LAWS OF THOUGHT

THE LAWS OF
THOUGHT
A Thematic Compilation
Avi Sion, Ph. D.
© AVI SION, 1990-2013.
PROTECTED BY INTERNATIONAL COPYRIGHT CONVENTIONS.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY MANNER
WHATSOEVER, OR STORED IN A RETRIEVAL SYSTEM OR TRANSMITTED,
WITHOUT EXPRESS PERMISSION OF THE AUTHOR-PUBLISHER, EXCEPT
IN CASE OF BRIEF QUOTATIONS WITH DUE ACKNOWLEDGEMENT.
First Published 2008. Expanded 2014.
By Avi Sion. Geneva, Switzerland.
Library Cataloguing Information:
Sion, Avi.
The Laws of Thought.
A Thematic Compilation.
ISBN: 978-1495973147
Abstract
The Laws of Thought is an exploration of the deductive
and inductive foundations of rational thought. The author
here clarifies and defends Aristotle’s Three Laws of
Thought, called the Laws of Identity, Non-contradiction
and Exclusion of the Middle – and introduces two more,
which are implicit in and crucial to them: the Fourth Law
of Thought, called the Principle of Induction, and the
Fifth Law of Thought, called the Principle of Deduction.
This book is a thematic compilation drawn from past
works* by the author over a period of twenty-three years.

Note: Some chapters have been merged here. Also, some chapters
have been split up into smaller sections.

Contents

Chapter One ……………………………………………………. 7
The Law of Identity ………………………………………. 7
The Law of Contradiction ……………………………… 9
The Law of the Excluded Middle………………….. 12
Chapter Two ………………………………………………….. 16
True or False ……………………………………………… 16
Branches of Logic ………………………………………. 18
Tools of Logic ……………………………………………. 20
Axioms of Logic ………………………………………… 21
Chapter Three ………………………………………………… 24
Ground of the Laws …………………………………….. 24
Functions of the Laws …………………………………. 29
More on Credibility …………………………………….. 31
Chapter Four …………………………………………………. 34
Internal Inconsistency …………………………………. 34
The Stolen Concept Fallacy …………………………. 37
Chapter Five ………………………………………………….. 41
Definition ………………………………………………….. 41
The Liar Paradox ………………………………………… 43
The Barber Paradox…………………………………….. 46
Chapter Six ……………………………………………………. 49
The Tetralemma …………………………………………. 49
Nagarjuna’s Misinterpretation ……………………… 52
Neither Real Nor Unreal ……………………………… 54
Common Way vs. Middle Way …………………….. 59
Chapter Seven ……………………………………………….. 63
Phenomenology ………………………………………….. 63
Knowledge is Based On Appearance …………….. 66
Chapter Eight ………………………………………………… 71
Appearance and Other Large Concepts………….. 71
Material, Mental, Intuitive, Abstract ……………… 76
Chapter Nine …………………………………………………. 82
Apprehension …………………………………………….. 82
Explications ……………………………………………….. 84
Negation ……………………………………………………. 89
Primaries …………………………………………………… 91
Chapter Ten …………………………………………………… 95
Logical Attitudes ………………………………………… 95
Unity In Plurality ……………………………………….. 97
Chapter Eleven …………………………………………….. 102
Dialectical Reasoning………………………………… 102
Genesis of Axioms ……………………………………. 104
Paradoxical Propositions ……………………………. 106
Chapter Twelve ……………………………………………. 112
Contradiction ……………………………………………. 112
Varieties of Contradiction ………………………….. 114
Double Standards ……………………………………… 116
Chapter Thirteen…………………………………………… 119
Special Status of the Laws …………………………. 119
Motors of Rational Thought ……………………….. 122
Cogito, Ergo Sum ……………………………………… 123
Concerning Identity. ………………………………….. 125
Chapter Fourteen ………………………………………….. 127
Appearance, Reality and Illusion ………………… 127
Existence and Non-existence ……………………… 129
Chapter Fifteen …………………………………………….. 133
Poles of Duality ………………………………………… 133
On the Liar paradox ………………………………….. 135
Non-Aristotelian “Logic” …………………………… 137
Postmodern “Logic” ………………………………….. 140
Chapter Sixteen ……………………………………………. 142
Negation in Adduction ………………………………. 142
Positive and Negative Phenomena ………………. 146
Positive Experience Precedes Negation ……….. 149
Negation is an Intention …………………………….. 152
Chapter Seventeen ………………………………………… 156
Formal Consequences ……………………………….. 156
Negation and the Laws of Thought ……………… 159
Pure Experience ……………………………………….. 163
Consistency is Natural……………………………….. 165
Chapter Eighteen ………………………………………….. 169
The Uniformity Principle …………………………… 169
The Principle of Induction …………………………. 173
Regarding Husserl …………………………………….. 176
The Flexibility of Induction ……………………….. 178
Chapter Nineteen ………………………………………….. 181
Briefly Put ……………………………………………….. 181
Antagonism to the Laws…………………………….. 183
Counterarguments …………………………………….. 185
Our Pedestrian Path …………………………………… 187
Chapter Twenty ……………………………………………. 191
Ontological Status of the Laws …………………… 191
The Need for a Subject ………………………………. 193
Fuzzy Logic ……………………………………………… 194
Stick to Logic …………………………………………… 199
Chapter Twenty One …………………………………….. 200
Ups and Downs of Aristotelianism ……………… 200
Aristotle Bashing ………………………………………. 202
Aristotle’s Dynamism ……………………………….. 207
Chapter Twenty Two…………………………………….. 210
How to Validate Logic? …………………………….. 210
The Inductive Nature of Knowledge ……………. 212
The Crucial Role of Negation …………………….. 214
Chapter Twenty Three…………………………………… 216
Truth vs. Proof …………………………………………. 216
Double Negation ………………………………………. 220
Chapter Twenty Four ……………………………………. 223
Misleading Symbolism ……………………………… 223
Upping the Ante ……………………………………….. 226
Chapter Twenty Five …………………………………….. 229
Mitigating Clarifications ……………………………. 229
Examples …………………………………………………. 231
Doing Rather than Talking …………………………. 233
Imprecise Language ………………………………….. 234
Chapter Twenty Six………………………………………. 239
Cognitive Virtues ……………………………………… 239
The Absurdity of the Antitheses …………………. 243
Chapter Twenty Seven ………………………………….. 246
Adapting the Laws of Thought …………………… 246
Two More Laws of Thought ………………………. 254
Chapter Twenty Eight …………………………………… 258
Zen’s Anti-logic ……………………………………….. 258
The Vanity of the Tetralemma ……………………. 272
Chapter Twenty Nine ……………………………………. 281
Modern Symbolic Logic ……………………………. 281
The Existential Import Doctrine …………………. 292
Chapter Thirty ……………………………………………… 318
The Liar Paradox (Redux) ………………………….. 318
The Russell Paradox (Redux)……………………… 333
Main References………………………………………………….. 364

CHAPTER 1
7

CHAPTER ONE
Drawn from Future Logic (1990),
Chapter 2.
THE FOUNDATIONS OF LOGIC
Logic is founded on certain ‘laws of thought’, which
were first formulated by Aristotle, an ancient Greek
philosopher. We shall describe them separately here, and
later consider their collective significance.
The Law of Identity
The Law of Identity is an imperative that we consider all
evidence at its face value, to begin with. Aristotle
expressed this first law of thought by saying ‘A is A’,
meaning ‘whatever is, is whatever it is’.
There are three ways we look upon phenomena, the
things which appear before us, however they happen to
do so: at their face value, and as real or illusory.
We can be sure of every appearance, that it is, and is
what it is. (i) Something has presented itself to us,
whether we thereafter judge it real or illusory, and (ii)
this something displays a certain configuration, whether
we thereafter describe and interpret it rightly or wrongly.
The present is present, the absent is absent.
Every appearance as such is objectively given and has a
certain content or specificity. We can and should and
commonly do initially regard it with a simple attitude of
8
THE LAWS OF THOUGHT
receptiveness and attention to detail. Every appearance is
in itself neutral; the qualification of an appearance (thus
broadly defined) as a ‘reality’ or an ‘illusion’, is a
subsequent issue.
That statement is only an admission that any
phenomenon minimally exists and has given
characteristics, without making claims about the source
and significance of this existence or these characteristics.
The moment we manage to but think of something, it is
already at least ‘apparent’. No assumption need be made
at this stage about the nature of being and knowledge in
general, nor any detailed categorizations, descriptions or
explanations of them.
Regarded in this way, at their face value, all phenomena
are evident data, to be at least taken into consideration.
The world of appearances thus offers us something to
work with, some reliable data with which we can build
the edifice of knowledge, a starting point of sorts. We
need make no distinctions such as those between the
physical/material and the mental, or sense-data and
hallucinations, or concrete percepts and abstract
concepts; these are later developments.
The law of identity is thus merely an acknowledgement
of the world of appearances, without prejudice as to its
ultimate value. It defines ‘the world’ so broadly, that
there is no way to counter it with any other ‘world’.
When we lay claim to another ‘world’, we merely expand
this one. All we can ever do is subdivide the world of
appearances into two domains, one of ‘reality’ and one of
‘illusion’; but these domains can never abolish each
other’s existence and content.
What needs to be grasped here is that every judgment
implies the acceptance, at some stage, of some sort of
CHAPTER 1
9
appearance as real. There is no escape from that; to claim
that nothing is real, is to claim that the appearance that
‘everything is illusory’ is real. We are first of all
observers, and only thereafter can we be judges.
Reality and illusion are simply terms more loaded with
meaning than appearance or phenomenon — they imply
an evaluation to have taken place. This value-judgment is
a final characterization of the object, requiring a more
complex process, a reflection. It implies we went beyond
the
immediately apparent. It implies a broader
perspective, more empirical research, more rigorous
reasoning. But what we finally have is still ‘appearance’,
though in a less pejorative sense than initially.
Thus, ‘real’ or ‘illusory’ are themselves always,
ultimately, just appearances. They are themselves, like
the objects of consciousness which they evaluate, distinct
objects of consciousness. We could say that, there is a bit
of the real in the illusory and a bit of the illusory in the
real; what they have in common is appearance. However,
these terms lose their meaning if we try to equate them
too seriously.
On what basis an appearance may or should be classified
as real or illusory is of course a big issue, which needs to
be addressed. That is the overall task of logic, to set
precise guidelines for such classification. But the first
step is to admit the available evidence, the phenomenal
world as such: this gives us a data-base.
The Law of Contradiction
The Law of Contradiction (or Non-contradiction) is an
imperative to reject as illusory and not real, any apparent
presence together of contradictories. This second law of
10
THE LAWS OF THOUGHT
thought could be stated as ‘Nothing is both A and not-A’,
or ‘whatever is, is not whatever it is not’.
We cannot say of anything that it is both present and
absent at once: what is present, is not absent. If the world
of appearance displays some content with an identity,
then it has effectively failed to display nothing.
Contradictory appearances cannot coexist, concur,
overlap: they are ‘incompatible extremes’.
We can say of something that it ‘is’ something else, in
the sense of having a certain relation to something
distinct from itself, but we cannot say of it that it both
has and lacks that relation, in one and the same respect,
at one and the same place and time.
It
is
evident, and therefore incontrovertible (by the
previous law), that appearances are variegated, changing,
and diverse. Phenomena have a variety of aspects and are
usually composed of different elements, they often
change, and differ from each other in many ways.
However, for any respect, place and time, we pinpoint,
the appearance as such is, and is whatever it is — and not
at once otherwise.
The law of contradiction is not a mere rephrasing of the
law of identity, note well, but goes one step further: it
sets a standard for relegating some appearances to the
status of illusions; in a sense, it begins to define what we
mean by ‘illusion’. It does not, however, thereby claim
that all what is leftover in the field of appearance is real
with finality; nor does it deny that some of the leftovers
are real (as is assured us by the law of identity).
By the law of identity, whatever appears is given some
credence: therefore, one might suggest, the coexistence
of opposites has some credence. The law of contradiction
interposes itself at this point, and says: no, such events
CHAPTER 1
11
carry no conviction for us, once clearly discerned. The
first law continues to function as a recognition that there
is an apparent contradiction; but the second law imposes
on us the need to resolve that contradiction somehow.
The law of contradiction is itself, like anything else, an
appearance among others, but it strikes us as an
especially credible one, capable of overriding the initial
credibility of all other considerations. It does not conflict
with the message of the law of identity, since the latter is
open to any event, including the event that some
appearances be more forceful than others. The law of
contradiction is precisely one such forceful appearance,
an extremely forceful one.
Thus, though the world of appearances presents itself to
us with some seeming contradictions, they appear as
incredible puzzles — their unacceptability is inherent to
them, obvious to us. We may verbally speculate about a
world with real contradictions, and say that this position
is consistent with itself even if inconsistent with itself.
But the fact remains that whenever we are face to face
with a specific contradiction (including that one) we are
unavoidably skeptical — something seems ‘wrong’.
The way we understand the apparent existence of
contradictions is by viewing the world of appearances as
layered, or stratified. Our first impressions of things are
often superficial; as our experience grows, our
consciousness penetrates more deeply into them. Thus,
though each level is what it is (law of identity), parallel
levels may be in contradiction; when a contradiction
occurs, it is because we are superimposing different
layers (law of contradiction). In this way, we resolve the
‘general contradiction’ of contradiction as such — we
separate the conflicting elements from each other.
12
THE LAWS OF THOUGHT
(Note in passing, as an alternative to the metaphor of
‘depth’, which likens consciousness to a beam of light,
we also sometimes refer to ‘height’. Here, the suggestion
is that the essence of things is more elevated, and we
have to raise ourselves up to make contact with it.)
That resolution of contradiction refers to the diversity
and change in the world of appearance as due to the
perspectives of consciousness. Thus, the appearance of
the phenomena we classified as ‘illusory’ is due to the
limitations of ordinary consciousness, its failure to know
everything.
This
restriction
in
the
power of
consciousness may be viewed as a ‘fault’ of our minds,
and in that sense ‘illusion’ is a ‘product’ of our minds.
For that reason, we regard the illusory as in some sense
‘imaginary’ — this is our explanation of it.
On a more objective plane, we may of course accept
diversity and change as real enough, and explain them
with reference to the space and time dimensions, or to
uniform and unchanging essences. In such cases, we are
able to meet the demands of the law of contradiction
without using the concept of ‘illusion’; only when space,
time, and respect, are clearly specified, does a
contradiction signify illusion.
The Law of the Excluded Middle
The Law of the Excluded Middle is an imperative to
reject as illusory and not real, any apparent absence
together of contradictories. This third law of thought
could be stated as ‘nothing is neither A nor not-A’, or
‘whatever is, either is some thing or is-not that thing’.
We cannot say of anything that it is at once neither
present nor absent: what is not present, is absent. If the
CHAPTER 1
13
world of appearance fails to display some content with an
identity, then it has effectively displayed nothing. There
is no third alternative to these two events (whence the
expression ‘excluded middle’): they are exhaustive.
We may well say that some parts or aspects of the world
are inaccessible to our limited faculties, but (as pointed
out in the discussion of identity) we cannot claim a world
beyond that of appearances: the moment we mention it,
we include it.
It may be that we neither know that something is so and
so, nor know that it is not so and so, but this concerns
knowledge only, and in reality that thing either is or is
not so and so. Whatever we consider must either be there
or not-there, in the specified respect, place and time,
even if we cannot discern things enough to tell at this
time or ever. There is an answer to every meaningful
question; uncertainty is a ‘state of mind’, without
‘objective’ equivalent.
Moreover, strictly speaking, ‘questions’ are artificial
attempts to anticipate undisplayed layers of appearance.
As things appear now, if nothing is being displayed, that
is the (current) ‘answer’ of the world of appearances; in
the world of appearances there are no ‘questions’.
‘Questions’ merely express our resolve to pursue the
matter further, and try to uncover other layers of
appearance; they are not statements about reality.
If we choose to, loosely speaking, regard doubts as kinds
of assertions, the law of the excluded middle enjoins us
to class them at the outset as illusory, and admit that in
reality things are definite. Problematic statements like ‘it
might or might not be thus’ are not intended to affirm
that ‘neither thus nor not-thus’ appeared, but that what
did appear (whether it was ‘thus’ or ‘not-thus’ — one of
14
THE LAWS OF THOUGHT
them did, for sure) was not sufficiently forceful to satisfy
our curiosity.
Even if no phenomenon is encountered which confirms
or discredits an idea, there must be a phenomenon
capable of doing so, in the world somewhere, sometime.
We have to focus on the evidence, and try and
distinguish the appearance or nonappearance of that
imagined phenomenon.
Thus, the law of the excluded middle serves to create a
breach of sorts between the ‘objective world’ and the
‘world of ideas’, and establishes the pre-eminence of the
former over the latter. The breach is not an unbridgeable
gap, but allows us to expand our language, in such a way
that we can discuss eventual layers of appearance besides
those so far encountered, even while we admit of the
evidence at hand.
Such an artifice is made possible by our general
awareness from past experience that appearances do
change in some cases, but should not be taken to mean
that any given appearance will change. It is only the
expression of a (commendable) ‘open-mindedness’ in
principle, with no specific justification in any given case.
What we have done, effectively, is to expand what we
mean by ‘appearance’, so as to include future
appearance, in addition to appearances until now in
evidence. Thus far, our implicit understanding was that
appearance was actual, including present realities and
present illusions. Now, we reflect further, and decide to
embrace our anticipations of ‘possible’ appearances as a
kind of actuality, too.
Such hypothetical projections are also, in a sense,
‘apparent’. But they are clearly imaginary, inventions of
the mind. Their status as appearances is therefore
CHAPTER 1
15
immediately that of ‘illusions’; that is their present status,
whatever their future outcome. However, they are
illusory with less finality than the phenomena so labeled
by the law of contradiction; they retain some degree of
credibility.
16
THE LAWS OF THOUGHT
CHAPTER TWO
Drawn from Future Logic (1990),
Chapter 3 (part of section 3 omitted).
LOGICAL RELATIONS
True or False
Reality and illusion are attributes of phenomena. When
we turn our attention to the implicit ‘consciousness’ of
these phenomena, we correspondingly regard the
consciousness
as
realistic
or
unrealistic.
The
consciousness, as a sort of peculiar relation between a
Subject (us) and an Object (a phenomenon), is essentially
the same; only, in one case the appearance falls in the
reality class, in the other it falls in the illusion class.
Why some thoughts turn out to be illusory, when
considered in a broader context, varies. For example, I
may see a shape in the distance, and assume it that of a
man, but as I approach it, it turns out to be a tree stump;
this latter conclusion is preferred because the appearance
withstands inspection, it is firmer, more often confirmed.
A phenomenon always exists as such, but it may ‘exist’
in the realms of illusion, rather than in that of reality. The
fact that I saw some shape is undeniable: the only
question is whether the associations I made in relation to
it are valid or not.
‘Propositions’ are statements depicting how things
appear to us. Understood as mere considerations (or
CHAPTER 2
17
‘hypothetically’), they contain no judgment as to the
reality or illusion of the appearance. Understood as
assertions (or ‘assertorically’), they contain a judgment
of the appearance as real or illusory.
Assertoric propositions must either be ‘true’ or ‘false’. If
we affirm a proposition, we mean that it is true; if we
deny a proposition, we mean that it is false. Our
definitions of truth and falsehood must be such that they
are mutually exclusive and together exhaustive: what is
true, is not false; what is false, is not true; what is not
true, is false; what is not false, is true.
Strictly speaking, we call an assertion true, if it verbally
depicts something which appears to us as real; and false,
if it verbally depicts something which appears to us as
illusory. In this ideal, absolute sense, true and false
signify total or zero credibility, respectively, and allow of
no degrees.
However, the expressions true and false are also used in
less stringent senses, with reference to less than extreme
degrees of credibility. Here, we call a proposition
(relatively, practically) true if the appearance is more
credible
than
any conflicting appearance; and
(effectively) false, if the appearance is not the most
credible of a set of conflicting appearances. Here, we can
speak of more or less true or false.
The ultimate goal of logic is knowledge of reality, and
avoidance of illusion. Logic is only incidentally
interested in the less than extreme degrees of credibility.
The reference to intermediate credibility merely allows
us to gauge tendencies: how close we approach toward
realism, or how far from it we stray. Note that the second
versions of truth and falsehood are simply wider; they
include the first versions as special, limiting cases.
18
THE LAWS OF THOUGHT
Propositions which cannot be classed as true or false
right now are said to be ‘problematic’. Both sets of
definitions of truth and falsehood leave us with gaps. The
first
system fails to address all propositions of
intermediate credibility; the second system disregards
situations where all the conflicting appearances are
equally credible.
If we indeed cannot tip the scales one way or the other,
we are in a quandary: if the alternatives are all labeled
true, we violate the law of contradiction; if they are all
labeled false, we violate the law of the excluded middle.
Thus, we must remain with a suspended judgment, and
though we have a proposition to consider, we lack an
assertion.
Branches of Logic
The concepts of truth and falsehood will be clarified
more and more as we proceed. In a sense, the whole of
the science of logic constitutes a definition of what we
mean by them — what they are and how they are arrived
at.
We shall also learn how to treat problematic
propositions, and gradually turn them into assertions.
The task of sorting out truth from falsehood, case by
case, is precisely what logic is all about. What is sure,
however, is that that is in principle feasible.
If thought was regarded as not intimately bound with the
phenomena it is intended to refer to, it would be from the
start disqualified. In that case, the skeptical statement in
question itself would be meaningless and self
contradictory. The only way to resolve this conflict and
paradox is to admit the opposite thesis, viz. that some
CHAPTER 2
19
thoughts are valid; that thesis, being the only internally
consistent of the two, therefore stands as proven.
This is a very important first principle, supplied to us by
logic, for all discussion of knowledge. We cannot
consistently deny the ultimate realism of (some)
knowledge. We cannot logically accept a theory of
knowledge which in effect invalidates knowledge. That
we know is unquestionable; how we know is another
question.
Now, logical processes are called deductive (or analytic)
to the extent that they yield indisputable results of zero or
total credibility; and inductive (or synthetic) insofar as
their results are more qualified, and of intermediate
credibility. Deductive logic is conceived as concerned
with truth and falsehood in their strict senses; inductive
logic is content to deal with truth and falsehood in their
not so strict senses.
This distinction is initially of some convenience, but it
ultimately blurs. Logical theory begins by considering
deductive processes, because they seem easier; but as it
develops, its results are found extendible to lesser truths.
Likewise, inductive logic begins with humble goals, but
is eventually found to embrace deduction as a limiting
case.
As we shall see, both these branches of logic require
intuition of logical relations, and both presuppose some
reliance on other phenomena. Both concern both
concrete percepts and abstract concepts. Both involve the
three faculties of experience, reason and imagination;
only their emphasis differs somewhat. There is, at the
end, no clear line of demarcation between them.
20
THE LAWS OF THOUGHT
Tools of Logic
The following are three logical relations which we will
often refer to in this study: implication, incompatibility,
and exhaustiveness. We symbolize propositions by letters
like P or Q for the sake of brevity; their negations are
referred to as notP (or nonP) and notQ, respectively.
a.
