 152, 160, 165-
166, 176-177, 181-182, 198,
202, 206, 208
plane 125-128, 131, 133
definition of 128
Plato 48
point 123-124, 126-127
definition of 124
possibility 34, 36-37, 47, 60, 72,
98, 104, 127, 153, 172-173, 221
postulate 1, 107, 130-131, 134, 168-
169
predicate 34, 39, 45, 51, 55, 135,
150-151, 158, 186-188, 192,
196, 219
predicate logic 105, 135, 161, 178,
184-190
premise 6, 8, 17, 31-32, 58, 77-78,
83, 93, 107, 134, 136-137, 139,
141, 150, 158-159, 163, 165,
172, 175-177, 180-186, 193,
195, 199-200, 202, 205-206,
210-211
private reality 24, 68
probability 72-73, 97-98, 196
proposition 15-16, 23, 45, 65, 108,
257
112-115, 135, 151, 158, 172-
173, 180, 190, 206
propositional logic 105, 135, 178-
184, 190
psychoanalysis 61
psychology 54, 64-65
public reality 24, 68
purely mental 54, 58, 68-69, 102
purpose 50-53, 59-60, 64-67, 94-
102, 217, 221-234, 238, 240,
247, 251
definition of 60
Pythagorean theorem 86-87, 210
quality 23-24, 28-30, 50-51
quantification 186-198
quantity 23-26, 104-105, 138, 146,
167, 169
quantum theory 98
questions 18, 25
rationalism 57, 63, 140
realism 48
reality 2-5, 7, 20, 23-26, 39, 43-
103, 135, 216-217, 219-220,
225, 228-229, 250-251
reason 2, 57, 61, 63, 140
reasons 35, 59, 61, 75, 138-139,
247
reciprocity in logic 165, 168-173,
193
reciprocity of love 239, 243-244
reductio ad absurdum 5, 109-110,
114, 119
reference 15
reflexivity 110, 117-118
relation in logic 177, 188-190, 197-
198
relativity, theory of 76-78, 80-93
mass-energy equivalence 76
space contraction 85
space-time 80-81
time dilation 85-87
twin paradox 88-89
responsibility, question of 52
rest 77-78, 81, 88-91
revealing connections 151-153,
157, 160, 176, 182, 200
rotation 89-92
Russell, Bertrand 115
sameness 17, 22, 52, 56, 69-70,
130, 166-168, 176, 189-190,
197-198
scholastic (medieval) logic 189
scientific method 2-3
self 12, 49-66, 69, 94, 99-102, 223,
227, 249-251
definition of 50
semantics 9
sense 15-17, 19, 21, 27
sense experience 2, 11
sense perception 23, 44, 57, 99,
228
senses 22-23, 48-49, 57, 67, 99
sentences 15-16, 18, 25, 34, 108,
no
set theory 118-121
countability 120-121
diagonal method 119
Sherlock Holmes 8
simple entities 28-29, 51
simultaneity 82-87, 93
size 22, 40, 70, 83-85, 92, 119-123
skepticism 24, 48
social science 60
Socrates 3, 13
sorites 211
soul 50, 53, 250
space 68, 80-81, 83-87, 89, 93,
124, 126-127
sphere 125, 127-129, 133
definition of 127
Spinoza 1
square 125
square of opposition 148
statements 16, 18, 25, 34, 64, 108,
110-115, 134-135, 139, 150-151,
175-176, 195-196
statistic 6, 73, 97-98
subaltern 212
subject 17, 34, 39, 45, 51, 55, 135,
139, 150-151, 158, 161, 165,
184-188, 190, 192-193, 195-197
subjectivity 68, 84, 87, 235, 247
substance 39-40, 48, 51, 54
substitution, rule of 163
substratum 40, 48
superposition 127-128, 130-131,
167, 171
surface 123-125, 127-128, 133
definition of 124
syllogism 29, 140-141, 163, 184,
211, 224-225
258
categorical 224
law of 163
mixed hypothetical 225
syllogistic 141, 148, 153
symbolic linkage 211-215, 224-225
symmetry 169-173
sympathy 246-248
synonym 27-28, 39, 42-43
definition of 42
synthetic connections 4, 11, 36
tautology 2, 136, 140, 150, 181
term 28, 42-43
definition of 42
Theorems, 1.1 41; 1.2 43; 11.1 53;
11.2 54; M.3 66; 11.4 70;
II. 5 74; 11.6 96; lll.l.i 142;
III. l.ii 143; lll.2.i 144;
111.2.11 145; lll.3.i 146;
111.3.11 146; lll.4.i 148;
111.4.11 148; lll.4.iii 148;
lll.4.iv 148; III.5 159;
III.6 160; III.7 163; III.8 165;
III.9 171
theory 5-6, 13, 93, 96, 217
thinking 25, 55
thought 10, 12, 22-23, 25-26, 38-
39, 45, 55, 63, 219-220
time 70-71, 80-85, 87, 92-93, 124,
142-143, 145, 152, 160, 164-
166, 176, 181-182, 206-208
transcendent realities 216-251
transitivity 31-33, 49, 163-165, 175,
184, 202-209, 212-214, 217
transposition 32, 43, 109, 114, 154,
159-162, 176, 180-182, 194,
199, 201, 203, 212-213
truism 31, 225-226
truth 2, 11, 18, 25-26, 47, 108-110,
112-115, 134-135, 151, 158,
172-173, 180-183, 190, 200,
206, 244-245, 247
truthfulness 240-241
truth table 108, 136, 170, 178-185
truth tables 180, 182
unanalyzability, supposition of
28, 106
unconscious, supposition of 54,
64, 238, 241
undefinability, supposition of 3,
28-29, 105-106, 123, 244
uninformative statements 17, 136
uninforming statements 107, 110-
111, 114, 135
unit 51-53, 69-70, 227
universal 20-22, 39, 48, 191-195
standards of 21-22
third-man argument 22
unprovability, supposition of 1-2,
5, 105, 111-113, 209-211, 244-
245
unrevealing connections 151, 153
unverifiability, supposition of 2,
106, 244
validity 134
variable 163, 181, 187-188, 190-191,
193-196, 198, 211
velocity 81, 89-91
volition 57-66, 96
volume 68-69, 80, 124, 127
weight 22, 75-77
whole 121-122, 168
will 37, 57-66, 223, 228-230
world 2-3, 7, 20, 22-23, 27, 39, 45,
47, 49-50, 52, 55-57, 59, 61,
67-70, 72, 74, 78, 85, 93, 96,
99, 101-102, 106, 124, 126-127,
143, 145, 152, 157, 216, 228-
229, 248-251
Zeno of Elea 122