Implication. One proposition (P) is said to imply
another (Q) if it cannot happen that the former is true and
the latter false. Thus, if P is true, so must Q be; and if Q
is false, so must P be — by definition. It does not follow
that P is in turn implied by Q, nor is this possibility
excluded. This relationship may be expressed as “If P,
then Q”, or equally as “If nonQ, then nonP”. We can
deny that Q is implicit in P by the formula “If P, not-then
Q”, or “If nonQ, not-then nonP”.
When we use expressions like ‘it follows that’, ‘then’,
‘therefore’, ‘hence’, ‘thence’, ‘so that’, ‘consequently’,
‘it presupposes that’ — we are suggesting a relation of
implication.
b.
Incompatibility (or inconsistency or mutual
exclusion). Two propositions (P, Q) are said to be
incompatible if they cannot both be true. This relation is
also called ‘exclusive disjunction’, and expressed by the
formula ‘P or else Q’. Thus, if either is true, the other is
false. The possibility that both be false is not excluded,
nor is it affirmed. This relation can be formulated as “If
P, then nonQ”, or equally as “If Q, then nonP”. The
denial of such a relation would be stated as “If P, not
then nonQ”., or “If Q, not-then nonP”.
We can also say of more than two propositions that they
are incompatible; meaning, if any one of them is true, all
CHAPTER 2
21
the others must be false (though they might well all be
false).
c.
Exhaustiveness. Two propositions (P, Q) are said
to be exhaustive if they cannot both be false. This
relation is also called ‘inclusive disjunction’, and
expressed by the formula ‘P and/or Q’. Thus, if either is
false, the other is true. The possibility that both be true is
not excluded, nor is it affirmed. This relation can be
formulated as “If nonP, then Q”, or equally as “If nonQ,
then P”. The denial of such a relation would be stated as
“If nonP, not-then Q”., or “If nonQ, not-then P”.
We can also say of more than two propositions that they
are exhaustive; meaning, if all but one of them is false,
the remaining one must be true (though they might well
be all true).
We note that whereas implication and its denial are
directional relations, incompatibility and exhaustiveness
and their denials are symmetrical relations.
Also, underlying them all is the concept of ‘conjunction’,
whether or not one can say one thing with or without the
other.
Consequently,
these
expressions
are
interconnected; we could rephrase any one in terms of
any other. For example, ‘P implies Q’ could be restated
as ‘P is incompatible with notQ’ or as ‘notP and Q are
exhaustive’.
Axioms of Logic
We can now re-state the laws of thought with regard to
the truth or falsehood of (assertoric) propositions as
follows. These principles (or the most primary among
them) may be viewed as the axioms of logic, while
22
THE LAWS OF THOUGHT
however keeping in mind our later comments (ch. 20) on
the issue of their development.
a.
The law of identity: Every assertion implies itself
as ‘true’. However, this self-implication is only a claim,
and does not by itself prove the statement.
More broadly, whatever is implied by a true proposition
is also true; and whatever implies a false proposition is
also false. (However, a proposition may well be implied
by a false one, and still be true; and a proposition may
well imply a true one, and still be false.)
b.
The law of contradiction: If an affirmation is
true, then its denial is false; if the denial is true, then the
affirmation is false. They cannot be both true. (It follows
that if two assertions are indeed both true, they are
consistent.)
A special case is: any assertion which implies itself to be
false, is false (this is called self-contradiction, and
disproves the assertion; not all false assertions have this
property, however).
More broadly, if two propositions are mutually exclusive,
the truth of either implies the falsehood of the other, and
furthermore implies that any proposition which implies
that other is also false
c.
The law of the excluded middle: If an affirmation
is false, then its denial is true; if the denial is false, then
the affirmation is true. They cannot both be false. (It
follows that if two assertions are indeed both false, they
are not exhaustive).
A special case is: any assertion whose negation implies
itself to be false, is true (this is called self-evidence, and
proves the assertion; not all true assertions have this
property, however).
CHAPTER 2
23
More broadly, if two propositions are together
exhaustive, the falsehood of either implies the truth of
the other, and furthermore implies that any proposition
which that other implies is also true (though propositions
which imply that other may still be false).
Thus, in summary, every statement implies itself true and
its negation false; it must be either true or false: it cannot
be both and it cannot be neither. In special cases, as we
shall see, a statement may additionally be self
contradictory or self-evident.
Some of these principles are obvious, others require more
reflection and will be justified later. They are hopefully
at least easy enough to understand; that suffices for our
immediate needs.
Note in passing that each of the laws exemplifies one of
the logical relations earlier introduced. Identity illustrates
implication, contradiction illustrates incompatibility,
excluded-middle illustrates exhaustiveness.
Although we introduced the logical relations before the
laws of thought, here (for the sake of clarity and since we
speak the same language), it should be obvious that,
conceptually, the reverse order would be more accurate.
First, come the intuitions of identity, contradiction, and
excluded-middle, with the underlying notions (visual
images, with velleities of movement), of equality (‘to go
together’), conflict (‘to keep apart’), and limitation (‘to
circumscribe’). Thereafter, with these given instances in
mind, we construct the more definite ideas of
implication, incompatibility, and exhaustion.
24
THE LAWS OF THOUGHT
CHAPTER THREE
Drawn from Future Logic (1990),
Chapter 20 (sections 1-3).
CREDIBILITY
Ground of the Laws
We began our study by presenting the laws of thought —
the Laws of Identity, of Contradiction, and of the
Excluded Middle — as the foundations of logic. We can
see, as we proceed, that these first principles are
repeatedly appealed to in reasoning and validation
processes. But in what sense are they ‘laws’?
a.
Many logicians have been tempted to compare
these laws to the axioms of geometry, or the top
postulates of natural sciences. According to this view,
they are self-consistent hypotheses, which however are
incapable of ultimate proof, from which all other
propositions of logic are derived.
There is some truth to this view, but it is inaccurate on
many counts. The whole concept of ‘systematization’ of
knowledge, ordering it into axioms and derivatives, is
itself a device developed and validated by the science of
logic. It is only ex post facto that we can order the
information provided by logic in this way; we cannot
appeal to it without circularity. If logic was based on so
tenuous a foundation, we could design alternative logics
CHAPTER 3
25
(and some indeed have tried), just as Euclidean geometry
or Newtonian mechanics were replaced by others.
Logic is prior to methodology. The idea that something
may be ‘derived’ from something else, depends for its
credibility on the insights provided by the ‘laws of
thought’. The ‘laws of thought’ ought not to be viewed as
general principles which are applied to particular cases,
because the process of application itself depends on
them.
Rather, we must view every particular occurrence of
identity, contradiction, and excluded-middle, as by itself
compelling, an irreducible primary independently of any
appeal to large principles. The principles are then merely
statements to remind us that this compulsion occurs; they
are not its source. This means that the ‘laws of thought’
are not general principles in the normal sense, but
recognitions that ‘there are such events’. The science of
logic is, then, not a systematic application of certain
axioms, but a record of the kind of events which have this
compelling character for us.
Note this well. Each occurrence of such events is self
sufficiently evident; it is only thereafter that we can
formulate statements about ‘all’ these events. We do not
know what to include under the ‘all’ beforehand, so how
could we ‘apply’ the laws to anything? These laws
cannot be strictly-speaking ‘generalizations’, since
generalization presupposes that you have some prior data
to generalize.
Thus, we must admit that first comes specific events of
identity, contradiction and excluded-middle, with a force
of their own, then we can say ‘these and those are the
kinds of situations’ where we experience that utter
26
THE LAWS OF THOUGHT
certainty, and only lastly can we loosely-speaking format
the information in the way of axioms and derivatives.
Nevertheless, it remains true that the laws of thought
have a compelling character on their own. There is no
way to put these laws in doubt, without implicitly
arousing doubt in one’s own claim. Sophisms always
conceal their own implications, and tacitly appeal to the
laws of thought for support, to gain our credulity. We
could, therefore, equally say that the principles as units in
themselves are entirely convincing, with utter finality —
provided we also say that every act of their ‘application’
is likewise indubitable. It comes to the same.
However, the previous position is more accurate, because
it explains how people unversed in the laws of thought,
can nonetheless think quite logically — and also how we
can understand the arguments here made about the laws
of thought. The inconsistency of denials of the laws of
thought is one instance of those laws, and not their whole
basis.
b.
What, then, is this ‘compulsion’ that we have
mentioned? It is evident that people are not forced to
think logically, say like physical bodies are forced to
behave in certain ways. This is given: we do make errors,
and these sometimes seem ‘voluntary’, and sometimes
accidental. In any case, if thought was a mechanistic
phenomenon, we would have no need of logical
guidelines. We may only at best claim that we can and
should, and sometimes do, think in perfect accord with
these laws.
The answer to this question was implicit in the above
discussion. It is or seems evident that things do present
themselves and that they do have certain contents
(identity), and that these presentations are distinct from
CHAPTER 3
27
their absences (contradiction), and that there is nothing
else to refer to (excluded-middle). Because these
statements concern appearances as such, it is irrelevant
whether we say ‘it is evident’ or ‘it seems evident’.
The concepts of reality and appearance are identical, with
regard to the phenomenal; the concept of illusion is only
meaningful as a subdivision of the phenomenal. These
laws are therefore always evident, whether we are
dealing with realities or illusions. We can wrongly
interpret or deliberately lie about what we ‘see’ (if
anything), but that we ‘saw’ and just what we ‘saw’ is
pure data. Thus, the ‘compulsion’ is presented to us an
intrinsic component of the phenomenal world we face.
The practical significance of this can be brought out with
reference to the law of contradiction. We are saying, in
effect, that whatever seems contradictory, is so. This
statement may surprise, since we sometimes ‘change our
minds’ about contradictions.
To understand it, consider two phenomena, say P1 and
P2, in apparent contradiction, call this C1. One way to
resolve C1, is to say that one or both of P1 and P2 are
illusory. But we might find, upon closer inspection, that
the two phenomena are not in contradiction; call this
noncontradiction C2. So we now have two new
phenomena, C1 and C2, in apparent contradiction; call
this new contradiction C3.
The question is, does C3 imply that one or both of C1
and C2 are illusory? The answer is, no — what happened
‘upon closer inspection’ was not a revision of C1, but a
revision of P1 and/or P2. So that in fact C2 does not
concern exactly the same phenomena P1 and P2, but a
slightly different pair of phenomena with the same
names.
28
THE LAWS OF THOUGHT
Thus, C1 and C2 could never be called illusory (except
loosely speaking), because they were never in conflict,
because they do not relate the same pair of phenomena.
Nor for that matter may C3 be viewed as now erroneous,
because the pair of phenomena it, in turn, related have
changed.
Which means that our ‘intuition’ of contradiction is
invariably correct, for exactly the data provided to it. A
similar argument can be made with regard to other
logical relations. The phenomena related may be unclear
and we may confuse phenomena (thinking them the same
when they are different) — but, at any level of
appearance, the logical relation between phenomena is
‘compulsively evident’, inflexibly fixed, given.
In other words, among phenomena, logical relations are
one kind which are always real; in their case, appearance
and reality are one and the same, and there are no
illusions. The laws of thought are presented as
imperatives, to urge us to focus on and carefully
scrutinize the phenomena related, and not to suggest that
the logical intuitions of thought are fallible, once the
effort is made to discern the relation.
This is not a claim to any prior omniscience, but a case
by case accuracy. As each situation arises, its logical
aspects are manifest to the degree that we inspect things
clearly. Note well, we do not need to know how the
intuition functions, to be able to know and prove that it
functions well. We have called it ‘intuition’ to suggest
that it is a direct kind of consciousness, which may well
be conceptual rather than perceptual, but these
descriptive issues are secondary.
Thus, with regard to the laws of thought, we have no
ground for wondering whether they are animal instincts
CHAPTER 3
29
imposed by the structure of the mind, or for wondering
whether they control the events external to it as well. In
either case, we would be suggesting that there is a chance
that they might be illusory and not real. If we claim that
the mind is distortive, one way or the other, we put that
very claim in doubt.
The mind is doubtless limited. It is common knowledge
that mental conditions, structural or psychological or
voluntary, can inhibit us from comparing phenomena
with a view to their logical relation — but that does not
mean that when the elements are brought together, the
comparison may fail.
Nervous system malfunctions, personality disorders,
drunkenness, fatigue — such things can only arrest,
never alter these intuitions. As for evasions and lies, we
may delude ourselves or others, to justify some behavior
or through attachment to a dogma — but these are after
the fact interventions.
Functions of the Laws
The laws of thought relate to the credibility, or
trustworthiness, of phenomena. They clarify things in
three stages. At the identity level, appearances are
acknowledged and taken as a data base. At the
contradiction level, we learn to discriminate clearly
between real and illusory appearances. At the excluded
middle level, we introduce a more tempered outlook,
without however ignoring the previous lessons. More
specifically, their functions are as follows:
The first law assigns a minimal credibility to any thought
whatsoever, if only momentarily; the evidence, such as it
is, is considered. If, however, the ‘thought’ is found to
30
THE LAWS OF THOUGHT
consist of meaningless words, or is overly vague or
obscure — it is as if nothing has appeared, and credibility
disappears (until and unless some improvement is made).
To the extent that a thought has some meaning, precision,
and clarity, it retains some credibility.
The second law puts in doubt any thoughts which
somehow give rise to contradictions, and thereby
somewhat enhances the credibility of all thoughts which
pass this test. In the case of a thought which is self
inconsistent (whether as a whole or through the conflicts
of its parts), its credibility falls to zero, and the
credibility of denial becomes extreme. In the case of two
or more thoughts, each of which is self-consistent, but
which are incompatible with each other, the loss of
credibility is collective, and so individually less final.
The third law sets bounds for any leftover thoughts
(those with more than zero and less than total credibility,
according to the previous two laws): if special ways be
found to increase or decrease their credibilities, the
overall results cannot in any case be such as to transgress
the excluded-middle requirement (as well as the no
contradiction requirement, of course). As we shall see,
the processes of confirmation and discrediting of
hypotheses are ways logic uses to further specify
credibilities.
We see that, essentially, the law of identity gives
credence to experience, in the widest sense, including
concrete perceptions and abstract conceptual insights.
The law of contradiction essentially justifies the logical
intuitions of reason. The law of the excluded-middle is
essentially directed at the projections of the imagination.
This division of labor is not exclusive — all three laws
CHAPTER 3
31
come into play at every stage — but it has some
pertinence.
The credibility of a phenomenon is, then, a measure of
how well it fits into the total picture presented by the
world of appearances; it is a component of phenomena,
like bodies have weight. This property is in some cases
fixed; but in most cases, variable — an outcome of the
interactions of phenomena as such.
The laws of thought are, however, only the first steps in a
study of credibility. The enterprise called logic is a
continual search for additional or subsidiary norms.
Logic theory develops, as we shall see, by considering
various kinds of situations, and predicting the sorts of
inferences which are feasible in each setting.
More broadly the whole of philosophy and science may
be viewed as providing us with more or less rough and
ready, practical yardsticks for determining the relative
credibility of phenomena. However, such norms are not
of direct interest to the logician, and are for him
(relatively speaking) specific world views. Logic has to
make do with the two broadest categories of reality and
illusion — at least, to begin with.
More on Credibility
Every phenomenon appears to us with some degree of
‘credibility’, as an inherent component of its appearance;
this is an expression of the law of identity. That initially
intuitive credibility may be annulled or made extreme,
through the law of contradiction; or it may be
incrementally increased or decreased, by various
techniques (yet to be shown), within the confines of the
laws of contradiction and of the excluded middle.
32
THE LAWS OF THOUGHT
Thus, credibility is primarily an aspect of the
phenomenal world, and a specific phenomenon’s degree
of credibility is a function of what other phenomena are
present in the world of appearances at that stage in its
development. Because phenomena interact in this way,
and affect each other’s credibilities, credibility may be
viewed as a measure of how well or badly any
phenomenon ‘fits in’ with the rest.
‘Reality’ and ‘illusion’ are just the extremes of
credibility and incredibility, respectively; they are
phenomena with that special character of total or zero
force of conviction. We cannot refer to a domain beyond
that of appearances, for good or bad, without thereby
including it within the world of appearances.
How do we know that all appearances must ultimately be
real or illusory? How do we know that median credibility
cannot be a permanent state of affairs in some cases, on a
par with the extremes of credibility and incredibility? We
answered this question, in broad terms, in our discussions
on the laws of thought, as follows. More will be said
about it as we proceed.
Reality and illusion are a dichotomy of actual
appearances: for them, whatever is inconsistent is
illusory, and everything else is real enough. Median
credibility only comes into play when we try to anticipate
future appearances, but has no equivalent in the given
world. In the actual field of concrete and abstract
experiences, things have either no credibility or
effectively total credibility; it is only through the
artificial
dimension of mental projections that
intermediate credibility arises.
Knowledge is merely consciousness of appearances; the
flip-side, as it were, of the event of appearance. Viewed
CHAPTER 3
33
in this perspective, without making claims to anything
but the phenomenal, knowledge is always a faithful
rendering of the way things appear. We may speak of
knowledge itself as being realistic or as unrealistic or as
hypothetical, only insofar as we understand that this
refers to the kind of appearance it reflects. These
characterizations refer primarily, not to knowledge, but
to its objects.
The difference between knowledge (in its narrower sense
of, knowledge of reality) and opinion (in the sense of, the
practically known), is thus merely one of degree of
credibility manifested by their objects (at that time); we
cannot point to any essential, structural difference
between them. However, this distinction is still
significant: it matters a lot that the objects carry different
weights of conviction.
Changes or differences in appearances and opinion are to
some extent explained by reference to variations in our
perspective, and breadth and depth of consciousness. But
this
explanation does not annul the primacy of
phenomena, in all their aspects.
In practice, median credibility is often not patiently
accepted, but we use our ‘wisdom’ to lean one way or the
other a bit, according to which idea seems to ‘hang
together’ the best. But a contrary function of wisdom is
the ability to see alternatives, or the remote possibility of
suggested alternatives, and thus keep an open mind. The
intelligent man is able to take positions where others
dither, and also to see problems where others see
certainties.
34
THE LAWS OF THOUGHT
CHAPTER FOUR
Drawn from Future Logic (1990),
Chapter 31 (sections 1 & 2).
PARADOXES
A very important field of logic is that dealing with
paradox, for it provides us with a powerful tool for
establishing some of the most fundamental certainties of
this science. It allows us to claim for epistemology and
ontology the status of true sciences, instead of mere
speculative digressions. This elegant doctrine may be
viewed as part of the study of axioms.
Internal Inconsistency
Consider the hypothetical form ‘If P, then Q’, which is
an essential part of the language of logic. It was defined
as ‘P and nonQ is an impossible conjunction’.
It is axiomatic that the conjunction of any proposition P
and its negation nonP is impossible; thus, a proposition P
and its negation nonP cannot be both true. An obvious
corollary of this, obtained by regarding nonP as the
proposition under consideration instead of P, is that the
conjunction of any proposition nonP and its negation not
nonP is impossible; thus, a proposition P and its negation
nonP cannot be both false.
CHAPTER 4
35
P,
So, the Law of Identity could be formulated as, “For any
proposition, ‘If P, then P’ is true, and ‘If nonP, then
nonP’ is true”. The Laws of Contradiction and of the
Excluded Middle could be stated: “For any proposition,
‘If
then not-nonP’ is true (P and nonP are
incompatible), and ‘If not-nonP, then P’ is true (nonP and
P are exhaustive)”.
Now, consider the paradoxical propositions ‘If P, then
nonP’ or ‘If nonP, then P’. Such propositions appear at
first sight to be obviously impossible, necessarily false,
antinomies.
But let us inspect their meanings more closely. The
former states ‘P and (not not)P is impossible’, which
simply means ‘P is impossible’. The latter states ‘nonP
and not P is impossible’, which simply means ‘nonP is
impossible’. Put in this defining format, these statements
no longer seem antinomial! They merely inform us that
the proposition P, or nonP, as the case may be, contains
an intrinsic flaw, an internal contradiction, a property of
self-denial.
From this we see that there may be propositions which
are logically self-destructive, and which logically support
their own negations. Let us then put forward the
following definitions. A proposition is self-contradictory
if
it denies itself, i.e. implies its own negation. A
proposition is therefore self-evident if its negation is self
contradictory, i.e. if it is implied by its own negation.
Thus, the proposition ‘If P, then nonP’ informs us that P
is self-contradictory (and so logically impossible), and
that nonP is self-evident (and so logically necessary).
Likewise, the proposition ‘If nonP, then P’ informs us
that nonP is self-contradictory, and that P is self-evident.
36
THE LAWS OF THOUGHT
The existence of paradoxes is not in any way indicative
of a formal flaw. The paradox, the hypothetical
proposition itself, is not antinomial. It may be true or
false, like any other proposition. Granting its truth, it is
its antecedent thesis which is antinomial, and false, as it
denies itself; the consequent thesis is then true.
If the paradoxical proposition ‘If P, then nonP’ is true,
then its contradictory ‘If P, not-then nonP’, meaning ‘P is
not impossible’, is false; and if the latter is true, the
former is false. Likewise, ‘If nonP, then P’ may be
contradicted by ‘If nonP, not-then P’, meaning ‘nonP is
not impossible’.
The two paradoxes ‘If P, then nonP’ and ‘If nonP, then
P’ are contrary to each other, since they imply the
necessity of incompatibles, respectively nonP and P.
Thus, although such propositions taken singly are not
antinomial, double paradox, a situation where both of
these paradoxical propositions are true at once, is
unacceptable to logic.
In
contrast
to
positive
hypotheticals,
negative
hypotheticals do not have the capability of expressing
paradoxes. The propositions ‘If P, not-then P’ and ‘If
nonP, not-then nonP’ are not meaningful or logically
conceivable or ever true. Note this well, such
propositions are formally false. Since a form like ‘If P,
not-then Q’ is defined with reference to a positive
conjunction as ‘{P and nonQ} is possible’, we cannot
without antinomy substitute P for Q here (to say ‘{P and
nonP} is possible’), or nonP for P and Q (to say ‘{nonP
and not-nonP} is possible’).
It follows that the proposition ‘if P, then nonP’ does not
imply the lowercase form ‘if P, not-then P’, and the
proposition ‘if nonP, then P’ does not imply the
CHAPTER 4
37
lowercase form ‘if nonP, not-then nonP’. That is, in the
context of paradox, hypothetical propositions behave
abnormally, and not like contingency-based forms.
This should not surprise us, since the self-contradictory is
logically impossible and the self-evident is logically
necessary. Since paradoxical propositions involve
incontingent theses and antitheses, they are subject to the
laws specific to such basis.
The implications and consistency of all this will be
looked into presently.
The Stolen Concept Fallacy
Paradoxical propositions actually occur in practice;
moreover, they provide us with some highly significant
results. Here are some examples:

denial, or even doubt, of the laws of logic conceals an
appeal to those very axioms, implying that the denial
rather than the assertion is to be believed;
denial of man’s ability to know any reality
objectively, itself constitutes a claim to knowledge of
a fact of reality;
denial of validity to man’s perception, or his
conceptual power, or reasoning, all such skeptical
claims presuppose the utilization of and trust in the
very faculties put in doubt;
denial on principle of all generalization, necessity, or
absolutes, is itself a claim to a general, necessary, and
absolute, truth.
38
THE LAWS OF THOUGHT
denial of the existence of ‘universals’, does not itself
bypass the problem of universals, since it appeals to
some itself, namely, ‘universals’, ‘do not’, and ‘exist’.
More details on these and other paradoxes, may be found
scattered throughout the text. Thus, the uncovering of
paradox is an oft-used and important logical technique.
The writer Ayn Rand laid great emphasis on this method
of rejecting skeptical philosophies, by showing that they
implicitly appeal to concepts which they try to explicitly
deny; she called this ‘the fallacy of the Stolen Concept’.
A way to understand the workings of paradox, is to view
it in the context of dilemma. A self-evident proposition P
could be stated as ‘Whether P is affirmed or denied, it is
true’; an absolute truth is something which turns out to
be true whatever our initial assumptions.
This can be written as a constructive argument whose left
horn is the axiomatic proposition of P’s identity with
itself, and whose right horn is the paradox of nonP’s self
contradiction; the minor premise is the axiom of
thorough contradiction between the antecedents P and
nonP; and the conclusion, the consequent P’s absolute
truth.
If P, then P — and — if nonP, then P
but either P or nonP
hence, P.
A destructive version can equally well be formulated,
using the contraposite form of identity, ‘If nonP, then
nonP’, as left horn, with the same result.
CHAPTER 4
39
If nonP, then nonP — and — if nonP, then P
but either not-nonP or nonP
hence, not-nonP, that is, P.
The conclusion ‘P’ here, signifies that P is logically
necessary, not merely that P is true, note well; this
follows from the formal necessity of the minor premise,
the disjunction of P and nonP, assuming the right horn to
be well established.
Another way to understand paradox is to view it in terms
of knowledge contexts. Reading the paradox ‘if nonP,
then P’ as ‘all contexts with nonP are contexts with P’,
and the identity ‘if P, then P’ as ‘all contexts with P are
contexts with P’, we can infer that ‘all contexts are with
P’, meaning that P is logically necessary.
We can in similar ways deal with the paradox ‘if P, then
nonP’, to obtain the conclusion ‘nonP’, or better still: P is
impossible. The process of resolving a paradox, by
drawing out its implicit categorical conclusions, may be
called dialectic.
Note in passing that the abridged expression of simple
dilemma, in a single proposition, now becomes more
comprehensible. The compound proposition ‘If P, then
{Q and nonQ}’ simply means ‘nonP’; ‘If nonP, then {Q
and nonQ}’ means ‘P’; ‘If (or whether) P or nonP, then
Q’ means ‘Q’; and ‘If (or whether) P or nonP, then
nonQ’ means ‘nonQ’. Such propositions could also be
categorized as paradoxical, even though the contradiction
generated concerns another thesis.
However, remember, the above two forms should not be
confused with the lesser, negative hypothetical, relations
‘Whether P or nonP, (not-then not) Q’ or ‘Whether P or
nonP, (not-then not) nonQ’, respectively, which are not
40
THE LAWS OF THOUGHT
paradoxical, unless there are conditions under which they
rise to the level of positive hypotheticals.
CHAPTER 5
41

CHAPTER FIVE
Drawn from Future Logic (1990),
Chapter 32.
DOUBLE PARADOXES
Definition
We have seen that logical propositions of the form ‘if P,
then nonP’ (which equals to ‘nonP’) or ‘if nonP, then P’
(which equals to ‘P’), are perfectly legal. They signify
that the antecedent is self-contradictory and logically
impossible, and that the consequent is self-evident and
logically necessary. As propositions in themselves, they
are in no way antinomial; it is one of their constituents
which is absurd.
Although either of those propositions, occurring alone, is
formally quite acceptable and capable of truth, they can
never be both true: they are irreconcilable contraries and
their conjunction is formally impossible. For if they were
ever both true, then both P and nonP would be implied
true.
We must therefore distinguish between single paradox,
which has (more precisely than previously suggested) the
form ‘if P, then nonP; but if nonP, not-then P; whence
nonP’, or the form ‘if nonP, then P; but if P, not-then
nonP; whence P’ — and double paradox, which has the
form ‘if P, then nonP, and if nonP, then P’.
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THE LAWS OF THOUGHT
Single paradox is, to repeat, within the bounds of logic,
whereas double paradox is beyond those bounds. The
former may well be true; the latter always signifies an
error of reasoning. Yet, one might interject, double
paradox occurs often enough in practice! However, that
does not make it right, anymore than the occurrence of
other kinds of error in practice make them true.
Double paradox is made possible, as we shall see, by a
hidden misuse of concepts. It is sophistry par excellence,
in that we get the superficial illusion of a meaningful
statement yielding results contrary to reason. But upon
further scrutiny, we can detect that some fallacy was
involved, such as ambiguity or equivocation, which
means that in fact the seeming contradiction never
occurred.
Logic demands that either or both of the hypothetical
propositions which constituted the double paradox, or
paradox upon paradox, be false. Whereas single paradox
is resolved, by concluding the consequent categorically,
without denying the antecedent-consequent connection
— double paradox is dissolved, by showing that one or
both of the single paradoxes involved are untrue,
nonexistent. Note well the difference in problem
solution: resolution ‘explains’ the single paradox,
whereas dissolution ‘explains away’ the double paradox.
The double paradox serves to show that we are making a
mistake of some kind; the fact that we have come to a
contradiction, is our index and proof enough that we
have made a wrong assumption of sorts. Our ability to
intuit logical connections correctly is not put in doubt,
because the initial judgment was too rushed, without
pondering the terms involved. Once the concepts
involved are clarified, it is the rational faculty itself
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43
which pronounces the judgment against its previous
impression of connection.
It must be understood that every double paradox (as
indeed every single paradox), is teaching us something.
Such events must not be regarded as threats to reason,
which put logic as a whole in doubt; but simply as
lessons. They are sources of information, they reveal to
us certain logical rules of concept formation, which we
would otherwise not have noticed. They show us the
outer limits of linguistic propriety.
We shall consider two classical examples of double
paradox to illustrate the ways they are dissolved. Each
one requires special treatment. They are excellent
exercises.
The Liar Paradox
An ancient example of double paradox is the well-known
‘Liar Paradox’, discovered by Eubulides, a 4th cent. BCE
Greek of the Megarian School. It goes: ‘does a man who
says that he is now lying speak truly?’ The implications
seem to be that if he is lying, he speaks truly, and if he is
not lying, he speaks truly.
Here, the conceptual mistake underlying the difficulty is
that the proposition is defined by reference to itself. The
liar paradox is how we discover that such concepts are
not allowed.
The word ‘now’ (which defines the proposition itself as
its
own subject) is being used with reference to
something which is not yet in existence, whose seeming
existence is only made possible by it. Thus, in fact, the
word is empty of specific referents in the case at hand.
The word ‘now’ is indeed usually meaningful, in that in
44
THE LAWS OF THOUGHT
other situations it has precise referents; but in this case it
is used before we have anything to point to as a subject
of discourse. It looks and sounds like a word, but it is no
more than that.
A more modern and clearer version of this paradox is
‘this proposition is false’, because it brings out the
indicative function of the word ‘now’ in the word ‘this’.
The word ‘this’ accompanies our pointings and
presupposes that there is something to point to already
there. It cannot create a referent for itself out of nothing.
This is the useful lesson taught us by the liar paradox.
We may well use the word ‘this’ to point to another word
‘this’; but not to itself. Thus, I can say to you ‘this “this”,
which is in the proposition “this proposition is false”‘,
without difficulty, because my ‘this’ has a referent, albeit
an empty symbol; but the original ‘this’ is meaningless.
Furthermore, the implications of this version seem to be
that ‘if the proposition is true, it is false, and if it is false,
it is true’. However, upon closer inspection we see that
the expression ‘the proposition’ or ‘it’ has a different
meaning in antecedents and consequent.
If,
for the sake of argument, we understand those
implications as: if this proposition is false, then this
proposition is true; and if this proposition is true, then
this proposition is false — taking the ‘this’ in the sense
of self-reference by every thesis — then we see that the
theses do not in fact have one and the same subject, and
are only presumed to be in contradiction.
They are not formally so, any more than, for any P1 and
P2, ‘P1 is true’ and ‘P2 is false’ are in contradiction. The
implications are not logically required, and thus the two
paradoxes are dissolved. There is no self-contradiction,
neither in ‘this proposition is false’ nor of course in ‘this
CHAPTER 5
45
proposition is true’; they are simply meaningless,
because the indicatives they use are without reference.
Let us, alternatively, try to read these implications as: if
‘this proposition is false’ is true, then that proposition is
false; and if that proposition is false, then that
proposition is true’ — taking the first ‘this’ as self
reference and the ‘thats’ thereafter as all pointing us
backwards to the original proposition and not to the later
theses themselves. In other words, we mean: if ‘this
proposition is false’ is true, then ‘this proposition is
false’ is false, and if ‘this proposition is false’ is false,
then ‘this proposition is false’ is true.
Here, the subjects of the theses are one and the same, but
the implications no longer seem called for, as is made
clear if we substitute the symbol P for ‘this proposition is
false’. The flavor of paradox has disappeared: it only
existed so long as ‘this proposition is false’ seemed to be
implied by or to imply ‘this proposition is true’; as soon
as the subject is unified, both the paradoxes break down.
We cannot avoid the issue by formulating the liar
paradox as a generality. The proposition ‘I always lie’
can simply be countered by ‘you lie sometimes (as in the
case ‘I always lie’), but sometimes you speak truly’; it
only gives rise to double paradox in indicative form.
Likewise, the proposition ‘all propositions are false’ can
be countered by ‘some, some not’, without difficulty.
However, note well, both the said general propositions
are indeed self-contradictory; they do produce single
paradoxes. It follows that both are false: one cannot
claim to ‘always lie’, nor that ‘there are no true
propositions’. This is ordinary logical inference, and
quite legitimate, since there are logical alternatives.
46
THE LAWS OF THOUGHT
With regard to those alternatives. The proposition ‘I
never lie’ is not in itself inconsistent, except for the
person who said ‘I always lie’ intentionally. The
proposition ‘all propositions are true’ is likewise not
inconsistent in itself, but is inconsistent with the logical
knowledge that some propositions are inconsistent, and
therefore it is false; so in this case only the contingent
‘some propositions are true, some false’ can be upheld.
The Barber Paradox
The Barber Paradox may be stated as: ‘If a barber shaves
everyone in his town who does not shave himself, does
he or does he not shave himself? If he does, he does not;
if he does not, he does’.
This double paradox arises through confusion of the
expressions ‘does not shave himself’ and ‘is shaved by
someone other than himself’.
We can divide the people in any town into three broad
groups: (a) people who do not shave themselves, but are
shaved by others; (b) people who do not shave
themselves, and are not shaved by others; (c) people who
shave themselves, and are not shaved by others. The
given premise is that our barber shaves all the people
who fall in group (a). It is tacitly suggested, but not
formally implied, that no one is in group (b), so that no
one grows a beard or is not in need of shaving. But, in
any case, the premise in fact tells us nothing about group
(c).
Next, let us subdivide each of the preceding groups into
two subgroups: (i) people who shave others, and (ii)
people who do not shave others. It is clear that each of
the six resulting combinations is logically acceptable,
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47
since who shaves me has no bearing on whom I can
shave. Obviously, only group (i) concerns barbers, and
our premise may be taken to mean that our barber is the
only barber in town.
Now, we can deal with the question posed. Our barber
cannot fall in group (a)(i), because he is not shaved by
others. He might fall in group (b)(i), if he were allowed
to grow a beard or he was hairless; but let us suppose not,
for the sake of argument. This still allows him to fall in
group (c)(i), meaning that he shaves himself (rather than
being shaved by others), though he shaves others too.
Thus, there is no double paradox. The double paradox
only arose because we wrongly assumed that ‘he shaves
all those who do not shave themselves’ excludes ‘he
shaves some (such as himself) who do shave
themselves’. But ‘X shaves Y’ does not formally
contradict ‘X shaves nonY’; there is no basis for
assuming that the copula ‘to shave’ is obvertible, so that
‘X shaves Y’ implies ‘X does not shave nonY’.
If the premise was restated as ‘he shaves all those and
only those who do not shave themselves’ (so as to
exclude ‘he shaves himself’), we would still have an out
by saying ‘he does not shave at all’. If the premise was
further expanded and restricted by insisting that ‘he
somehow shaves or is shaved’, it would simply be self
contradictory (in the way of a single paradox).
Further embellishments could be made to the above, such
as considering people who shave in other towns, or
making
distinctions
between
always,
sometimes/sometimes-not, and never. But I think the
point is made. The lesson learned from the barber
‘paradox’ is that without clear categorizations,
equivocations can emerge (such as that between ‘shaves’
48
THE LAWS OF THOUGHT
and ‘is shaved’), which give the illusion of double
paradox.
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49
CHAPTER SIX
Drawn from Buddhist Illogic (2002),
Chapters 1 & 2.
The Tetralemma
Western philosophical and scientific thought is based on
Aristotelian logic, whose founding principles are the
three “Laws of Thought”. These can be briefly stated as
“A is A” (Identity), “Nothing is both A and non-A”
(Non-contradiction) and “Nothing is neither A nor non
A” (Exclusion of the Middle). These are not claimed as
mere hypotheses, note well, but as incontrovertible
premises of all rational human thought1.
Religions like Judaism, Christianity and Islam, even
while adhering to these laws in much of their discourse
and paying lip-service to them, in their bids to interpret
their own sacred texts and to make their doctrines seem
reasonable to their converts, have often ignored these
same laws. This is especially true of mystical trends
within these religions, but many examples could be given
from mainstream writings. The same can be said of some
aspects of Buddhist philosophy.
1
See my Future Logic (Geneva: Author, 1996. Rev.
ed.), ch. 2 and 20, and later essays on the subject (published
on my website www.thelogician.net).
50
THE LAWS OF THOUGHT
The tetralemma2 is a derivative of the laws of thought,
with reference to any two terms or propositions, labeled
A and B, and their opposites non-A and non-B. Four
combinations of these four terms are conceivable,
namely “A and B” (both), “non-A and non-B” (neither),
“A and non-B” and “non-A and B” (one or the other
only). According to Aristotelian logic, these four
statements are incompatible with each other (only one of
them can be true, because if two or more were affirmed
then “A and non-A” or “B and non-B” or both would be
true, and the latter implications are self-contradictory)
and exhaustive (at least one of them must be true, since if
they were all denied then “not A and not non-A” or “not
B and not non-B” or both would be true, and the latter
implications go against the excluded middle).
Now, what Nagarjuna does is insert the term A in place
of B (i.e. he takes the case of B = A), and effectively
claim that the above four logical possibilities of
combination apply in that special case – so that “A and A
(=B)”, “non-A and non-A (=non-B)”, “A and non-A
(=non-B)”, “non-A and A (=B)” seem logically
acceptable. He then goes on to argue that there are
four existential possibilities: affirmation of A (A + A =
A), denial of A (non-A + non-A = non-A), both
affirmation and denial of A (A and non-A) and
neither affirmation nor denial of A (not A and not
non-A). He is thus apparently using the principles and
terminology of common logic to arrive at a very opposite
result. This gives him and readers the impression that it is
2
See Cheng, pp. 36-38, on this topic. He there refers to
MT opening statement, as well as XVII:12a and XXIII:1a.
Etym. Gk. tetra = four, lemma = alternatives. Term coined in
contrast to the dilemma “A or non-A”.
CHAPTER 6
51
quite reasonable to both affirm and deny or to neither
affirm nor deny.
But in Aristotelian logic, the latter two alternatives are at
the outset excluded – “both A and non-A” by the Law of
Non-contradiction and “neither A nor non-A” by the Law
of
the Excluded-Middle – and the only logical
possibilities left are “A” or “non-A”. The anti
Aristotelian position may be viewed, in a positive light,
as an anti-Nominalist position, reminding us that things
are never quite what they seem or that things cannot be
precisely classified or labeled. But ultimately, they intend
the death of Logic; for without the laws of thought, how
are we to distinguish between true and false judgments?
The law of identity “A is A” is a conviction that things
have some identity (whatever it specifically be) rather
than another, or than no identity at all. It is an affirmation
that knowledge is ultimately possible, and a rejection of
sheer relativism or obscurantism. Nagarjuna’s goal is to
deny identity.
It should be noted here that Aristotle is very precise in
his formulation of the law of contradiction, stating in his
Metaphysics “The same attribute cannot at the same time
belong and not belong to the same subject in the same
respect” (italics mine). Thus, an alternative statement of
the laws of thought would be the ‘trilemma’ (let us so
call it) “either wholly A, or wholly non-A, or both partly
A and partly non-A”, which excludes the fourth
alternative “both wholly A and wholly non-A”. The
Buddhist attack on the laws of thought draws some of its
credibility from the fact that people subconsciously refer
to this ‘trilemma’, thinking superficially that indeed
opposite things may occur in the same place at different
times or at the same time in different places or in various
52
THE LAWS OF THOUGHT
respects, without thereby giving rise to logical difficulty
incapable of resolution. But it should be clear that the
Buddhist position is much more radical than that,
accepting thoroughgoing antinomy.
Similarly with regard to the law of the excluded middle,
which affirms the situation “neither A nor non-A” to be
impossible in fact. People are misled by the possibility of
uncertainty in knowledge, as to whether A or non-A is
the case in fact, into believing that this law of thought is
open to debate. But it must be understood that the thrust
of this logical rule is inductive, rather than deductive; i.e.
it is a statement that at the end of the knowledge
acquisition process, either “A” or “non-A” will result,
and no third alternative can be expected. It does not
exclude that in the interim, a situation of uncertainty may
occur. Nagarjuna’s position exploits this confusion in
people’s minds.
Nagarjuna’s Misinterpretation
Nagarjuna interprets the limitation implied by the
dilemma “A or non-A” as an arbitrary ‘dualism’ on the
part of ordinary thinkers3. It only goes to show that he
3
It is misleading to call this a ‘duality’ or ‘dichotomy’, as
Buddhists are wont to do, because it suggests that a unitary
thing was arbitrarily cut into two – and incidentally, that it might
just as well have been cut into four. But, on a perceptual level,
there is no choice involved, and no ‘cutting-up’ of anything. A
phenomenon appearing is one single thing, call it ‘a’ (a proper
name, or an indicative ‘this’), and not a disjunction. The issue
of ‘dichotomy’ arises only on a conceptual level. Negation is a
rational act, i.e. we can only speak of ‘non-a’, of what does not
appear, by first bringing to mind something ‘a’, which
previously appeared (in sensation or imagination). In initial
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53
misunderstands formalization (or he pretends to, in an
attempt to confuse gullible readers). When logicians use
a variable like “B” and allow that “non-A and B” and “A
and non-B” are both in principle possible, they do not
intend that as a generality applicable to all values of B
(such as “A”), but only as a generic statement applicable
to any consistent values of B. In the specific case where
B = A, the said two combinations have to be eliminated
because they are illegal (i.e. breach two of the laws of
thought).
The above-stated property of symbols, i.e. their
applicability only conditionally within the constraints of
consistency, is evident throughout the science of formal
logic, and it is here totally ignored by Nagarjuna. His
motive of course was to verbalize and rationalize the
Buddha’s doctrine that the ultimate truth is beyond nama
and rupa, name and form (i.e. discrimination and
discourse),
knowable only by a transcendental
consciousness (the Twofold Truth doctrine). More
precisely, as Cheng emphasizes, Nagarjuna’s intent was
to show that logic is inherently inconsistent and thus that
reason is confused madness to be rejected. That is, he
was (here and throughout) not ultimately trying to defend
conceptualization, two phenomena are compared and
contrasted, to each other and to other things, in some
respect(s); the issue is then, are they similar enough to each
other and different enough from other things to be judged
‘same’ and labeled by a general term (say ‘A’), or should they
be judged ‘different’ or is there an uncertainty. At the later
stage of recognition, we have to decide whether a third
phenomenon fits in the class formed for the previous two (i.e.
falls under ‘A’) or does not fit in (i.e. falls under ‘non-A’) or
remains in doubt. In the latter case, we wonder whether it is
‘A’ or ‘non-A’, and forewarn that it cannot be both or neither.
54
THE LAWS OF THOUGHT
a tetralemma with B equal to A – or even to affirm that
things are both A and non-A, or neither A nor non-A –
but wished to get us to look altogether beyond the
distinctions of conceptualization and the judgments of
logic.
But as above shown he does not succeed in this quest.
For his critique depends on a misrepresentation of logical
science. He claims to show that logic is confused and
self-contradictory, but in truth what he presents as the
thesis of logical science is not what it claims for itself but
precisely what it explicitly forbids. Furthermore, suppose
logical theory did lead to contradictions as he claims, this
fact would not lead us to its rejection were there not also
a tacit appeal to our preference for the logical in practice.
If logic were false, contradictions would be acceptable.
Thus, funnily enough, Nagarjuna appeals to our logical
habit in his very recommendation to us to ignore logic. In
sum, though he gives the illusion that it is reasonable to
abandon reason, it is easy to see that his conclusion is
foregone and his means are faulty.
Neither Real Nor Unreal
But
Nagarjuna also conceives ultimate reality
(“emptiness”4) as a “middle way” 5 – so that the world of
experience is neither to be regarded as real, nor to be
regarded as unreal (“there is nothing, neither mental nor
non-mental, which is real” and it “cannot be conceived as
4
Beyond consciousness of “Shunyata” is a more vivid
awareness called “Mahamudra”, according to Chögyam
Trungpa,. But such refinements need not concern us here.
5
See Cheng, pp. 38-39, on this topic. He there refers to
MT XIII:9a and XVIII:7.
CHAPTER 6
55
unreal,” reports Cheng). In this context, Nagarjuna is
clearly relying on one of the above-mentioned logically
impossible disjuncts, namely “neither A nor non-A” (be
it said in passing). I want to now show why Nagarjuna’s
statement seems superficially reasonable and true.
As I have often clarified and explained6, knowledge has
to be regarded or approached phenomenologically (that is
the only consistent epistemological thesis). We have to
start by acknowledging and observing appearances, as
such, without initial judgment as to their reality or
illusion. At first sight all appearances seem real enough.
But after a while, we have to recognize that some
appearances conflict with other appearances, and judge
such appearances (i.e. one or more of those in conflict) as
illusory. Since there is nothing in our ‘world’ but
appearances, all remaining appearances not judged as
illusions (i.e. so long as they are not logically invalidated
by conflicts with other appearances) maintain their initial
status as realities.
That is, the distinction between appearances as realities
or illusions emerges within the world of appearances
itself, merely classifying some this way and the rest that
way. We have no concept of reality or illusion other than
with reference to appearance. To use the category of
reality with reference to something beyond appearance is
concept stealing, a misuse of the concept, an
extrapolation which ignores the concept’s actual genesis
in the context of appearance. To apply the concept of
illusion to all appearances, on the basis that some
appearances are illusions, is an unjustified generalization
ignoring how this concept arises with reference to a
6
See my Future Logic, ch. 60-62, and later essays on
the subject.
56
THE LAWS OF THOUGHT
specific event (namely, inconsistency between certain
appearances and resulting diminishment of their innate
credibilities). Moreover, to claim that no appearances are
real or that all are illusions is self-defeating, since such
claim itself logically falls under the category of
appearance.
The illusory exists even though it is not reality – it exists
as appearance. The real is also apparent – some of it, at
least. Therefore, appearance per se is neither to be
understood as reality (since some appearances are
illusory), nor can it be equated to illusion (since not all
appearances have been or can be found illusory).
Appearance is thus the common ground of realities and
illusions, their common characteristic, the dialectical
synthesis of those theses and antitheses. It is a genus,
they are mutually exclusive species of it. (The difference
between appearance and existence is another issue, I
have dealt with elsewhere – briefly put, existence is a
genus of appearance and non-appearance, the latter
concepts being relative to that of consciousness whereas
the former is assumed independent.)
None of these insights allows the conclusion that
appearances are “neither real nor unreal” (granting that
‘unreal’ is understood to mean ‘non-real’). All we can
say is that some appearances are real and some unreal.
Formally, the correct logical relation between the three
concepts is as follows. Deductively, appearance is
implied by reality and illusion, but does not imply them;
for reality and illusion are contradictory, so that they
cannot both be true and they cannot both be false.
Moreover, inductively, appearance implies reality, until
and unless it is judged to be illusion (by virtue of some
inconsistency being discovered).
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57
More precisely, all appearances are initially classed as
real.
Any appearance found self-contradictory is
(deductively)
illusory,
and
its
contradictory
is
consequently self-evident and (deductively) real. All
remaining appearances remain classed as real, so long as
uncontested. Those that are contested have to be
evaluated dynamically. When one appearance is belied
by another, they are both put in doubt by the conflict
between them, and so both become initially problematic.
Thereafter, their relative credibilities have to be
tentatively weighed in the overall context of available
empirical and rational knowledge – and repeatedly
reassessed thereafter, as that context develops and
evolves. On this basis, one of these appearances may be
judged more credible than the other, so that the former is
labeled probable (close to real) and the latter relatively
improbable (close to illusory). In the limit, they may be
characterized as respectively effectively (inductively)
real or illusory. Thus, reality and illusion are the
extremes (respectively, 100% and 0%) in a broad range
of
probabilities with many intermediate degrees
(including problemacy at the mid-point).
To be still more precise, pure percepts (i.e. concrete
appearances, phenomena) are never illusory. The value
judgment of ‘illusory’ properly concerns concepts (i.e.
abstract appearances, ‘universals’) only. When we say of
a percept that it was illusory, we just mean that we
misinterpreted it. That is, what we initially considered as
a pure percept, had in fact an admixture of concept,
which as it turned out was erroneous. For example, I see
certain shapes and colors in the distance and think ‘here
comes a girl on a bike’, but as I get closer I realize that
all I saw was a pile of rubbish by the roadside. The pure
58
THE LAWS OF THOUGHT
percept is the shapes and colors I see; the false
interpretation is ‘girl on bike’, the truer interpretation is
‘pile of rubbish’. The initial percept has not changed, but
my greater proximity has added perceptual details to it.
My first impression was correct, only my initial
judgment was wrong. I revise the latter concept, not
through some superior means to knowledge, but simply
by means of further perception and conception.
Strictly speaking, then, perception is never at issue; it is
our conceptions that we evaluate. It is in practice,
admittedly, often very difficult to isolate a percept from
its interpretation, i.e. from conceptual appendages to it.
Our perception of things is, indeed, to a great extent
‘eidetic’. This fact need not, however, cause us to reject
any perception (as many Western philosophers, as well
as Buddhists, quickly do), or even all conception. The
conceptual ‘impurities’ in percepts are not necessarily
wrong. We know them to have been wrong, when we
discover a specific cause for complaint – namely, a
logical or experiential contradiction. So long as we find
no such specific fault with them, they may be considered
right. This just means that we have to apply the rules of
adduction7 to our immediate interpretations of individual
percepts, just as we do to complex theories relative to
masses of percepts. These rules are universal: no
judgment is exempt from the requirement of careful
scrutiny and reevaluation.
7
Adduction treats all conceptual knowledge as
hypothetical, to be tested repeatedly – in competition with all
conceivable alternative hypotheses – with reference to all
available logic and experience.
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59
Common Way vs. Middle Way
Now, judging by Cheng’s account and certain quotations
of Nagarjuna therein, we could interpret the latter as
having been trying to say just what I have said. For
instance, Cheng writes8: “What Nagarjuna wanted to
deny is that empirical phenomena… are absolutely
real…. However, [this] does not mean that nothing
exists. It does not nullify anything in the world” (my
italics).
I
interpret
this
non-nullification as an
acknowledgment of appearance as the minimum basis of
knowledge. Nagarjuna may have had difficulties
developing an appropriate terminology (distinguishing
existence, appearance and reality, as I do above),
influenced no doubt by his penchant for paradoxical
statements seeming to express and confirm Buddhist
mystical doctrine.
But if that is what he meant, then he has not succeeded to
arrive at a “middle way” (a denial of the Law of the
Excluded Middle), but only at a “common way” (a
granted common ground). As far as I am concerned, that
is
not a meager achievement – the philosophical
discovery of phenomenology! But for him that would be
trivial, if not counterproductive – for what he seeks is to
deny ordinary consciousness and its inhibiting rationales,
and to thereby leap into a different, higher consciousness
capable of reaching transcendental truth or ultimate
reality.
It is interesting to note that the Madhyamika school’s
effective denial of reality to all appearance was not
accepted by a later school of Mahayana philosophy, the
8
P. 42.
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THE LAWS OF THOUGHT
Yogachara (7th-8th cent. CE). Cheng describes the latter’s
position as follows9: “Every object, both mental and non
mental, may be logically or dialectically proven illusory.
But in order to be illusory, there must be a certain
thought that suffers from illusion. The very fact of
illusion itself proves the existence and reality of a certain
consciousness or mind. To say that everything mental
and non-mental is unreal is intellectually suicidal. The
reality of something should at least be admitted in order
to make sense of talking about illusion” (italics mine).
That is the tenor of the phenomenological argument I
present above, although my final conclusion is clearly
not like Yogachara’s, that everything is consciousness or
mind (a type of Idealism), but leaves open the possibility
of judging and classifying appearances as matter or mind
with reference to various considerations.
The Madhyamika rejection of ‘dualism’ goes so far as to
imply that “emptiness” is not to be found in nirvana, the
antithesis of samsara (according to the earlier Buddhist
viewpoint), but in ‘neither samsara nor nirvana’. In truth,
similar statements may be found in the Pali Canon, i.e. in
the much earlier Theravada schools, so that it is not a
distinctly Mahayana construct. The difference is one of
emphasis, such statements, relatively rare in the earlier
period, are the norm and frequently repeated in the later
period. An example may be found in the Dhammapada, a
sutra dating from the 3rd cent. BCE10, i.e. four or five
9
10
P. 25.
London: Penguin, 1973. This is supposedly the date
of composition, though the translator, Juan Mascaro, in his
Introduction, states “compiled” at that time, thus seeming to
imply an earlier composition. It is not clear in that commentary
when the sutra is estimated to have been first written down.
And if it was much later, say in the period of crystallization of
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61
hundred years before Nagarjuna. Here, samsara is
likened to a stream or this shore of it, and nirvana to the
further shore; and we are told to get beyond the two.
When you have crossed the stream of Samsara,
you will reach Nirvana… He has reached the
other shore, then he attains the supreme vision
and all his fetters are broken. He for whom there
is neither this nor the further shore, nor both….
Such a formula is legitimate if taken as a warning that
pursuing nirvana (enlightenment and liberation) is an
obstacle to achieving it, just a subtle form of samsara
(ignorance and attachment); there is no contradiction in
saying that the thought of nirvana as a goal of action
keeps us in samsara – this is an ordinary causal
statement. The formula is also logically acceptable if
taken as a reminder that no word or concept – not even
‘samsara’ or ‘nirvana’ – can capture or transmit the full
meanings intended (i.e. ‘not’ here should more precisely
be stated as ‘not quite’). There is also no contradiction in
saying that one who has attained nirvana does not need to
leave the world of those locked in samsara, but can
continue to exist and act in it though distinctively in a
way free of attachment.
But it would be a contradiction in terms to speak of
‘emptiness’ as ‘neither samsara nor nirvana’, given that
nirvana as a concept is originally defined as non
samsara; the truth cannot be a third alternative. At best,
one could say that emptiness is a higher level of nirvana
(in an enlarged sense), which is not to be confused with
the lower level intended by the original term nirvana, nor
Mahayana thought, say in 100 BCE to 100 CE, the latter may
have influenced the monks who did the writing down. See ch.
26 (383-5) for the quotation.
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THE LAWS OF THOUGHT
of course with samsara. In that case, nirvana (in a generic
sense of the term, meaning literally non-samsara)
includes both a higher species and a lower one; and the
statement ‘neither samsara nor lower-nirvana’ is then
compatible with the statement ‘higher nirvana’. There is
a big difference between rough, poetic, dramatic
language, and literal interpretation thereof.
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63
CHAPTER SEVEN
Drawn from Phenomenology (2003),
Chapter 1 (sections 1 & 2).
Phenomenology
Phenomenology may be defined as the study of
appearances as such. By an ‘appearance’ is meant any
existent which impinges on consciousness, anything
cognized, irrespective of any judgment as to whether it
be ‘real’ or ‘illusory.’ The evaluation of a particular
appearance (an existent within the field of consciousness)
as an illusion (existing only in consciousness) or a reality
(existing not merely in consciousness, but also before it,
after it, without it or beyond its range) is a complex
process, involving inductive and deductive logical
principles and activities. Opinion has to earn the status of
strict knowledge. To begin with, appearance must be
taken neutrally, at face value, as the common ground of
reality and illusion (i.e. one of a triad).
An appearance is whatever it seems to be. At this level of
consideration, the verbs ‘to seem’ and ‘to be’ are one and
the same. It is only at the next level, where an assessment
of status is involved, that they have to be separated.
Since appearing is being known, phenomenology can be
regarded as a branch of both Ontology (the study of
being as such; or more restrictively, of real being) and
Epistemology (the study of knowledge as such; or more
restrictively, of true knowledge). Phenomenology differs
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THE LAWS OF THOUGHT
from ontology in being less presumptive as to the nature
or status of the object dealt with, and it is for this reason
a study essential to epistemology. The basic insight or
premise of phenomenology is that knowledge develops
from neutral appearance. The common-sense view of
knowledge would seem to be that knowledge develops
from data considered at the outset as ‘sensory,’ but as we
shall see this view involves logical difficulties. The
phenomenological approach is an attempt to overcome
these difficulties, and propose a more coherent order of
development.
As I have shown in my work Future Logic, no item of
apparent knowledge, not even a percept, is ever
immediately and definitively ‘true’ all by itself. An item
may initially seem to be true, or contain some truth; but it
is only in relation to all other items, which likewise seem
to be true, that the judgment as to whether it is really or
entirely true can be made. Even the various criteria and
tests involved in such terminal judgments are themselves
to start with merely seemingly true. The science of
phenomenology is built on the same basic insight.
In
this
volume, we shall understand the term
‘appearance’ very broadly as including: (a) objects of
perception, i.e. concretes or phenomena in the physical or
mental domains; (b) objects of intuition, i.e. one’s
subjective self, cognitions, volitions and valuations;
and/or (c) objects of conception, i.e. simple or complex
abstracts of preceding appearances. Abstraction relies on
apprehensions of sameness and difference between
appearances
(including
received
or
projected
appearances, and projected negations of appearances).
Abstracts are firstly simply summaries of information,
and at a later stage more complex hypothetical entities.
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65
Coherence in knowledge (perceptual, intuitive and
conceptual) is maintained by apprehensions of
compatibility or incompatibility.
With regard to terminology, the reader is advised to keep
in mind that in philosophy, and in this particular
philosophical treatise, we use words somewhat
differently or more specifically than in common
parlance. Contrary to the impression given by the term
‘phenomenology,’ it should be understood as a study not
merely of ‘phenomena,’ but of all appearances, including
intuited particulars and abstract data11. The word
‘appearance’ is often confused with ‘illusion,’ but here
includes ‘reality.’ It is about equivalent in scope to the
term ‘object’ (content of consciousness) or ‘thing’ in
logic (anything existing or thought of). Note well that
here ‘experiences’ refers not only to the phenomena of
physical perception, but includes mental percepts, and
even intuited data. In common parlance, the term can be
more restrictive (limited to sensory inputs) or even
coextensive with ‘appearances’ (e.g. ‘my life
experiences’ includes my abstract thoughts). And so
forth – all terms will be made clear in due course. See
Illustrations at the end of the book.
Phenomenology is a science based primarily on
attentive detailed observation of one’s own experience
and discursive behavior, and only secondarily on
11
There is no point in coining a new term, even though
the term phenomenon is in the present volume used in its
primary sense of material or mental concrete particular, in
contradistinction to intuited objects or abstracts. But note that
in practice the term is often used more loosely with reference
to complex appearances like ‘a social phenomenon’ – which
include not only concretes, but also intuitive experiences and
even abstracts.
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THE LAWS OF THOUGHT
careful logical analysis and ordering of such
observations. Thus, practice of meditation is a
prerequisite to development of this philosophical
discipline, and our success in the latter depends on our
skills in the former. Although philosophical awareness
and thinking are ultimately obstacles to meditation
(which rises above intellectual pursuits), the former can
in the interim still draw significant lessons from the
latter. Labeling phenomena as “phenomena”, or making
distinctions between them, or distinguishing them from
intuitive experiences or from abstractions – such acts are
all non-meditative; but they may well occur and be
remembered in the course of meditation.
Knowledge is Based On Appearance
Our primary consideration ought to be just what is
apparent to our awareness at each and every moment.
Nothing can be granted offhand except this first given.
Appearance is immediately granted – because there is
nothing else to discuss or refer to, because discourse
arises solely in reaction and in relation to it. Thereafter,
we may stage by stage show how knowledge in general,
including our alleged knowledge of those stages,
develops.
The core thesis of phenomenology, thus, is that
knowledge is based on appearance. This is in stark
contrast to other approaches to epistemology, which
propose that knowledge is based on ‘external reality’ or
on ‘subjective truth’ or some such premature thesis.
Moreover, phenomenology regards as essential that the
sequence in which knowledge arises and develops out of
appearance be clarified. A notion or suggestion may be
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67
appropriate if intelligently placed in the ‘order of things,’
but very misleading if misplaced.

Consider, for instance, Naïve Realism (or
Materialism or Objectivism)12. This philosophy
proposes that we have a body with sense-organs, that
when these come in contact with external objects
sensations are produced, which in turn produce
primary ideas (images) in the mind, which are what
we experience and build more complex ideas
(abstract concepts) from. At first glance, this thesis
may appear obvious and worthy of universal belief.
But upon reflection, we see that it leads to serious
logical problems. If, as it suggests, ideas ‘represent’
external reality, how do we know that they indeed
‘correspond’ to it? If, as this theory implies, all we
know are ideas (sense-data and their combinations),
how can we even get to know that there is an external
reality at all, let alone a body with sense organs in
which our minds reside? Thus, surprisingly enough,
this approach to knowledge is internally inconsistent.
In reaction to this conundrum, some philosophers
have opted for the opposite extreme, a Mentalism (or
Idealism or Subjectivism)13. They have, in fact,
12
Historically, at least in its modern version in the West,
we owe this philosophy to John Locke (English, 1632-1704).
The difficulties inherent in it were noticed implicitly by his
predecessor René Descartes (French, 1596-1650), and later
by the likes of David Hume (Scottish, 1711-76) and Immanuel
Kant (German, 1724-1804). Notwithstanding, Naïve Realism
has remained a basic belief, and a source of considerable
confusion, for many people, including philosophers and
scientists.
13
For example, the Yogachara school of Buddhist
philosophy.
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THE LAWS OF THOUGHT
accepted the core tenet of Naïve Realism that what
we perceive and build knowledge on are mental
substances called ideas, while simply dropping its
thesis that these ideas originate in physical sensations
in response to stimuli from external objects. The
trouble with this thesis is that it involves a stolen
concept, since it would be hard put to define
mentality after having done away with that of
materiality. Moreover, it does not really explain the
mass of data at hand – it merely explains it away as
illusory happenstance. It does not elucidate why there
would appear to be an enormous universe of matter
15 billion years old, composed of innumerable
galaxies, stars, atoms, quarks, including on a small
planet called Earth apparent human beings, with
apparent bodies, with apparent sense organs.
Mentalism just ignores all this, or discards it as sheer
fantasy; it does not make it comprehensible. It is
therefore incomplete.
Having grasped the problem inherent in the former
theory, we might be tempted to opt for the latter,
however imperfect, were it not for the possibility of
another approach, that of Phenomenology, which
presents neither the flaw of internal inconsistency nor
that
of
incompleteness. Phenomenology brings
together the best in both those theories, while
weeding out their faulty elements.
Phenomenology starts like Mentalism with the given
content of consciousness, but identifies that content
neutrally as ‘appearance,’ instead of taking up the
prejudice that it is something mental (idea). For it
must be realized that the concept of mind was built in
contrast to that of matter; it has no meaning by itself,
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69
and would not have arisen were it not for the concept
of matter. Phenomenology therefore posits a concept
of appearance, which leaves the question of mind or
matter open to begin with, a question to be answered
in a larger context.
Phenomenology ends like Naïve Realism with a
belief in matter as well as mind, but it does not get
to that thesis in the same manner. The error of
Naïve Realism is not essentially its notion of a
physical body having sensations that generate ideas,
but the fact that it takes this notion for immediately
granted, treating it effectively as a mere
observation. Phenomenology avoids this error by
understanding the notion in question as a
hypothetical model, through which we manage to
organize appearances into an orderly and consistent
whole called knowledge.
Our premise is that the starting point of epistemology is
never a blank mind in a social vacuum, but the belief
framework of ordinary persons in a given historical and
geographical
cultural
context.
Researchers
in
epistemology are themselves such ordinary persons in a
given societal climate, with their particular viewpoints,
though hopefully outstanding intellectual capacities.
Any theory such researchers propose must ultimately
convincingly explain the genesis of the ordinary
frameworks. Whether the latter are thus wholly
justified, or demonstrated to be aberrant to some extent,
they can neither be ignored nor entirely rejected without
logical absurdity.
It is worth making a comment here, parenthetically,
about the cultural context. A man like me, born in the
20th Century and educated in the West, normally takes
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THE LAWS OF THOUGHT
the Realist viewpoint for granted, and assumes that
everyone else in the world naturally does too. People
with an opposite perspective seem at first unnatural
(philosophical nitpickers or weirdo mystics), if not
nonexistent. But it must be kept in mind that in other
regions of the world and in other periods of history, there
have been humans who sincerely held very different
worldviews (consider animism or shamanism, for
instances). One should remain open minded.
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71

CHAPTER EIGHT
Drawn from Phenomenology (2003),
Chapter 2 (sections 2 & 3).
Appearance and Other Large Concepts
By ‘appearance’ is meant, first of all, anything and
everything – but upon reflection, more specifically
anything which ‘comes to mind,’ by whatever means.
This is not a definition, but an indication. The term
appearance is too fundamental to be definable without
circularity, we can only ‘point to’ its instances; indeed,
whatever we can point to, in any sense of the term
(physically with a finger, mentally by projecting a
boundary, verbally by defining or intentionally by
focusing on), is an appearance. Thus, ‘appearance’ refers
to any object – of consciousness (but of course,
‘consciousness’ is itself too basic to be definable – see
further on).
The concept of appearance differs from that of
‘existence’ as of when we assume that things exist before
or after we are aware of them, and therefore by
extrapolation that things exist that we are never aware of.
This assumption that there are things (existents) we are
not conscious of, serves to explain or integrate, among
others, the appearance that things disappear and
reappear (signifying continuity of existence in the
interim – granting reliability to memory). It also
expresses our belief that other selves beside oneself exist
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THE LAWS OF THOUGHT
(as opposed to solipsism), each of which is aware of (and
reports) some things one is not aware of, or unaware of
some things one is aware of.
Thus, although the two concepts may initially coincide,
at some stage we come to regard appearance as a
subcategory of existence, implying that whereas all
appearances exist, some existents are not apparent. Non
apparent existents are, note well, hypothetical; i.e.
‘nonappearance’ is a word whose content is by definition
unknown but not in principle unknowable. Non-existents
do not, of course, exist; which means that the word
‘nonexistence’ has no ideational content, but is just a
verbal construct by negation (an artifice we use as a sort
of garbage can for incoherent hypothetical concepts or
propositions).
We may here also mention, in passing, the subsidiary
concept of actuality, or ‘present existence,’ which arises
in the specific context of natural modality, to distinguish
between potentiality with present existence and that
without present existence.
The concept of appearance likewise to begin with
coincides with that of ‘reality.’ But as of when we come
to the conclusion, as a way to explain certain illogical
appearances (like contradictions between experiences or
between our beliefs/predictions and experiences) that
some things are illusory, i.e. that consciousness errs
occasionally, we posit that reality is a mere subcategory
of
appearance, and therefore of existence. The
complementary subcategory of appearance, unreality or
‘illusion,’ also has the status of existence, note well.
There are also appearances that we are at a given time
unable to classify as reality or illusion; these are
temporarily problematic.
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One cannot claim that all appearance is illusion, without
thereby contradicting oneself, since such a claim is itself
an appearance that is being assumed a reality; it is
therefore logically self-evident that some appearances
are realities. The deductive relation between these
concepts is therefore this: appearance is the common
ground of reality and illusion, i.e. implied by both but not
implying either. Reality and illusion are mutually
contradictory concepts – both cannot be true/applicable,
but one of them must ultimately be so. Thus, every object
of awareness can be claimed as appearance offhand,
without prejudicing the issue as to whether it is real or
illusory. However, appearance and reality are also
inductively related, as follows: every appearance may be
assumed a reality unless (or until, if ever) it is judged
(for logical reasons, as mentioned) to be an illusion. Just
as the concepts of appearance and reality are initially (at
an uncritical, naïve level) the same, so in every instance
they remain equal except where illusion is demonstrated
(or at least, doubt is instilled).
Note well that the above differentiations between
existence, appearance and reality are not immediately
obvious, neither in the development of an individual’s
knowledge nor in the history of human thought. They are
not a priori givens, or self-evident deductive certainties
or an axiomatic absolute truths, but conclusions of
rational (conceptual and logical) process. That is, they
express a set of hypotheses which inductively, over time,
have been found to satisfactorily integrate and explain a
mass of appearances, i.e. to fit-in in a comprehensive and
convincing world-view. Thus, to mention these
differentiations ab initio, as we do here, may be
misleading – they are only at this stage vague notions
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THE LAWS OF THOUGHT
and assumptions, which are in the long run further
defined and found confirmed by the absence of any
equally credible hypotheses, any other conceptual
constructs which prove as coherent and consistent both
internally (as theoretical postulates) and externally (in
relation to cumulative appearance, and especially
experience). Their being hypotheses does not per se
invalidate them, for the claim that all hypothesizing is
invalid is itself equally hypothetical and so self
invalidating.
We shall again anticipate, with reference to what we
mean by ‘consciousness’ or ‘awareness’ or ‘cognition.’
This may be defined as the relation between Subject and
Object, whatever activities or states either may undergo
within such relation14. The fundamental given is
appearances – but we have no reason to believe that all
appearances appear to each other, i.e. we seem to have a
privilege among existents in being aware of other
existents. We suppose thereby that the fact of
‘appearance’ is different from mere ‘existence,’ and
occurs only relative to a conscious Subject.
The ‘Subject’ of this relation is identified with the
intuited self (me, in my case – you, in yours), but such
intuition has at first only the status of an appearance; it is
initially a vague and uncertain notion rather than a fully
developed and justified concept. The other pole in the
putative relation of consciousness, the ‘Object,’ refers to
the appearances involved (which are here given another
14
Whereas ‘consciousness’ refers to the relation,
‘cognition’ is conceived rather as an ‘act,’ and ‘awareness’ as
a state – but for our purposes we shall regard them as
equivalent terms. The point is that the essence is relational,
irrespective of activities or states that may often attend it.
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75
name to stress their being taken into consideration
specifically within the said relation).
To posit such a relation does not tell us anything much
about it, admittedly – we merely have a word for it,
referring to something supposedly too primary in
knowledge to be definable. But the trilogy Subject
consciousness-Object is posited by us in a bid to
understand and explain how and why appearance differs
from existence. The meaning and validity of this
hypothesis, including the new ideas of a Subject and
consciousness, are not immediate, but established with
reference to the cumulative thrust of experience and
reasoning,
including
consideration
of
conflicting
hypotheses. It is only after the latter are found less
coherent and consistent than the former that we
inductively conclude that our hypothesis is convincing
and reliable.
Let me emphasize preemptively that to postulate that
appearance signifies existence within awareness is not
meant to imply that the existence of appearances is
caused by awareness, but only to differentiate putative
non-apparent existents from appearances. The relation of
consciousness is postulated as per se neutral, affecting
neither the Subject nor the Object. Existents remain
essentially unchanged by it when they enter the field of
awareness and are labeled more specifically as
‘appearances.’ To presume the contents of consciousness
‘subjective’ (in the pejorative sense of the term),
implying a dependence (creation or modification) of the
Object by the Subject, is a very different hypothesis; one,
indeed, hard to uphold, since if we apply it to itself we
put it in doubt. Moreover, if such subjectivist hypothesis
were claimed true, there would be no need for it, for
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THE LAWS OF THOUGHT
‘appearance’ and ‘existence’ would be coextensive. So
our hypothesis of consciousness is inherently rather
‘objectivist.’ Evidently, there is lots of reasoning behind
such concepts and postulates; they are not arbitrary
assertions (as some philosophers contend). Also, such
reflections and clarifications are not and need not be
consciously made before at all embarking on the
enterprise of knowledge; they flower gradually in
response to specific doubts and questions.
Material, Mental, Intuitive, Abstract
Now, of all appearances, those labeled ‘concrete’ are the
most manifest, the most evidently present to our
consciousness. They are also called ‘phenomena,’ to
stress that we should not immediately take for granted
their apparent reality, having over time become aware
that
some are best judged illusory after due
consideration. Concrete objects seem more directly or
immediately knowable than others – apart from the issue
of reality or illusion just mentioned – so we assign them
a special kind of consciousness or cognition called
perception and label them ‘percepts.’
Among concretes, some are more ostentatious and
permanent than others and seem relatively far and
independent of us – these we refer to as ‘material’ or
‘physical.’ The remainder we label ‘mental’ or
‘imaginary,’ distinguishing them by their relative
poverty, transience, intimacy and dependence on us.
Most of our common ‘world’ (cumulative appearance) is
composed of material phenomena, and all or most mental
phenomena seem to be derivative replicas of them or of
parts of them. Among material phenomena, some are
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considered ‘in our own body’ or ‘physiological,’ and the
others ‘outside our body,’ our ‘body’ being distinguished
by its relative proximity (to the observer) and the
peculiar events occurring in it (sensations and
sentiments). Some bodily phenomena (such as sentiments
and ‘actions’) seem to have mental origins, and so are
called
‘psychosomatic.’ Conversely, many mental
phenomena are regarded as having bodily causes.
In addition to mental phenomena, we should distinguish
the non-concrete appearances we may call ‘intuitive’
appearances, which are our impressions of self
knowledge (one’s self, cognitions, valuations, volitions).
These differ from imaginations, in that they per se have
no concrete expressions, yet they share with mental
phenomena the appearance of intimacy and being in our
power to some degree. They are assigned a specific kind
of consciousness called intuition (whence their name
here) or apperception.
Concretes (mental or material) and intuited objects have
in common a status of immediate evidence, which we
express by calling them ‘empirical’ or ‘experiential.’
Experiences are ‘givens’ in a way other appearances
(namely abstracts) cannot match. Considered purely in
and for themselves, without interpretation or inference,
they are unassailable, not requiring any proof.
‘Abstract’ appearances may be classed as last in that
they seem derived, by various means, from the
preceding, experiential varieties of appearance. These
means are collectively labeled ‘rational’ (implying they
proceed from a faculty of reason). The term abstract
refers to the primary act of reason, namely abstraction
(which depends on identification of sameness or
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THE LAWS OF THOUGHT
difference, i.e. on comparison and contrast between two
or more appearances).
Abstract appearances share with intuitive ones the lack of
concrete manifestation; we have nothing to directly show
for them, they are phenomenally blank. But abstracts
differ from intuitive appearances, in that getting to know
the former requires a process (comparison and contrast),
whereas the latter are directly known (in self-experience).
Furthermore, abstract objects are ‘universals’ and
essentially ‘external to us,’ whereas intuitive objects are
‘particulars’ and very much ‘part of us.’
Consciousness of abstracts is called conception, so they
are also called ‘concepts.’ But the processes leading to
concepts (our discourse) are far from simple and seem
subject to many rules; the latter are labeled ‘logic.’
Abstracts require proof, and ultimately some sort of
empirical grounding. The only exception to this rule is
the case of self-evident propositions, which cannot
logically be denied without committing a self
contradiction. But even in the latter cases, the concepts
involved are never entirely ‘a priori,’ but require some
preceding experience to have at all arisen.
Let me summarize here: perception is knowledge of
concretes, i.e. material or mental phenomena; intuition is
self-knowledge; perception and intuition are experiences;
conception is knowledge of abstracts, derived with the
aid of logic from phenomenal or intuitive data.
‘Knowledge,’ of course, at first simply means
consciousness or cognition – the term is rendered more
precise later with reference to cumulative Appearance.
‘Thought’ and ‘idea’ are, by the way, catchall terms that
may include a mix of conception (concept formation,
conceptualization),
imagination
(visualization,
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verbalization, forming hypotheses) and logical discourse
(inductive and deductive), all of course implying some
experience (sensory or intuitive).
As I have indicated earlier, I am not convinced that
qualitative differences alone suffice to distinguish
material from mental phenomena. We tend to think of the
latter as less clear or vivid than the former, but this is not
always the case. Dreams are sometimes extremely vivid
and colorful, and the physical world is sometimes misty
and
unclear.
For this reason, I suggest that
phenomenology must suppose that introspection is to
some extent involved in making this fundamental
distinction. We are presumably somehow aware of the
direction of input of the concrete data. Material data is
‘felt’ as coming from or via the body, whereas mental
data is ‘felt’ as coming from a closer source (called the
mind). Granting that such ‘feelings’ of direction of
source are not themselves phenomenal marks (otherwise
we would be begging the question), we must interpret
them more precisely as intuitions. To be consistent we
must say that we do not intuit where the data comes
from, but rather intuit in what direction we turn our
attention to gain access to the data.
It
should be noted that we have above effectively
distinguished three substances or stuffs of existence,
matter, mind and spirit. We have based their
differentiation partly on the fact that some experiences
(those intuited) do not have phenomenal characteristics;
and partly (as regards the distinction between material
and
mental phenomena) on the differences in
phenomenal properties and locations combined with
assumed intuited differences. All three of these
substances may give rise to concepts. We may also
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THE LAWS OF THOUGHT
presume souls, i.e. spiritual entities, other than our own
through their apparent phenomenal effects and by
conceptual means.
Just as the phenomenal modalities and qualities and their
behaviors are considered as mere varieties of matter and
mind, so the cognitions, volitions and affections of the
soul need not be assigned yet another substance, but may
be considered as events or properties of that same
substance. Abstracts relating to material, imaginary or
spiritual givens do not, likewise, require a further
substance, but may be considered as mere expressions of
these
three
substances.
There
is
nothing
epistemologically unreasonable in assuming substantial
differences between the said three classes of object. It
remains possible that the three substances are ultimately
different versions or degrees of one and the same stuff.
The concept of substance is introduced relative to those
of static attributes and dynamic movements, implying a
presumed substratum for them. It allows us to presume
continuity of something, an individual entity, in the
midst of motion or change. The various attributes and
movements are thus conceived not as mere
happenstances but as all ‘belonging’ to and ‘caused’ by
an abiding, unifying entity15. We also assume that
different instances of that kind of entity remain
essentially the same (i.e. of same substance) although
some of their attributes and movements may differ. Note
well that both ‘substance’ and ‘entity’ are abstracts.
15
In the case of a human Subject/Agent, causality is
usually meant as ‘volition’ (implying some consciousness and
responsibility) not as mere mechanical ‘causation,’ note well.
Similarly, ‘possession’ of attributes may in some cases be
voluntary.
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Although material and mental phenomena have concrete
character, while soul has not, the latter may nonetheless
equally legitimately be conceptually posited.
These beliefs, in substances and entities, are not
immediate
certainties
but
constitute
conceptual
hypotheses. This fact alone does not disqualify them,
contrary to what some philosophers suggest. If a
hypothesis gives rise to a world-view that is always, all
things considered, consistent and confirmed, and no
alternatives serve the same purpose as well or better, then
it is inductively worthy of adoption. This seems to be the
case with regard to the concepts of substance and entity.
Without them, we would find ourselves unable to ‘make
sense’ of (integrate, explain) all our experiences and
intuitions; no one has to my knowledge managed to
construct in detail equally credible and useful counter
hypotheses.
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THE LAWS OF THOUGHT
CHAPTER NINE
Drawn from Phenomenology (2003),
Chapter4 (section 2).
COMPATIBILITY OR INCOMPATIBILITY
Apprehension
Allied to sameness and difference are the concepts of
compatibility or incompatibility, which underlie what
Aristotle has called the three ‘laws of thought’ – identity,
non-contradiction and exclusion-of-the-middle. How do
we apprehend things (percepts, intuitions, concepts and
propositions about them) as able to coexist (compatible)
or as unable to do so (incompatible) or problematic (not
established as either compatible or incompatible)? We
must answer this question urgently, if we admit that these
logical processes of confrontation (or facing-off) are as
basic as those of identifying sameness or difference. The
whole of logical science is built on their assumption, and
we must explain how we know two things to be
harmonious or mutually exclusive or of undecided
correlation.
An important insight or principle we may suggest at the
outset is that consistency is not something we
apprehend – it is inconsistency we apprehend;
consistency is just the absence of inconsistency. Thus,
despite the polarities we have given the words,
compatibility
is
something
negative,
whereas
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incompatibility is something positive. Everything seems
harmonious to us, till we discern some conflict. We judge
things consistent, so long as we have no logical insight of
inconsistency between them. Thus, strictly speaking,
inconsistency can be directly ‘seen’, whereas consistency
is normally assumed till found lacking. In some cases,
consistency is indirectly put in doubt, without some
direct inconsistency having been found, so that an
uncertainty arises.
Aristotle formulated his three ‘laws’ firstly with
reference to percepts or concepts by stating them as ‘A is
A’, ‘A cannot be non-A’ and ‘Either A or non-A’. In a
later stage, they are formulated with reference to
propositions. As I argue extensively in Future Logic16,
these laws are not laws in the sense of a-priori principles
or arbitrary axioms, as some have claimed, though they
are self-evident in that to deny them is self
contradictory17, but have to be regarded as given in their
objects somehow. Psychologically, they are profound
impulses (which may be ignored or followed), which
make humans rational; ethically (in the ethics of
knowledge gathering), they are indispensable tools and
imperatives to actively respond to certain epistemic
situations in certain ways (though one can be dishonest
or unaware and ignore the facts, or evasive or lazy and
ignore the imperative).
16
17
See Future Logic, chapters 2 and 20.
See Future Logic, chapter 31.
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THE LAWS OF THOUGHT
Explications
Identity brings to mind the visual image and sensation of
calm or attraction or a tendency to merge of two things
(equation), contradiction that of conflict or repulsion or
explosive collision between them (because they cannot
occupy the same place), while exclusion of the middle
refers to a gap or deficiency between them (raising
doubts and awakening questions). These may be
imaginative representations for philosophical discussion
like here, but they are not always (if ever) involved in
concrete identification of identity, contradiction or
research needs. Their involvement is more technical or
abstract, straddling as it were the experiential domain and
the conceptual knowledge domain. Although formulated
as a triad, the laws of thought are three aspects of
essentially one and the same necessity.
The law of identity, simply put, tells us “what you see is
what you get” – it is a mere acknowledgment that the
data of phenomenal experience are the fundamental
givens of any knowledge enterprise; that there is
ultimately no other data to base inference on, so that all
their details must be paid attention to and taken into
consideration in any inference. With respect to its
formulation as ‘A is A’, with reference to terms rather
than propositions, this law would simply mean that, if we
for instance compare the constituent points in any two
material or mental complex phenomena, we have to
acknowledge that wherever dots appear (or fail to
appear) to us, we can definitively say that there are (or
are not, respectively) dots (at least phenomenal dots) – at
least for now, until if ever the situation changes or further
scrutiny tends to belie the first observation (because
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many later observations supplant the first, by their
statistical weight).
Identity is a law, because there is no other way to
conceive things – at this phenomenal level to ‘seem’ is to
‘be’. You can deny your phenomenon’s reality, but not
its very occurrence or existence. If you try to deny your
actual phenomenon by immediately hypothesizing some
invisible conflicting ‘phenomenon’ behind it (a
noumenon, to use Kant’s word), you are condemned to
being basically unempirical and therefore without
epistemological justification for your own act. You have
nothing to show for your case, since by definition you
appeal to the unseen, whereas you must acknowledge the
seen as seen to at all deny it. The baselessness and
circularity of such refusal to accept the phenomenon (as
a phenomenon, no more, at least) merely reflects that the
phenomenon experienced is the given to deal with in the
first place (for this reason any denial of it is bound to
admit it, implicitly and explicitly by referring to it). All
such argumentation is of course very conceptual, and so
only at best lately and peripherally significant in any
actual act of acceptance of the phenomenon as such.
Phenomenologically, the law of identity means that an
image of a material entity, mentally projected externally
onto that entity, does not blank out the entity (being as it
were in a parallel space, transparent). When such mental
image seemingly shares outer space with the material
body it is projected on, then the phenomenon as a whole
has changed, though the material entity stays on
(perseveres as an appearance), having been augmented in
respect of a mental image. That is, the new phenomenon
is enlarged (by an additional image) in comparison to the
originally
given
phenomenon. This means that
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THE LAWS OF THOUGHT
postulation of a noumenon merely adds a mental
component (including additional phenomena) to the first
presented phenomenon, and does not succeed in erasing
the first phenomenon, precisely because it is introduced
in relation to the first phenomenon (specifically, as an
attempt to explain it or explain it away).
The law of identity is an impulse, a call to empiricism,
which we normally obey without doubt or question. It
acknowledges that appearances might in the long run
change or prove misleading, taking into consideration all
other appearances. It does not deny, nor accept ab initio,
that behind the seen appearance there might be unseen or
invisible events or things; but such outcome can only be
arrived at through an overall consideration of all
experiences and much pondering. That is, ‘noumena’
might well exist beyond a given field of phenomena –
but they would have to be end products of an evaluative
process and could not be first assumptions. Since
evoking noumena does not in itself annul phenomena
(merely adding more phenomena to them), the questions
inherent in phenomena and their apparition to us remain
unanswered.
The reason why the thesis of noumena seems at first
sight credible, is that we have experience of different
sense-modalities, each implying that the others are
incomplete, and we have memory of changes in our
experience and/or its interpretation over time, so that our
conceptual knowledge (or its suppositions) has naturally
come to conclusions that ‘things are not quite or always
what they seem’. But in such case, the term noumenon is
trivially but another name for abstracts or concepts. In
Kant’s coinage and use of the term, however, the
noumenon is not a hidden extension of the phenomenon,
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but purports to discard and replace the phenomenon
altogether. The noumenon is by definition unknowable
(universally) – though Kantians never tell us how come
they themselves have the privilege to even know enough
about it to know that it exists and is unknowable! The
correct statement would rather be that noumena (i.e. less
abstrusely, abstracts, concepts) are not concrete
experiences, but merely logically assumed derivatives of
percepts. They are hoped to be ontologically ‘more real’
than percepts, digging deeper into reality than the visible
surface of things (to which we are supposedly restricted
somewhat by the limited range of sense-modalities open
to
cognition), even as they are epistemologically
admitted to be less reliable.
The laws of non-contradiction and of the excluded
middle are intertwined with that of identity, as evident in
the arguments above. But how do we know that ‘A is not
non-A’ or that it is either-or between them? Consider our
basic dot of light or its absence (darkness) in the visual
field – such a dot is evidently never in contradiction with
itself. We never simultaneously perceive a dot and not
perceive it – in any given place we mentally chose to
focus on, there either appears or does not appear a lighted
(or dark) dot. At this level, where the object is reduced to
a single character (light) and precise place (the smallest
possible size), we cannot honestly, sincerely answer ‘yes
and no’ or ‘neither yes nor no’ to the question. It is there
or it is not. If it seems there, it is. If it does not seem
there, it is not. We cannot even pretend we don’t see
what we see – at least not in words, for we would have to
acknowledge their meanings, and therefore the actual
phenomenon.
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THE LAWS OF THOUGHT
These laws are indeed in the phenomenal world, insofar
as positively no phenomena ever appear in contradiction
or as neither-nor, i.e. by absence of empirical evidence to
the contrary. They are in, because their negations are not
in. But they relate to mind, inasmuch as when a dot A
appears and we start speaking of the unseen non-A, we
are in fact imagining non-A in our heads, and so bring a
new (mental) element into the picture. By the law of
identity, this non-A phenomenon (which is mental) must
be distinguished from its alleged opposite A (the given,
which may or may not be mental), and admitted as an
addition in the experiential field. But it remains true that
A and non-A themselves are not in fact coexisting or
both absent in the field – rather what we experience is
coexistence of the given A with a projected non-A.
The law of contradiction does not deny the possibility
that two different things might coexist, like a dot of light
and the imagination (or memory) of absence of such dot
of light; such things are merely contrary. The law of the
excluded middle does not deny the possibility for
something and the idea of its absence to be both absent
from a field of experience; in such case, we can still
suppose, as we indeed see as experience, that the thing
itself is absent (even though the idea of its absence is
allegedly absent – until mentioned as absent, that is!)18.
18
Our minds seem so made that, indeed, we might
consider that we always think non-A when we see A. This is
not a mere perversion of the mind, it is rather an expression of
the fact that concept-formation involves not only reference to
perceived similarities between two objects, but also to
perceived dissimilarities between other objects and them.
Thus, in order to classify something as A, we must
simultaneously declassify it from non-A. That is, the thought of
A automatically calls forth the thought of non-A, for purposes
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Thus, these laws are empirical, in the sense that they do
not impose anything on the phenomenon, but accept it as
is. They merely push the observer back into the fold of
experience, should he venture to stray. They do not
involve a modification or manipulation of the
phenomenon, but on the contrary make the observer
openly and carefully attentive to what is phenomenal.
They involve a distinction between primary phenomena
(be they ‘material’ or ‘mental’), as given ab initio, and
imaginary alleged representations (ideas, mental
phenomena) of eventual phenomena, which merely
introduce additional phenomena.
Negation
It is very important to emphasize again that negation is a
logical act. It is never a pure experience, but always
involves conceptual interference by the Subject. In
formal logic, terms like A and non-A are neutral and
formally indistinguishable. That is, they function in
interchangeable ways, so that the negation of non-A
(non-non-A) is technically equivalent to A (by
obversion); and we might label non-A as ‘B’ and A as
of distinction. It is not that A per se implies non-A (though in
most cases, A in one thing implies non-A in others, otherwise
neither A nor non-A would be distinguishable in the first
place), rather it is that A cannot be fully delimited or
understood without bringing to mind non-A as a possible
alternative (except perhaps ‘non-existence’ – though in that
ultimate case, we can say that the term is merely verbal,
without conceivable concrete referent). Furthermore, concepts
formed by negation (like darkness) presuppose some
relatively positive phenomena (like light), whose absence they
express, having been conceived first.
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THE LAWS OF THOUGHT
‘non-B’ without affecting inferential processes. But at
the phenomenological level, these labels are quite
distinct.
Something appearing would be labeled
positively (say, A), whereas something not-appearing
would be labeled negatively (as non-A).
What we here labeled A is a phenomenon or percept.
What we here labeled non-A is not apparent per se, but
only effectively ‘apparent’ in that A did not appear. Non
A signifies that we have asked a question ‘is A there (i.e.
in the phenomenal field)?’ and after further scrutiny
answered it by ‘no, I do not find it there’. The former
(presence) is directly known, the latter (absence) is
indirectly known through a mental projection (imagining
A, i.e. inventing it or remembering it from previous
perceptions) coupled with an experimental search (whose
result is unsuccessful). Clearly these are very different
cognitions – one being purely passive and empirical, the
other involving an active inquiry and referring to
observation only by the failure to confirm an anticipated
equivalent of one’s imagination. The later is useful and
informative, but it is a construct.
Negative concepts or statements are thus never strictly
speaking empirical, and negation is a fundamental
building block of reason. A negation is at the outset, by
its very definition when introduced by the Subject as a
cognitive artifice, logically contradictory to something. It
cannot then be said empirically that both percepts A and
non-A occur (since saying I ‘see’ non-A in the present
field of perception just means I looked for and did not
see A in it), nor that neither A nor non-A occur (since if I
look and do not see A in the present field of perception, I
would conclude non-A for it – though I may remain
open-minded about other eventual fields of perception
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containing A)19. A negative concept or statement is
therefore fundamentally different from a positive one,
and can at best only indirectly ever be characterized as
‘empirical’.
Primaries
The three laws of thought are logical primaries, involved
in all discourse about any phenomenon (and similarly
relative to intuitive data, and at a later stage with respect
to conceptual discourse itself). They jointly operate in
identical ways in every observation, pushing us to admit
what we see (identity), not to contradict what we see
(non-contradiction), and not to ignore and add
possibilities to what we see (exclusion of a middle). To
fail to apply them is simply to confuse the given data
with additional mental ingredients (fantasies), which
neurotically either deny the evidence (mentally replacing
it with its contradiction) or question it (by mentally
proposing a ‘middle’ term). These laws can be stated as
propositions, but they nevertheless have no conceivable
alternatives. Any doctrine proposed has to be reconciled
with experience somehow, since all discourse is a
reaction to experience, an attempt to solve the mystery it
19
Of course, at a conceptual level, i.e. when dealing
with abstracts, we may encounter contradictions (i.e. both A
and non-A seeming true) and doubts (i.e. neither A nor non-A
seeming true). Here, both the positive and negative concepts
are mental constructs, and so there is no guarantee that the
issue can immediately be resolved by one look. That is of
course where the whole science of logic comes into play; it is
needed to deal with just such issues with reference to a
plurality of experiences.
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THE LAWS OF THOUGHT
presents, so merely ignoring experience does not qualify
as reconciliation.
In that sense, it is accurate to say that these laws are laws
of thought; they are laws for the mind (the observer). We
may say that something is A and not A, or neither A nor
not A. But these words have no meaning in experience,
no phenomenal referents. They are just words, sounds or
drawings that signify nothing, not even an imaginable
circumstance. The way we ‘imagine’ them is to stupidly
or deliberately confuse a thing and an image of a thing,
and project the idea of non-A (instead of non-A itself)
next to A (or next to the idea of A) or some such artifice.
In other words, the propositions claiming to deny the
laws of thought have only a superficial meaningfulness
and credibility, due to in fact having referents (ideas)
other than those they pretend to have (things). With
regard to the original objects of perception, they are in
fact silent.
Note well that application or obedience the laws of
thought does not involve an imaginative act (a volition);
it is on the contrary attempts to ignore or deny them
which do, requiring interference of the observer’s
imagination in the cognitive process (preempting
experience). That is, the laws of thought themselves are
objective, it is only their denials that are subjective (in
the pejorative sense). The laws of thought thus remain
empirically,
and
epistemically,
and
therefore
epistemologically, undeniable. So much with regard to
applications of the laws of thought to perceptual
evidence.
contrasts,
With regard to concepts (which derive from comparisons
and
or
recombining
such
from subsequent imaginations
concepts)
and
propositions
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(imaginations of relations between concepts), they
remain always open to doubt, hypothetical, so long as
equally credible alternatives are imaginable. Credibility
is found in everything experienced or thought, it is
merely admittance that such and such has been
experienced or thought (thought being a sort of
experience, though mental). Ab initio, any two concepts
or
propositions are compatible, having both been
thought. Incompatibility is a later judgment, which
follows realization that the concept or proposition
somehow directly or indirectly contradicts experiential
evidence or leads to internal inconsistency in knowledge
or is inherently self-contradictory.20
If two such ideas or thoughts are found or not found to be
in utter conflict, they both retain the minimal credibility
of being at least imaginable, at least till one or both of
them is found incoherent with some experience(s) or for
some reason unimaginable. If for some reason they are
considered to be in conflict, they separately retain some
credibility, though their interaction raises a doubt and it
is understood that we have to ultimately eliminate at least
one of them, removing its temporary credibility with
reference
to
further
experiences
or
abstract
considerations. During the phase of doubt, we may refer
to their frequencies of confirmation in experience, and
regard one as more credible (or likely or probable) than
the other.
20
We consider concepts or propositions compatible until
and unless we find some incompatibility between them. As I
already pointed out in Future Logic, in opposition to the claims
of certain modern logicians, we do not ‘prove consistency’ but
rather ‘find inconsistencies’.
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THE LAWS OF THOUGHT
The job of Logic is, note well, not to exclude as much as
possible, but to find ways to include as much as possible,
so that all opinions and points of view (which all have
some basis and so represent some kind of experience) are
accounted for and explained or explained away. Logic is
thus
not
merely, as some contend, search for
contradictions, but (this in order to) search for
harmonizations.
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CHAPTER TEN
Drawn from Phenomenology (2003),
Chapters 7 (sect. 3) and 4 (sect. 5).
Logical Attitudes
Logic is usually presented for study as a static
description and prescription of forms of proposition and
arguments, so that we forget that it is essentially an
activity, a psychic act. Even the three Laws of Thought
have to be looked at in this perspective, to be fully
understood. To each one of them, there corresponds a
certain mental attitude, policy or process…
a) To the Law of Identity, corresponds the attitude of
acknowledgement of fact, i.e. of whatever happens
to be fact in the given context. Here, the term ‘fact’ is
meant broadly to include the fact of appearance, the
fact of reality or illusion, or even the fact of
ignorance or uncertainty. Also, the attention to
eventual conflicts (contradictions, incompatibilities,
paradoxes, tensions) and gaps (questions, mysteries);
and by extension, other forms of oppositional
relations.
b) To the Law of Non-contradiction, corresponds the
policy of rejection of contradictions. Contradictions
occur in our knowledge through errors of processing
of some kind (e.g. over-generalization, uncontrolled
adduction,
unsuccessful
guessing),
which
is
ultimately due to the gradual presentation of
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THE LAWS OF THOUGHT
information to the human observer and to his limited,
inductive cognitive means. The Law is an insight that
such occurrence, once clearly realized, is to be
regarded not as a confirmation that contradiction can
occur in reality, but as a signal that a mere illusion is
taking place that must be rejected.
c) To the Law of the Excluded Middle, corresponds the
process of searching for gaps or conflicts in
knowledge and pursuing their resolution. This is
the most dynamic cognitive activity, an important
engine in the development of knowledge. And when
a contradiction or even an uncertainty arises, it is this
impulse of the human thinking apparatus that acts to
ask and answer the implicit questions, so as to
maintain a healthy harmony in one’s knowledge.
Thus, the exercise of logic depends very much on the
human will, to adopt an attitude of factualism and
resolve to check for consistency, look for further
information and issues, and correct any errors found. The
psychological result of such positive practices, coupled
with opportunity and creativity, is increasing knowledge
and clarity. The contraries of the above are avoidance or
evasion of fact, acceptance of contradictions, and
stupidity and laziness. The overall result of such illogical
practices is ignorance and confusion.
Whereas ‘consciousness’ refers to the essentially static
manifestation of a Subject-Object relation, ‘thought’ is
an activity with an aim (knowledge and decision
making). The responsibility of the thinker for his thought
processes exists not only at the fundamental level of the
three Laws, but at every level of detail, in every
cognitive act. Reasoning is never mechanical. To see
what goes on around us, we must turn our heads and
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focus our eyes. To form a concept or formulate a
proposition or construct an argument or make an
experiment or test a hypothesis, we have to make an
effort. The more attentive and careful our cognitive
efforts, the more successful they are likely to be.
Unity In Plurality
The … ‘wave’ theory of universals, granting its premise
that everything is ultimately reducible to ‘waves,’ i.e.
mobile vibrations in some sort of continuum, leads to the
very radical conclusion that ‘all things are one.’
The world as it appears to our touch-organs or to the
naked eye – or even the eye aided by microscope or
telescope – may give the impression that dimensionless
points, lines or surfaces exist in nature, but as Physics
has evolved it has become clearer that physical objects
do not have precise corners, sides or facades – but fuzzy
limits, arbitrarily defined by the visibility to our senses
(specifically, sight and touch), aided or unaided, of
concentrations of matter or energy.
For example, the tip of my penknife may seem like a
sharp “point” to my touch or sight, but it is really –
according to physical science (i.e. upon further
investigation and reflection) – a rough, voluminous
conglomerate of atoms, which are themselves complexes
of smaller and smaller particles (electrons, protons and
neutrons, seemingly some distance ‘apart’ from each
other, etc.), which are themselves without beginning or
end being really vague clusters of waves. Similarly with
regard to the cutting edge or flat sides of my penknife.
Indeed, if one takes these considerations to their extreme
conclusion, one could say that no object has a beginning
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THE LAWS OF THOUGHT
or end, every object stretches to the ends of the universe
or to infinity, and what we refer to as a specific
individual object is merely the most humanly visible or
concentrated part of that whole, which we arbitrarily or
conventionally consider a separable unit (and habitually
name, to solidify our viewpoint). So that ultimately,
there are in fact no individual objects, but only ripples
in the single object that is the universe as a whole.
Where does an atom (or any other body) begin or end,
granting that all consists of waves? If we see a star
billions of miles away, on what basis do we say that the
star ends over there, while the “light from the star” is
here? Rather, we ought to say that the light we see is part
of the star, i.e. that it extends all the way to us (at and
through our visual sense organs, and on to our memory)
and beyond. At what distance from the star do the gases
or the light it emits cease to ‘belong’ to it, and are to be
considered as ‘separate’ bodies? The cut-off point can
only be arbitrary, i.e. mere convention. Gravity operates
at astronomical distances. What objective ground do we
have for distinguishing a field from its apparent origin?
Furthermore, stars are in constant flux, arising in time
and disappearing in time. At what point in time (as well
as space) may we claim that the matter and energy we
now call a star is ‘not yet’ or ‘no longer’ a star? Surely,
the quarks from which the star emerged were already ‘the
star’ and when the star bursts or is absorbed into a black
hole it is still ‘the star.’ We ourselves are stardust – does
that mean that the stars in question became us, or that
being a star – from the beginning of time to its end –
includes eventual human forms?
In this view, every entity in the universe stretches out
with every other to fill the whole space and time of the
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universe! And if we say this, we might as well say –
without any mystical intent, though in agreement with
Buddhist mystics – that all things are one. There are just
more intense concentrations of matter or energy here and
there, now and then, in one continuous field, but nowhere
dividing lines. Because we perceive only fractions of the
totality, only the aspects involving the sense-modalities,
we isolate small blobs of the whole as individual
phenomena. All phenomena perceived are centers of
complex wave activities in the universal fabric; We
‘individuate’ phenomena with reference to the sense
modalities they exhibit which are accessible to our
senses. We regard as delimiting an individual object in
space and time such perceivable fraction (visible to the
senses) of the wave activity stretching to the ends of the
universe – ignoring its larger invisible extensions, later
induced by reason. Thus, all individuation is fantasy
(this can be known by rational considerations, as here),
reinforced
by naming (itself a sense-modality
phenomenon, by the way). In which case, strictly
speaking, nothing is divisible at all.
That would seem to be a correct view of our physical
world in the context of present knowledge – the
hypothesis most consistent with experience, experiment
and current scientific theorizing. We thus, provided we
anticipate the results of Physics and claim that some sort
of unified field theory is sure to be established, and
provided we stretch that assumption to include wave
explanations of the mental and spiritual domains, arrive
at a concept of the world as ‘unity in plurality’ – a
harmonious marriage of the philosophies of Pluralism
and Monism. Heraclitus was right – everything is
ultimately motion (i.e. waves) and Parmenides was right
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THE LAWS OF THOUGHT
too – everything is ultimately one thing (i.e. the medium
subject to waves).
We could even view this conclusion as a justification of
the Buddhist view that “all things are empty!” For
instance, the message of The Diamond Sutra seems to be
that all objects material or spiritual are infinite vortices
with no beginning and no end. They are neither
categorical as they seem; nor can they be surely declared
hypothetical, being delimited merely by our naming of
them, but having no sure limits in themselves so far as
we know so that they are therefore effectively boundless.
We have already, inspired by Buddhist doctrine,
concurred with them that individuation is a man-made
artifice. But even granting that we might legitimately, out
of mere convenience, focus on specific places and
durations of the universe, because a disturbance ‘stands
out’ there and then in relation to our senses – we are still
left with the question as to what it is that is disturbed?
What is the medium or substratum of all wave motions?
We are tempted to view it as a stuff and call it
“existence,” or like Descartes call it “the ether.” The
problem is that since the Michelson-Morley experiment
on the velocity of light such a substance underlying
waves has apparently been discredited. These physicists
measured the velocity of light in the same direction as
our planet’s motion and in the opposite direction. To
everyone’s surprise, they found the velocity identical
either way. This was eventually explained by Albert
Einstein as indicative that there is no absolutely
stationary substratum or “ether” relative to which wave
motions occur, and he built his famous theory of
Relativity as an alternative world-view (such that space
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and time coordinates are depend on the velocity of the
observer relative to what he measures).
Thus, although when we think of waves, and
mathematically work out their motions and interactions,
we regard them as disturbances within some medium, it
turns out that there is no such medium according to
experimental indices! On this basis, we can agree with
Buddhist
philosophers
that
(surprisingly,
incomprehensibly) nothing is being waved – i.e. that the
ultimate nature of “existence” is “emptiness.” And there
is no need of high meditation or mystical insight to arrive
at this conclusion – it is seemingly justified by ordinary
experience and reason (scientific experiment and theory).
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CHAPTER ELEVEN
Drawn from Ruminations (2005),
Chapter 1 (sections 1-3).
Dialectical Reasoning
The three “Laws of Thought” may be briefly explicated
as follows:
Thesis: there are certain appearances; appearances
appear.
Antithesis: there are incompatibilities between
certain of these appearances; in such cases, one or
both of them must be false.
Synthesis: some remaining appearances must be
true; find out which!
We can in this perspective consider dialectic as a
fundamental form of thought, through which knowledge
is made to progress on and on. It is not a mere detail, an
occasional thought-process, but a driving force, an
engine, of thought.
The laws are not mere information, but calls to cognitive
action. They enjoin proactive and curative cognitive
measures, to ensure (as much as possible at any given
time)
continued
completeness.
verification,
consistency
and
(i) The law of identity tells us to seek out the facts and
sort them out as well as we can. The purpose of this law
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is to instill in people a healthy respect for facts, in the
course of observation and judgment. It is essentially a
call to honesty, and submission to the verdict of truth.
People often think, or act as if they think, that ignoring or
denying unpleasant facts or arguments will make them
‘go away’ – the law of identity says ‘no, they will not
disappear, you must take them into consideration’.
Some people think that it is impossible for us to ignore
that “A is A”. Far from it! All of us often do so – as
when we refuse to look at or admit the evidence or a
logical demonstration; when we avoid reality or evade it
having glimpsed it; when we lie to ourselves or to others;
and so forth. If the law of identity were always obeyed
by us, there would be no need to formulate it. Logic
states the obvious, because it is often shunned.
(ii) When the law of non-contradiction says to us “you
cannot at once both affirm and deny a proposition”, it is
also telling us that if we ever in the course of discourse
encounter a situation where a proposition seems both true
(for some reason) and false (for other reasons), we have
to go back upstream in our discourse and find out
where we went wrong in the course of it21, and we have
to effect an appropriate correction such as to eliminate
the difficulty.
We are not just saying: “ah, there is a contradiction”, and
leaving it at that, nonplussed. No, we are impelled to
seek a solution to the problem, i.e. to resolve the
contradiction. We are inferring that there must be
something wrong in our earlier thinking that led us to this
conundrum, some error of observation or reasoning that
requires treatment. So long as this situation is tolerated,
and we cannot pinpoint the source of error, the credibility
21
“Check your premises”, Ayn Rand would say.
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of all related knowledge is proportionately diminished.
Consistency must be restored as soon as possible, or we
risk putting all subsequent knowledge in doubt.
(iii) Similarly, the law of the excluded middle does not
just inform us that “no proposition can be claimed neither
true nor false”. This law insists that if we find ourselves
in such a situation, and it is indeed the case that both a
proposition and its exact negation both seem false, we
cannot let the matter rest or hope to find some
compromise position – we have to eventually, as soon as
possible, find good reason to opt for one side or the
other. There is no logically acceptable middle ground, no
avenue of escape.
These action implications inherent in the laws of thought
may also be characterized as dialectical thinking. In this
perspective, the “thesis” is our knowledge (or opinion) as
it happens to be at a given time; the “antithesis” is the
discovery of a logical flaw in that thesis, which causes us
to have doubts about it and seek its review; and finally,
the “synthesis” is the corrections we make in our
premises, so as to resolve the difficulty encountered and
obtain a less problematic new state of knowledge.
Genesis of Axioms
Axioms are not arbitrary, a-priori starting points of true
human knowledge. They may be deductive or inductive,
but in either case are to some extent empirical (in the
large sense of ‘phenomenological’, i.e. without
depending on any materialist or mentalist assumption
concerning what is experienced).
Deductive axioms are established using certain positive
or negative logical arguments, which we naturally find
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convincing. But even a deductive axiom relies on certain
experiences, those that gave rise to the concepts and
logical techniques involved in the proposition and its
acknowledgment as an axiom.
The positive argument for an axiom is essentially
dilemmatic: “whether this or that, so and so is true”. An
example is the axiom that diversity exists. The mere
seeming of diversity is itself a case of diversity, sufficient
to establish the fact of diversity. It is no use arguing (like
Parmenides or the Buddha) that this apparent diversity is
an “illusion”, and that “all is really one” – because the
coexistence of illusion and reality is itself an event of
diversity. Thus, diversity truly exists, and cannot just be
ignored. We might still try to uphold the thesis that
reality is ultimately unitary, but only if we convincingly
account for the fact of diversity.
Deductive axioms are also justified negatively through
paradoxical
logic,
i.e.
by
showing that their
contradictories are self-contradictory. For example,
“There is no diversity” is a claim to diversity (since it
involves many words, many letters, many sounds, etc.),
and therefore self-contradictory; whence, it is self-evident
that “There is some diversity”. This argument may also
be construed (as above) as dilemmatic in form: “whether
you deny or affirm diversity, you affirm it”.
Inductive axioms rely on some generalization, or (more
broadly) adduction, from experience; but such inductive
process in their case is not ever likely to be in need of
revision. Many truths of utility to epistemology are
inductive, and yet once realized remain immutable; they
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thus behave largely like deductive axioms, and may by
analogy be classed as inductive axioms.22
For example, the fact that most of our beliefs are
contextual is a non-contextual truth, though based on
common observation. The awareness that most of our
knowledge is empirical, and subject to revision as new
experiences are encountered, that it is in constant flux,
altering and growing – this is a broad observation that
once realized will not be affected by any further
empirical data. This observation is not useless, note well:
it logically affects pursuit of knowledge, teaching us to
remain aware of the non-finality of most of our beliefs.
But note also, the said principle of contextuality is pretty
vague; it cannot by itself put specific knowledge in doubt
(i.e. without some other more specific reason for doubt).
Another example of such general but unspecific truth is
the principle (derived from the law of the excluded
middle) that “there is always some explanation”. This
optimistic principle serves to encourage research, but
does not tell us what the solution of the problem is
specifically.
Paradoxical Propositions
A (single) paradoxical proposition has the form “if P,
then notP” or “if notP, then P”, where P is any form of
proposition. It is important to understand that such
propositions are logically quite legitimate within
discourse: a (single) paradox is not a contradiction. On
22
Indeed, it could be argued that, since ‘deductive’
axioms all have some empirical basis (as already explicated),
they are ultimately just a special case of ‘inductive’ axiom.
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the other hand, a double paradox, i.e. a claim that both “if
P, then notP” and “if notP, then P” are true in a given
case of P, is indeed a contradiction.
The law of non-contradiction states that the conjunction
“P and notP” is logically impossible; i.e. contradictory
propositions cannot both be true. Likewise, the law of the
excluded middle states that “notP and not-notP” is
logically unacceptable. The reason for these laws is that
such situations of antinomy put us in a cognitive
quandary – we are left with no way out of the logical
difficulty, no solution to the inherent problem.
On the other hand, single paradox poses no such threat to
rational thought. It leaves us with a logical way out –
namely, denial of the antecedent (as self-contradictory)
and affirmation of the consequent (as self-evident). The
proposition “if P, then notP” logically implies “notP”,
and the proposition “if notP, then P” logically implies
“P”. Thus, barring double paradox, a proposition that
implies its own negation is necessarily false, and a
proposition that is implied by its own negation is
necessarily true.
It follows, by the way, that the conjunction of these two
hypothetical propositions, i.e. double paradox, is a breach
of the law of non-contradiction, since it results in the
compound conclusion that “P and notP are both true”.
Double paradox also breaches the law of the excluded
middle, since it equally implies “P and notP are both
false”.
These various inferences may be proved and elucidated
in a variety of ways:

Since a hypothetical proposition like “if x, then y”
means “x and not y is impossible” – it follows that “if
P, then notP” means “P and not notP are impossible”
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(i.e. P is impossible), and “if notP, then P” means
“notP and not P are impossible” (i.e. notP is
impossible). Note this explanation well.
We know that the negation of P is the same as
notP, and the negation of notP equals P, thanks to
the laws of non-contradiction and of the excluded
middle. Also, by the law of identity, repeating the
name of an object does not double up the object:
it remains one and the same; therefore, the
conjunction “P and P” is equivalent to “P” and
the conjunction “notP and notP” is equivalent to
“notP”.
Notice that the meaning of “if P, then notP” is “(P
and not notP) is impossible”. Thus, although this
implies “notP is true”, it does not follow that “if notP
is true, P implies notP”. Similarly, mutatis mutandis,
for “if notP, then P”. We are here concerned with
strict implication (logical necessity), not with so
called material implication.
The reason why this strict position is necessary is that
in practice, truth and falsehood are contextual – most
of what we believe true today might tomorrow turn
out to be false, and vice-versa. On the other hand,
logical necessity or impossibility refer to a much
stronger relation, which in principle once established
should not vary with changes in knowledge context:
it applies to all conceivable contexts.
Since a hypothetical proposition like “if x, then y”
can be recast as “if x, then (x and y)” – it follows that
“if P, then notP” equals “if P, then (P and notP)”, and
“if notP, then P” equals “if notP, then (notP and P)”.
In this perspective, a self-contradictory proposition
implies a contradiction; since contradiction is
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logically impermissible, it follows that such a
proposition must be false and its contradictory must
be true. This can be expressed by way of apodosis, in
which the laws of thought provide the categorical
minor premise, making it possible for us to
exceptionally draw a categorical conclusion from a
hypothetical premise.
If P, then (P and notP)
but: not(P and notP)
therefore, not P
If notP, then (notP and P)
but: not(notP and P)
therefore, not notP
We can also treat these inferences by way of
dilemma, combining the given “if P, then notP” with
“if notP, then notP” (the latter from the law of
identity); or likewise, “if notP, then P” with “if P,
then P”. This gives us, constructively:
If P then notP – and if notP then notP
but: either P or notP
therefore, notP
If notP then P – and if P then P
but: either notP or P
therefore, P
Paradox sometimes has remote outcomes. For instance,
suppose Q implies P, and P implies notP (which as we
saw can be rewritten as P implies both P and notP).
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Combining these propositions in a syllogism we obtain
the conclusion “if Q, then P and notP”. The latter is also
a paradoxical proposition, whose conclusion is “notQ”,
even though the contradiction in the consequent does not
directly concern the antecedent. Similarly, non-exclusion
of the middle may appear in the form “if Q, then neither
P nor notP”. Such propositions are also encountered in
practice.
It is interesting that these forms, “Q implies (P and notP),
therefore Q is false” and “Q implies (not P and not notP),
therefore Q is false”, are the arguments implicit in our
application of the corresponding laws of thought. When
we come across an antinomy in knowledge, we
dialectically seek to rid ourselves of it by finding and
repairing some earlier error(s) of observation or
reasoning. Thus, paradoxical argument is not only a
derivative of the laws of thought, but more broadly the
very way in which we regularly apply them in practice.
That is, the dialectical process we use following
discovery of a contradiction or an excluded middle (or
for that matter a breach of the law of identity) means that
we believe that:
Every apparent occurrence of antinomy is in
reality an illusion.
It is an illusion due to paradox, i.e. it means that some of
the premise(s) that led to this apparently contradictory or
middle-excluding conclusion are in error and in need of
correction. The antinomy is never categorical, but
hypothetical; it is a sign of and dependent on some
wrong previous supposition or assumption. The apparent
antinomy serves knowledge by revealing some flaw in its
totality, and encouraging us to review our past thinking.
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Contradiction and paradox are closely related, but not the
same thing. Paradox (i.e. single not double paradox) is
not equivalent to antinomy. We may look upon them as
cognitive difficulties of different degrees. In this
perspective, whereas categorical antinomy would be a
dead-end, blocking any further thought––paradox is a
milder (more hypothetical) degree of contradiction, one
open to resolution.
We see from all the preceding (and from other
observations below) the crucial role that paradox plays in
logic. The logic of paradoxical propositions does not
merely concern some far out special cases like the liar
paradox. It is an essential tool in the enterprise of
knowledge, helping us to establish the fundaments of
thought and generally keeping our thinking free of
logical impurities.
Understanding of the paradoxical forms is not a
discovery of modern logic23, although relatively recent
(dating perhaps from 14th Cent. CE Scholastic logic).
23
For instance, Charles Pierce (USA, 1839-1914)
noticed that some propositions imply all others. I do not know
if he realized this is a property of self-contradictory or logically
impossible propositions; and that self-evident or necessary
propositions have the opposite property of being implied by all
others. I suspect he was thinking in terms of material rather
than strict implication.
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CHAPTER TWELVE
Drawn from Ruminations (2005),
Chapter 1 (sections 4-6).
Contradiction
Many people misunderstand what we logicians mean by
‘contradiction’. The contradictory of a term ‘A’ is its
negation, ‘not A’, which refers to anything and
everything in the universe other than A, i.e. wherever
precisely A is absent in the world. The relation of
contradiction between A and not-A is mutual, reversible,
perfectly symmetrical.
The presence of something (A) excludes its absence (i.e.
not A) in that very same thing, and vice versa, if all
coordinates of space and time are identical. However,
this does not exclude the logical possibility that the same
thing may be partly A and partly not A. Thus, the law of
thought ‘either A or not A’ can also be stated more
quantitatively as “either ‘all A’ or ‘all not A’ or ‘part A
and part not A”.
Some people appeal to this possibility of three
alternatives as an argument against the laws of thought!
But that is a misunderstanding – or worse, deliberate
sophistry.
If something, e.g. ‘B’, implies but is not implied by not
A, it (i.e. B) is as ‘incompatible’ with A as not-A is, but
it is not contradictory to A: it is merely contrary to A.
The contradictory not-A of A differs from A’s contraries
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in that the absence of not-A implies A, whereas in the
case of mere contraries like B (or B1 or B2… etc.) this
added logical relation of ‘exhaustiveness’ does not apply.
When contradictories are placed in a disjunction, ‘either
A or not-A’, the disjunction involved signifies both
mutual exclusion (‘or’, meaning ‘not together’) and
exhaustiveness (‘either’, meaning ‘and there is no other
alternative’). It intends: if ‘A’, then not ‘not-A’; and if
not ‘A’, then ‘not-A’.
On the other hand, any number of contraries can be
placed in a disjunction: ‘A or B or B1 or B2… etc.’, so
that the presence of any disjunct implies the absence of
all the others; but such disjunction is not exhaustive,
unless we specify that the list of contraries in it is
complete. If that list is indeed complete, then the
negation of all but one of the disjuncts implies the
affirmation of the remaining one. Thus, ‘not-A’ can be
equated to the exhaustive disjunction of all things in the
world ‘contrary to A’.
Something different from A, e.g. ‘C’, is not necessarily
contradictory or even contrary to A. The mere fact of
difference does not imply incompatibility. Different
things (like A and C) may be compatible, i.e. capable of
coexistence in the same thing, at the same time and
place. ‘Difference’ simply signifies that we are able to
distinguish between the things concerned: i.e. they are
not one and the same when they appear before our
consciousness. ‘Similar’ things may be the same in
appearance, but not one (e.g. two instances of the same
kind); or they may be one (i.e. parts of a single whole),
yet not the same.
Thus, for example, the logical relation between the colors
black and white depends on how precisely we focus on
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them. They are different, since distinguishable. Since
they may coexist on different parts of the same surface,
they are broadly compatible. However, as such or per se,
they are contrary; that is to say: if I perceive a surface or
part of surface as totally white, and you perceive the very
same place and time as totally black, our claims are
incompatible24.
This
irreconcilability
is
a
contradiction, however, because it is possible for a
surface to be neither black nor white.
not
Varieties of Contradiction
The expression ‘contradiction in terms’ refers to a
compound term composed of incompatible elements,
such as ‘A and not A’ or ‘A and B (where B is contrary
to A)’. Such a mixed-up term may be said to be
paradoxical, as well as internally inconsistent, since it
implies that contradiction is possible, so that the laws of
thought are denied by it, and then (by generalization, if
you like) ‘anything goes’ including denial of the ‘A and
not A’ conjunction.
For example, the term “illusory reality” is a
contradiction in terms. On the other hand, note,
terms like ‘an inhuman human’ or ‘an anti
Semitic Jew’ are not strictly speaking
contradictions in terms; they refer to natural
possibilities of conjunction, only the terminology
used makes them superficially seem contradictory
24
Our disagreement is not terminological, note. We
have in the past agreed as to what experiences ‘black’ and
‘white’ correspond to; here, we suddenly diverge.
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(i.e. there are people who behave inhumanly, or
Jews that hate their own people).
The proposition ‘A is not A’ (or ‘some thing that is A is
also not A’), being self-contradictory, implies ‘A is A’,
its
contradictory form. This statement should be
explicitly acknowledged, though obvious, because it
correlates two important concepts, viz. ‘internal
inconsistency’ and ‘the logic of paradoxes’.
The statement ‘A is not A’ is logically impossible,
because it both affirms and denies the same thing.
Therefore, the opposite statement is true. That statement,
i.e. ‘A is A’, is logically necessary, because even its
contradictory ‘A is not A’ implies it.
Whoever claims ‘A is not A’ is admitting ‘A is A’ – ipse
dixit, he himself said it! Whereas, whoever claims ‘A is
A’ is consistent with himself.
Self-contradiction consists of three items:
The proposition in question, call it P.
The admission that it is an assertoric statement,
i.e. one that affirms or denies something.
The admission that all assertoric statements
involve claims to consciousness, to knowledge, to
truth, etc.
Thus, given P (e.g. “reality is unknowable”), admit that P
implies “this is an assertion” – but all assertions imply
some knowledge of reality – therefore, P implies non-P.
There is a process from P to its negation, which Logic
demands we acknowledge. That demand cannot be
refused without committing the very same self
contradiction. This is not a circular or ad infinitum proof,
but an appeal to honesty, without which no dialogue is
possible.
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That all assertoric propositions assert is an aspect of the
Law of Identity. The Law of Non-contradiction may be
discerned in the argument: All assertions assert
something; P is an assertion; therefore, P asserts;
whence, if P denies asserting, P implies non-P. The Law
of the Excluded Middle is also implicit here, in the
awareness that we have no choice but to firmly disown P.
Double Standards
Contradictions appear in discourse in many guises. They
are not always overt, but may be hidden in the fact of
making a statement or in the standards of judgment used.
A claim may be paradoxical because it inherently
entails its own contradiction, although it does not on the
surface seem to be self-inconsistent. Such implication is
not always formal but requires awareness of the meaning
of the terms used. This form of indirect self-contradiction
has been called “the Stolen Concept fallacy”25.
For instance, the skeptical claim “I know nothing” may
be rejected as self-contradictory, because as soon as
someone makes it – someone who understands and
intends the meaning of the terms “I”, “know” and
“nothing” – that is by itself proof absolute that the person
concerned “knows” something, whence the original claim
(of total ignorance) is shown up to be unavoidably
contradictory and thus necessarily false.
Thus, in cases of this sort, the tacit implication involved
is that one of the terms used (knowing nothing) implicitly
includes the act in question (knowing that I know
nothing), as a case in point contradictory to the explicit
25
By Ayn Rand and (I think) Nathaniel Branden.
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claim. (Rephrasing the said statement as “I do not know
anything” does not change its underlying assumptions,
needless to say.)
There are countless examples of such inherent self
contradiction. Saying “I have nothing to say” is saying
something. Claiming “We have no memory” is self
contradictory, because each term in it presupposes a
word, concept and background experiences remembered
by the speaker – and the hearer too. An amusing common
example is “I do not speak a word of English”!
Another important form of covert self-inconsistency is
the use of a double standard. This consists in applying
less stringent standards of judgment to one’s own
discourse than to the discourse of one’s intellectual
opponents. A lot of philosophical, and particularly
political and religious, discourse resorts to such
inequitable methodology.
The contradiction involved in a double standard is
apparent the moment we step back and view its user’s
knowledge and methodology as a whole. In this wider
perspective, the user of a double standard is clearly
inconsistent with himself, even if his discourse viewed
piecemeal may superficially seem self-consistent.
Whole philosophies may be based on such
fallacious
reasoning.
For
instance,
Phenomenalism sets as a general standard a
limitation of knowledge to sensory data without
allowing extrapolations from them to assumed
external material objects – yet it does not criticize
its own adductions using the same rigid standard.
There are two ways this fallacy may be committed: one
may use relaxed standards on one’s own discourse, while
seemingly applying universal norms to one’s opponents’
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discourse; or one may appear to apply universal norms to
oneself, while concocting overly strict norms for them.
One may exempt oneself from the usual logical rules, or
one may make unusual logical demands on others.
In either case, the holder of a double standard is in
conflict with logic’s requirement of uniformity. An
assumption of reason is that all humans are
epistemologically on the same plane. Equity is an aspect
of ‘common sense’. Experience and logic have to be used
to
convince oneself and others, not sophistical
manipulation or authority.
Standards of judgment have to be fair and universal; all
discourse must be equally treated. If differences are
advocated, they have to be convincingly justified. The
principle of equality admittedly involves generalization;
but the onus of proof is on any proposed particularization
of it.
An example of a double standard is the appeal to cultural
relativism. One may seek to rationalize ideas or thought
processes that are contrary to ordinary reason, by
claiming them to belong to a different cultural
framework. Such tolerance seems on the surface friendly
and open-minded, but it is proposed without full
consideration of its negative human and epistemological
implications.
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CHAPTER THIRTEEN
Drawn from Ruminations (2005),
Chapter 1 (sections 7-10).
Special Status of the Laws
The three Laws of Thought must not be construed as
some prejudice of Aristotle’s, which some scientific
discovery – like the particle-wave duality or the relativity
of space-time measurements – could conceivably raise
doubt about or displace. These laws of thought are
intended as perfectly neutral; they make no direct,
specific ontological or epistemological claim, but
rationally sort out the very act and concept of such
claims – whence their name.
These laws express the ways we assimilate complex
experiences, and resolve difficulties in the course of
thought (concepts, propositions and arguments). Only by
such logic can we ‘make sense’ of the world around us
and in us. By making these truths explicit, Aristotle made
it possible for humans to henceforth consciously practice
the logic they were already unconsciously tending to.
These laws exclude, ab initio, the notion that something
could both have and lack some property, or neither have
nor lack it – at the same place and time and in the same
respects. The latter specification, which Aristotle clearly
and repeatedly stressed, is often ignored by those who
consider these laws expendable.
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That, say, a stone is blue on one side and red on the
other, is not a contradiction, since the different colors are
in different parts of it. That over time the colors may
change is not an antinomy either: the concept of time is
intended to ensure that. That you and I view the same
object from different angles, and see different aspects of
it, is no surprise. That my view of the world and yours
are not quite identical, is quite understandable in view of
the different context of experience and thought we each
have.
The laws of thought do not evade or deny the
appearance of contradictions or unsolved problems; they
just tell us that such appearances are illusions, not
realities. They are designed precisely to help us take
such apparent discrepancies into consideration and
resolve them in some way. We continue to need the same
laws of thought in the more complex cases uncovered by
modern physics.
The theory of relativity is precisely an attempt to
rationalize the surprising empirical constancy in the
velocity of light, whichever direction we measure it
from. The theory is not a statement that there are no
absolute truths, but a statement that such and such a way
of looking at the surprising events discovered makes
them rationally comprehensible. The theory affirms that
this
way is probably (i.e. inductively) the best
explanation, and effectively denies those who contradict
it (unless they come up with an inductively better
explanation, more in line with the empirical findings). It
does not deny the laws of thought, but is an application
of them.
Similarly, the discovery that the same things may behave
occasionally as particles and occasionally as waves does
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not constitute an argument against the laws of thought.
Whether we interpret this duality epistemologically or
ontologically, as due to different circumstances of
observation or different material circumstances, it is
affirmed to be a mysterious finding that must be faced.
This realist attitude is precisely what the laws of thought
demand. Any attempt to interpret the finding, one way or
the other, is again an attempt to make the finding
rationally comprehensible, so that we do not feel them
logically impossible.
Under no circumstances may scientists or philosophers
seriously claim the laws of thought to be abrogated. Such
a claim is self-contradictory – because then its opposite is
equally acceptable. It is therefore as if nothing has been
said. It is the denial of reason, the institution of madness.
The three laws of thought thus together constitute the
most incontrovertible and universal frame of reference of
rational thought.
Note also, the emphasis the laws of thought lay on
existence. A common error of deniers of these laws is to
regard ‘non-existence’ as just some other sort of
existence, a parallel world or a location beyond space
and time from which new existents come and to which
finished existents go! These people are misled by
linguistic habit into a reification of the word ‘non
existence’.
Whatever positively appears, exists to that extent.
Existence becomes open to doubt to the extent that we
add assumptions to appearance – i.e. we adductively
guess what might lie beyond them. At this stage, the
reality vs. illusion dichotomy arises. At this stage, too,
the rational act of negation comes into play – when we
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say: this is apparent, but (since it gives rise to some
antinomy) it is not real, it is illusory.
The ‘concept’ of non-existence thus has no direct
empirical basis of its own. It is based on a rational act
relative to experiences of existence. It is just a figment of
the imagination, a mental dumping place for ideas that
have failed the test of existential basis.
Motors of Rational Thought
It is important to realize that the laws of thought are the
motors of rational thought. They generate questions and
the pursuit of answers; they feed curiosity and fuel
research. If we are satisfied with the way things seem,
however contradictory or incomplete they seem, thought
is arrested. We lose perspective and become ignorant.
We lose intelligence and become stupid. We lose touch
with reality and become insane.
Consider the irrelevancy to science of a hypothetical
denial of the laws of thought. For instance, according to
Einstein’s theory of relativity, nothing can travel faster
than light, yet it has been found that particles may affect
each other instantaneously even though they are far apart.
If in the face of such an apparent contradiction we just
said: “oh, well, I guess the law of contradiction must be
wrong!” and left it at that – would we be consoled?
Clearly, not – this would not honestly solve the problem
for us, but merely sweep it under the carpet. Our minds
would not rest till some deeper, more convincing
explanation was found.
Accepting contradiction is just simplistic and evasive.
Similarly, with breaches of the law of the excluded
middle: if you ask me a question, and inquire is X the
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answer or not X? and I reply, it is neither, but some third
thing: will you be satisfied with such reply? Your
knowledge of the issue at hand is not made complete by
such reply; a gap remains, which can only be filled by
either X or nonX. The law of the excluded middle is just
a recognition of the inadequacy of such neither-nor
replies.
Cogito, Ergo Sum
Descartes’ “cogito, ergo sum”26 is composed of two self
evident propositions: “I think” (in the sense, I am
conscious) and “I am” (I exist). For the contradictory of
each of these propositions is self-contradictory, i.e.
involves a stolen concept and gives rise to a paradox.
Thus, “I am not conscious” could not be thought or said
(or for that matter heard or understood) without being
conscious. Similarly, “I am not” could not be expressed
(or observed) without existing. Thus, Descartes was quite
right in regarding these propositions as axioms; i.e. as
first principles, which do not depend on prior principles.
Note moreover that these two clauses are axiomatically
true independently of each other – So what about the
ergo, which suggests that the sum follows from the
cogito? Is the “therefore” perhaps meant to imply an
order of knowledge, rather than an inference? One could
formally deduce existence from consciousness, in the
sense that a conscious being is a fortiori an existent
26
See Hamlyn, p. 137. The comments made here are
not intended as an exhaustive analysis of the cogito
statement, needless to say.
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THE LAWS OF THOUGHT
being; but one would never in practice resort to such
inference.
In practice, in my opinion, we are conscious of other
things before we become conscious that we are conscious
of them – so it would not be correct to place the “I think”
before the “I am”. It could be argued that a baby may
first experience inner states, but I would reply that such
states are results of prior sensations. We may however
support Descartes’ order, by considering it a logical one,
in the sense that if the Subject did not have the power of
consciousness, he or she would not be aware of
existence. That is, it perhaps means: “I can think,
therefore I can know that I am”.
But I think the correct interpretation is the following:
when we are aware of something, any thing, this provides
an occasion to become aware of oneself, i.e. that there is
a Subject who is being conscious of that thing, whatever
it is. Thus, the first clause of the sentence is not strictly:
“I think”, but: “consciousness of things is taking place”
(or “thought is occurring”). Whence the second clause is
truly inductively inferred, i.e. we may well hypothesize
that “there is something being conscious of things”, i.e.
“thought has a Subject as well as an Object”, i.e. “there is
an I” (or “I exist”).
It is the self that is inferred from the appearance of
objects – reason argues: they must appear before
someone. This is what distinguishes appearance from
mere existence: it occurs through ‘cognition’ by
‘someone’. Thus, Descartes is justifying our habitual
assumption of a cognizing Subject from the fact of
cognition. It is not mere grammatical convention, he tells
us, but “think” implies “I”.
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Concerning Identity27.
Where does a material object begin or end, in view of the
constant flow of particles and energy in and out of it,
even (over a long enough time) in the case of apparent
solids? We have to use the apparent limits of things as
their space-time definition. Or more precisely, in
acknowledgment of the above difficulties, their illusory
limits. Thus, knowledge of matter is built on arbitrary,
knowingly inaccurate, delimitations of “things”.
We can similarly argue concerning mental objects (i.e.
images, sounds, etc.). At first thought, their limits seem
obvious; but upon reflection, they become doubtful –
imprecise and insecure. And this being the case, we
cannot convincingly argue that the limits of material
bodies are mental projections. If the limits of mental
lines are unsure, then the limits of whatever they are
intended to delimit are still unsure.
Ultimately, then, since we cannot even mentally delimit
mental or material things, all delimitations are merely
verbal artifices, i.e. claims we cannot substantiate. This
remark concerns not only ‘borderline’ cases, but all
material or mental objects.
These are very radical queries, productive of grave
skepticism. They are principles of vagueness and doubt
much more unsettling than the Uncertainty Principle,
since they more basically question the validity of any
geometry (and therefore, more broadly, of mathematics
and physics).
When some Greek or Indian philosophers expressed
skepticism at the possibility of human knowledge, this is
27
I
have already discussed this ontological issue in
chapter 10, section 2 (above).
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THE LAWS OF THOUGHT
perhaps what they were referring to. If one cannot
delimit things, how can one produce precise concepts and
propositions? And without precision, how can we judge
them true or false?
Whereas denial of knowledge as such is self
contradictory, denial of accurate knowledge is not so. It
is
possible to observe the general vagueness of
experience without denying the law of identity. If
cloudiness is the identity of things, or we are simply
incapable of sufficiently focusing our senses to get past
such cloudiness, we simply remain stuck at that level of
experience, like it or not.
The best counterargument I can muster is that
phenomenological knowledge is still knowledge of sorts,
and this can be used as a springboard to arrive at deeper
knowledge, by means of adduction. That is, we can still
formulate ontological hypotheses, capable of ongoing
confirmation or rejection with reference to reason and
experience, even if the epistemological status of the latter
is at the outset merely phenomenological.
This does not directly overcome the difficulty of
measurement, but it gives us some hope that we might
succeed indirectly. I leave the issue open, and move on.
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127
CHAPTER FOURTEEN
Drawn from Ruminations (2005),
Chapter 2 (section 17).
Appearance, Reality and Illusion
Phenomenology results from a realization that the
building blocks of knowledge are appearances. This
realization is obtained through a dialectic, comprising
thesis, antithesis and synthesis, as follows.
(a) At first, one naturally regards everything one comes
across in experience or thought as ‘real’ (this is the
‘naïve realist’ stance).
(b) Then, faced with evident contradictions and gaps in
one’s knowledge, one logically realizes that some
things that seemed real at first must or at least may
eventually be considered unreal – i.e. ‘illusory’ (this
constitutes a cognitive crisis).
(c) Finally, one realizes that, whether something is real
or illusory (and ultimately remains so or turns out to
be the opposite), at least it can immediately
(unconditionally and absolutely) be acknowledged as
‘apparent’ (this is the ‘phenomenological’ stance,
which resolves the crisis).
Knowledge of reality can then be inductively built up
from knowledge of appearances, thanks to the following
principle (d): One may credibly assume something that
appears to be real is indeed real, until and unless it is
proved illusory or at least put in doubt for some specific
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THE LAWS OF THOUGHT
reason. This may be characterized ‘subtle realism’, and
proceeds from the realization that the mere fact of
appearance is the source of all credibility.
Thus, phenomenology follows the natural flow of
knowledge, which is to initially accept individual
appearances as real, while remaining ready to reclassify
them as illusory if they give rise to specific logical
problems that can only be solved in that specific way.
The concept of ‘appearance’ is therefore not strictly
primary, but a transitional term for use in problematic
cases. Since it refers to the common ground between
‘reality’ and ‘illusion’, it is deductively primary. But
since the latter are in practice attained before it, it is
inductively secondary.
The concepts appearance, reality and illusion are to begin
with concerned with experiences; and only thereafter, by
analogy, they are applied to abstractions, i.e. conceptual
products of experience arrived at through rational
considerations, such as comparison and contrast (i.e.
affirmation or negation, and measurement).
The term ‘fact’ is usually intended to refer to purely
experiential data, i.e. the raw material of knowledge, in
which case the opposite term ‘fiction’ refers to other
items of knowledge, i.e. those tainted by interpretative
hypotheses. (But note that in practice of course we do not
always abide by such strict definitions, and may use the
terms more broadly or narrowly.)
The concepts of truth, falsehood and uncertainty
correspond in scope to those of reality, illusion and
appearance. The latter triad is applied to the contents of
propositions, while the former concerns the propositions
as such. For example, considering “dogs bark”, the fact
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of dogs barking is ‘a reality’, while the proposition that
dogs bark is ‘true’; similarly in other cases.
Once we understand all such concepts as signifying
different epistemological and ontological statuses, it
becomes clear why they need to be distinguished from
each other. They are all used as logical instruments – to
clarify and order discourse, and avoid confusions and
antinomies.
Note well that phenomenology is not a skeptical
philosophy that denies reality to all appearances and
claims them all to be illusions. Such a posture (which too
many philosophers have stupidly fallen into) is logically
self-contradictory, since it claims itself true while
rejecting all possibility of truth. The concept of illusion
has no meaning if that of reality is denied; some credulity
is needed for incredulity. Doubt is always based on some
apparent contradiction or gap in knowledge; i.e. it is
itself also an item within knowledge.
Existence and Non-existence
What is the relation between the concepts of existence
and non-existence (or being and non-being), and those
just elucidated of appearance, reality and illusion, one
might ask?
At first, the term existence may be compared to that of
reality, or more broadly to that of appearance (to admit
the fact that illusions occur, even if their status is not
equal to that of realities). However, upon reflection, an
important divergence occurs when factors like time and
place are taken into consideration.
We need to be able to verbally express changes in
experience over time, space and other circumstances. An
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THE LAWS OF THOUGHT
appearance, be it real or illusory, ‘exists’ at the time and
place of its appearance – but may ‘not exist’ at some
earlier or later time, or in another place. The ‘existence’
of appearances is transient, local, conditional and
relative.
What appears today may cease to appear tomorrow,
although it might (or might not) continue to appear less
manifestly, through someone’s memory of it or through
the appearance of exclusive effects of it. Something may
appear here within my field of vision, but be absent
elsewhere. You may see this in some circumstances, and
then notice its absence in others.
We thus need to distinguish different ways of
appearance. With reference to time: in actuality, or
through memory or anticipation; or with reference to
spatial positioning. Or again, with regard to modality: in
actuality, only through potentiality (i.e. in some
circumstances other than those currently operative), or
through necessity (i.e. in all circumstances).
Time and place also incite a distinction between
‘existence’ and ‘reality’ (or ‘truth’), in that when
something ceases to exist at a given time and place, the
reality of its having existed at the previous time and
place is not affected.
Furthermore, appearances are apparent to someone,
somewhere – they are contents of consciousness, objects
of cognition. The concept of existence is differentiated
also with reference to this, by conceiving that what may
be apparent to one Subject, may not be so to another.
Moreover, we wish to eventually acknowledge that
something may conceivably exist even without being
experienced by anyone (though of course, in defining
such a category, we must admit for consistency’s sake
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that we are thereby at least vaguely and indirectly
conceptually cognizing the object concerned).
We thus come to the realization that the concept of
appearance is a relatively subjective one, involving two
distinct factors: an object of some kind with specific
manifestations, on the one hand, and an awareness by
someone of that object at a given time and place. The
concept of existence is intended to separate out the
objective factor from the factor of consciousness implicit
in the concept of appearance.
‘Existence’ is thus needed to objectify ‘appearance’, and
allow us to conceive of the object apart from any
subject’s consciousness of it. We need to be able to
conceive of the objects appearing to us as sometimes
‘continuing on’ even when we cease to be aware of them.
Furthermore, we need to be able to consider objects that
we have not yet personally experienced, and even may
never experience. In this manner, we can project our
minds beyond mere appearance, and through conception
and adduction hope to grasp existence in a larger sense.
The concept of existence and its negation are thus
additional instruments of logic, facilitating rational
discourse, without which we would not be able to
mentally express many distinctions. Consequently,
saying ‘existence exists’ and ‘non-existence does not
exist’ is not mere tautology, but an acknowledgement
that the words we use have certain useful intentions.
These statements constitute one more way for us to
express the laws of thought. Existence cannot be denied
and non-existence cannot be affirmed.
We do not make the distinction between ‘existents’ and
non-existents’ by mentally lining up two kinds of things,
like apples and things other than apples. The
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THE LAWS OF THOUGHT
epistemological scenario applicable to most of our
concepts is not applicable to such basic ones, which are
of a more broadly pragmatic nature. Discernment rather
than distinction is involved.
Whereas the concept ‘existence’ has some ultimate
experiential content, ‘non-existence’ has none – because
factual denial is not based on the same mental process as
affirmation. We never experience non-existence – we
only (in certain cases) fail to experience existence. The
concept of existence is not built up by contrast to that of
non-existence, since (by definition) the former relates to
‘all things’ and the latter to ‘nothing’, and nothing is not
some kind of something. There is no time, place or
circumstance containing nothingness. The word ‘non
existence’ is just a dumping place for all the words and
sentences that have been identified as meaningless or
false.
Terms like ‘existence’ and ‘non-existence’ are not
ordinary subjects, copulae or predicates; they are too
broad and basic to be treated like any other terms. Those
who construct a theory of knowledge, or an ontology,
which concludes that ‘existence does not exist’ or that
‘non-existence exists’ have not understood the logic of
adduction. When there is a conflict between theory and
observed facts, it is the theory (or the ‘reasoning’ that led
up to it) that is put in doubt and is to be dismissed, not
the facts.
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CHAPTER FIFTEEN
Drawn from Ruminations (2005),
Chapters 3 (sect. 5), 5 (sect. 1) and 6 (sect. 3 & 4).
Poles of Duality
Concerning the principle, advocated by many, especially
oriental, philosophers, that poles of duality (e.g. good
bad, light-dark, etc.) arise together – certain comments
are worth making.
Oriental philosophers pursue a non-sorting mode of
consciousness, the awareness prior to the making of
distinctions; for this reason, dualities are obstacles in
their eyes. Such Monist consciousness is, however, rarely
if ever attained.
I would reply, ontologically: since we can conceive of
Monism, then we can also conceive of a universe with
only good or only light, etc.; i.e. a world with one
polarity of such dualities is logically possible. Of course,
this would only be strict Monism, if this quality was
quite alone and no other quality was found in the world
(i.e. not just not the other polarity of that quality). Of
course, also, we – those now conceiving of that world –
would not be distinguishable in it, since then there would
be two things in it – viz. object and subject.
But note such solitude of existence could not apply to
just any quality. Negative concepts like ‘imperfect’
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cannot exist alone28; i.e. an only imperfect world is
inconceivable, as some part of it must remain perfect to
exist at all. However, this remark may rather concern the
next observation.
From an epistemological and psychological (rather than
ontological) viewpoint, there is some truth in the said
oriental belief. That is, the idea of good or light is not
possible without the idea of bad or dark. Imaging one
pole necessitates our also bringing to mind the other pole
for the purpose of contrast. This is due to the mechanics
of concept formation: it functions by making distinctions
as well as by identification of the things distinguished.
Because it is only by way of contrast to dissimilars that
similars can be classified, every word, every concept, has
to
make some room for its opposite; we cannot
comprehend a term without having to think of its
opposite. Thus, one might suggest: although logically, X
totally excludes nonX – psychologically, “X” may be
said to be say 99% “X” and 1% “nonX”.
Another point worth making, here: contradictory terms,
such as X and not-X, have equal logical status, i.e. their
formal
treatment
is
identical;
however,
phenomenologically, affirmation and denial are very
different: the first signifies an actual experience
(phenomenal, through the senses or mentally, or non
phenomenal, intuitively) – whereas the latter signifies a
rational act, a conceptual report that some anticipated
experience has not occurred. Strictly, perhaps,
experiences should be verbalized affirmatively, while
negations should be cast in negative terms. In practice,
this is rarely followed.
28
As Alan Watts pointed out, somewhere.
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A positive word like ‘silence’ or ‘stillness’ may indicate
a negative event (no sound, no move). However, even in
such cases, there may be an underlying positive event; in
our examples, although silence refers to the non
perception of any sound phenomenon – we may by this
term mean rather to refer to our will to block sounds,
which volition is something positive, though without
phenomenal character, known intuitively.
Similarly, I suspect, some negatively cast words may in
fact refer to positive experiences, although there may be
a good reason why the negative form is preferred. For
example, ‘unabashed’ simply means without apology,
but viewed more closely refers to certain behavior
patterns; so, though negative in form, it is rather positive
in intent. However, the negative form is not accidental,
but serves to indicate the missing ingredient in the
behavior
patterns,
questionable.
which makes them socially
On the Liar paradox29
Once we grasp that the meaning of words is their
intention, singly and collectively – the solution of the liar
paradox becomes very obvious. Self-reference is
meaningless, because – an intention cannot intend itself,
for it does not yet exist; an intention can only intend
something that already exists, e.g. another intention
directed at some third thing.
In view of this, the proposition “this proposition is false”
is meaningless, and so is the proposition “this proposition
is true”. Both may freely be declared equally true and
29
Further to chapter 5, section 2 (above).
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THE LAWS OF THOUGHT
false, or neither true nor false – it makes no difference in
their case, because the words “this proposition” refer to
nothing at all30.
Although the words used in these sentences are
separately meaningful, and the grammatical structure of
the sentences is legitimate – the words’ collective lack of
content implies their collective logical value to be nil.
Self-reference is syntactically cogent, but semantically
incoherent. It is like circular argument, up in the air,
leading nowhere specific.
Regarding the exclusive proposition “Only this
proposition is true”, it implies both: “This proposition is
true” and “All other propositions are false” – i.e. it is
equivalent to the exceptive proposition “All propositions
but this one are false”. The latter is often claimed by
some philosopher; e.g. by those who say “all is illusion
(except this fact)”.
My point here is that such statements do not only involve
the fallacy of self-reference (i.e. “this proposition”). Such
statements additionally involve a reference to “all others”
which is open to criticism, because:

To claim knowledge of “all other propositions” is a
claim to omniscience, a pretense that one knows
everything there is to know, or ever will be. And
generally, such statements are made without giving a
credible justification, though in contradiction to all
prior findings of experience and reason.
Surely, some other propositions are in fact regarded
and admitted as true by such philosophers. They are
generally rather talkative, even verbose – they do not
30
See Future Logic, chapter 32.2.
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137
consistently only say that one statement and refuse to
say anything else.
And of course, formally, if “this” is meaningless (as
previously shown), then “all others”, which means
“any other than this” is also meaningless!
The liar paradox, by the way, is attributed to the ancient
Greeks, either Eubulides of Miletus (4th Cent. BCE) or
the earlier Epimenides of Crete (6th Cent. BCE). I do not
know if its resolution was evident to these early
logicians, but a (European?) 14th Cent. CE anonymous
text reportedly explained that the Liar’s statement is
neither true nor false but simply meaningless. Thus, this
explanation is historically much earlier than modern
logic (Russell et alia, though these late logicians certainly
clarified the matter).31

Non-Aristotelian “Logic”
As already stated, many “modern logicians” – since the
late 19th Century – have yearned to do for (or to) Logic,
what Copernicus did in Astronomy, or later what
Einstein did in Physics. Each one of them was, it seems,
fired by the grandiose desire to be the equivalent great
modern revolutionary in the field of logic.
They thus inaugurated a persistent assault on Reason, a
veritable carnival of Unreason, which has lasted for over
a hundred years, with disastrous consequences for many
a poor mind and for social peace and wellbeing.
Their conceptual model was non-Euclidean geometry.
Just as modern mathematicians came to consider certain
Euclidean axioms to be debatable, if not arbitrary, so
31
See Future Logic, chapter 63, sections 3 and 6.
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THE LAWS OF THOUGHT
these modern logicians sought to put in doubt or discard
the Aristotelian “laws of thought”, and found some new
system – a “non-Aristotelian logic”.
But this is an impossible exercise, because32 the laws of
thought are more fundamental to reason than Euclid’s
axioms (in particular, that regarding parallels). The
geometrical model of axioms and theorems is only
superficially applicable to logic, because it is itself an
aspect or teaching of (Aristotelian) logic.
When mathematicians decided to review the traditional
axioms of geometry, they were using reasoning by means
of the laws of thought. They argued: “we see no self
contradiction, or doctrinal inconsistency, or even
(eventually) contradiction to experience in proposing
some alternative axioms and systems; therefore, Euclid’s
assumptions are not exclusive and irreplaceable.”
The same cannot be argued in the case of logic itself,
without self-contradiction. We cannot, say, point to the
particle-wave duality and say “it seems that
contradictions do exist in the world, therefore we shall
review the logical axiom of non-contradiction” – we
cannot do so, for the reason that such review is motivated
and rendered credible precisely by the law of non
contradiction, in the way of an attempt to restore an
apparently lost consistency.
The very method used of reviewing one’s premises in the
face of contradiction and abandoning or at least
modifying one or more of them to recover consistency –
this very methodology is a teaching of Aristotelian logic!
We cannot say: “I understand that if I advocate
contradiction, I open myself to being contradicted; but
32
As I have explained repeatedly in Future Logic.
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139
that does not bother me, because it is a consistency of
sorts – I accept self-contradiction.”
In the very act of making such a superficially reasonable
proposal, we are reasserting the universality of the laws
of thought, their being at the very root of reason, inherent
in the very act of reasoning. The only way we could
conceivably abandon these laws would be to give up all
thought, all attempt at rational knowledge. Logic cannot
be used against itself: it is the very paradigm and paragon
of consistency.
We can suggest: “A can be non-A”, or some such “new
axiom” for logic, but the resulting discourse will still be
nonsense – however nicely wrapped up and ordered,
however well “systematized” stealing the methods of
Aristotelian logic. Such proposals are an imposture.
Those who propose such ideas are swindlers, profiting
from the gullibility and intimidation of many people. It is
like in the story of the emperor’s new clothes, in which
con men sold the emperor invisible clothes, which no one
dared to deny were clothes – till a child pointed out he
was naked.
There simply is no such thing as “non-Aristotelian logic”
(i.e. a logical system that denies one, two or all three
laws of thought). To come forward with such a system is
merely to pronounce words. These words have no
collective content, no meaning; there is nothing behind
them other than the imagination that there might be
something behind them because the phrase is composed
of individually meaningful words.
No “Copernican revolution” is conceivable in the field of
logic: it would not merely be anti-Aristotelian but anti
rational. Logicians must abandon such vain ambitions,
and more modestly continue to expand the scope of
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THE LAWS OF THOUGHT
logical analysis and the depth of understanding of logic.
The role of logicians is to do logic, not undo it. Reason is
a precious value for mankind, and logicians ought to be
its guardian.
Would you entrust your life to, say, an airplane built by
engineers practicing “non-Aristotelian logic”, people
who feel cozy in the midst of contradictions and in
between truth and falsehood? Similarly, in all fields of
human endeavor and interaction: logic is a guarantee of
sanity and safety.
Postmodern “Logic”
As if such irrational currents were not enough, there is (I
gather) a new generation of “postmodern” logicians and
philosophers who eschew even the pretense of
accountability, considering that any discourse that seems
to be about “logic” is acceptable. These are of course part
of a wider trend, not limited to our field.
Being relativists, these people are not directly attacking
anything or anyone. They are not mere anti-rationalists:
they are so indifferent to the niceties of reason that they
feel no need to justify themselves. They are of course the
natural offspring of the moderns, taking their teachings to
their ‘logical’ conclusion. They are more consistently
illogical than their predecessors, no longer owing a
semblance of allegiance to reason, not needing even to
pay lip service to it. Absurdity does not bother them, so
they need no logical window dressing for their doctrines.
Indeed, these people take pride in their fashionable
madness. They strive to be as confusing and
incomprehensible as possible, considering that what
others cannot possibly understand must be very deep
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indeed. They have only a very vague notion of what logic
is
about, but seek to impress other people with
meaningless symbolic constructs and use of fancy
pseudo-scientific terminology. They prattle away,
eruditely formulating fake theories immune to any
empirical or rational review. They function as (con)
artists rather than scientists.
Yes, such people do exist; some even have teaching
positions in prestigious universities. Because most people – including some in high academic positions, including
some who are hired to teach logic – know or understand
little about logic, they are easily intimidated by such
intellectual posturing and imposture. They fear to reveal
their own poverty in the course of questioning or debate.
Besides, it is no use denouncing the swindle; no one
apparently cares, because few people realize the
importance of logic (apart from some simple formulas
needed in computer programming). Reason is out of
fashion, has been for generations. Logic is too abstract;
you cannot show artistic footage of it on TV. It cannot be
very entertaining: it requires an effort of thought.
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THE LAWS OF THOUGHT
CHAPTER SIXTEEN
Drawn from Ruminations (2005),
Chapter 9 (sections 1-4).
ON NEGATION
Negation in Adduction
Concepts and theories are hypothetical constructs. They
cannot (for the most part) be proven (definitely, once and
for all), but only repeatedly confirmed by experience.
This is the positive side of adduction, presenting
evidence in support of rational constructs. This positive
aspect is of course indispensable, for without some
concrete evidence an abstraction is no more than a
figment of the imagination, a wild speculation. The more
evidence we adduce for it, the more reliable our concept
or theory.
But, as Francis Bacon realized, the account of adduction
thus far proposed does not do it justice. Just as important
as the positive side of providing evidence, is the negative
aspect of it, the rejection of hypotheses that make
predictions conflicting with experience. As he pointed
out, even if a hypothesis has numerous confirmations, it
suffices for it to have one such wrong prediction for it to
be rejected.
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Stepping back, this means that the process of adduction
is
concerned with selection of the most probable
hypothesis among two or more (already or yet to be
conceived) explanations of fact. Each of them may have
numerous ‘positive instances’ (i.e. empirical evidence
that supports it); and so long as they are all still
competitive, we may prefer those with the most such
instances. But, the way we decisively advance in our
conceptual/theoretical knowledge is by the successive
elimination of propositions that turn out to have
‘negative instances’ (i.e. empirical evidence against
them).
Now all the above is well known and need not be
elucidated further. This theory of inductive logic has
proven
extremely successful in modern times,
constituting the foundation of the scientific method.
But upon reflection, the matter is not as simple and
straightforward as it seems at first!
Consider, for example, the issue of whether or not there
is water on Mars. It would seem that the proposition
“There is water on Mars” is far easier to prove
inductively than the contradictory proposition “There is
no water on Mars”. Both propositions are hypotheses.
The positive thesis would be somewhat confirmed, if it
was discovered using certain instruments from a distance
that there are serious indices that water is present; the
thesis would be more solidly confirmed, if a sample of
Mars was brought back to Earth and found upon analysis
to contain water. In either case, the presence of water on
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Mars would remain to some (however tiny) degree
unsure, because some objection to our instrumental
assumptions might later be raised or the sample brought
back may later be found to have been contaminated on
the way over. Nevertheless, something pretty close to
certainty is conceivable in this matter.
The negative thesis, by contrast, is much more difficult to
prove by experience. We can readily assume it to the
extent that the positive thesis has not so far been greatly
confirmed. That is, so long as we have not found
evidence for the positive thesis (i.e. water on Mars), we
should rather opt for the negative thesis. But the latter is
only reliable to the degree that we tried and failed to
confirm the former. If we earnestly searched for water
every which way we could think of, and did not find any,
we can with proportionate confidence assume there is no
water.
Thus, in our example, the negative thesis is actually more
difficult to establish than the positive one. It depends on a
generalization, a movement of thought from “Wherever
and however we looked for water on Mars, none was
found” to “There is no water on Mars”. However, note
well, it remains conceivable that a drop of water be found
one day somewhere else on Mars, centuries after we
concluded there was none.
Granting this analysis, it is clear that Bacon’s razor that
“What is important is the negative instance” is a bit
simplistic. It assumes that a negative is as accessible as
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(if not, indeed, more accessible than) a positive, which is
not always the case.
In practice, a negative may be inductively more remote
than a positive. Granting this conclusion, the question
arises – is the negative instance ever more empirically
accessible than (or even as accessible as) the positive
one? That is, when does Bacon’s formulation of
induction actually come into play?
If we look at major historical examples of rejection of
theories, our doubt may subsist. For example, Newtonian
mechanics was in place for centuries, till it was put in
doubt by the discovery of the constancy of the velocity of
light (which gave rise to Relativity theory) and later
again by the discovery of various subatomic phenomena
(which gave rise to Quantum mechanics). In this
example, the ‘negative instances’ were essentially
‘positive instances’ – the only thing ‘negative’ about
them was just their negation of the Newtonian
worldview!
Such reflections have led me to suspect that the
‘negation’ referred to by Bacon is only meant relatively
to some selected abstraction. His razor ought not be
taken as an advocacy of absolute negation. If we look at
the matter more clearly, we realize that the data used to
thus negate an idea is essentially positive. A deeper
consideration of the nature of negation is therefore
patently called for.
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Positive and Negative Phenomena
People have always considered that there is a difference
between a positive and a negative term. Indeed, that is
why logicians have named them differently. But
logicians have also found it difficult to express that
difference substantially. Yet, there are significant
phenomenological differences between positive and
negative phenomena.
a.
The concrete material and mental world is
evidently
composed only of positive particular
phenomena, some of which we perceive (whether
through the bodily senses or in our minds). These exist at
least as appearances, though some turn out to seem real
and
others
illusory.
This
is
an
obvious
phenomenological, epistemological and ontological truth.
To say of phenomena that they are ‘particular’ is to
express awareness that they are always limited in space
and time. They have presence, but they are finite and
transient, i.e. manifestly characterized by diversity and
change.
We do not ordinarily experience anything concrete
that stretches uniformly into infinity and eternity
(though such totality of existence might well exist,
and indeed mystics claim to attain consciousness of it
in deep meditation, characterizing it as “the eternal
present”). We do commonly consider some things as
so widespread. ‘Existence’ is regarded as the
substratum of all existents; ‘the universe’ refers to the
sum total of all existents; and we think of ‘space
time’ as defining the extension of all existents. But
only ‘existence’ may be classed as an experience (a
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quality found in all existents); ‘the universe’ and
‘space-time’ must be admitted as abstractions.
However, the limits of particulars are perceivable
without need of negation of what lies beyond them,
simply
due to the variable concentration of
consciousness, i.e. the direction of focus of attention.
That is, though ‘pointing’ to some positive phenomenon
(e.g. so as to name it) requires some negation (we mean
“this, but not that”), one can notice the limits of that
phenomenon independently of negation.
b.
Negative phenomena (and likewise abstracts,
whether positive or negative), on the other hand, do
depend for their existence on a Subject/Agent – a
cognizing ‘person’ (or synonymously: a self or soul or
spirit) with consciousness and volition looking out for
some remembered or imagined positive phenomenon and
failing to perceive it (or in the case of abstracts,
comparing and contrasting particulars).
Thus, negative particular phenomena (and more
generally, abstracts) have a special, more ‘relative’ kind
of existence. They are not as independent of the Subject
as positive particular phenomena. That does not mean
they
are,
in
a Kantian sense, ‘a priori’ or
‘transcendental’, or purely ‘subjective’ – but it does
mean that they are ontological potentials that are only
realized in the context of (rational) cognition.
Another kind of experience is required for such
realization – the self-experience of the Subject, his
intuitive knowledge of his cognitions and volitions. This
kind of experience, being immediate, may be positive or
negative without logical difficulty. The Subject reasons
inductively as follows:
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I am searching for X;
I do not find X;
Therefore, X “is not” there.
The negative conclusion may be ‘true’ or ‘false’, just like
a positive perception or conclusion. It is true to the
degree that the premises are true – i.e. that the alleged
search for X was diligent (intelligent, imaginative, well
organized, attentive and thorough), and that the alleged
failure to find X is not dishonest (a lie designed to fool
oneself or others).
Whence it is fair to assert that, unlike some positive
terms, negative terms are never based only on perception;
they necessarily involve a thought-process – the previous
mental projection or at least intention of the positive term
they negate.
This epistemological truth does reflect an ontological
truth – the truth that the ‘absences’ of phenomena lack
phenomenal aspects. A ‘no’ is not a sort of ‘yes’.
Note well the logical difference between ‘not perceiving
X’ and ‘perceiving not X’. We do not have direct
experience of the latter, but can only indirectly claim it
by way of inductive inference (or extrapolation) from the
former. In the case of a positive, such process of
reasoning is not needed – one often can and does
‘perceive X’ directly.
Suppose we draw a square of opposition for the
propositions (labeling them by analogy to standard
positions) – “I perceive X” (A), “I do not perceive not X”
(I), “I perceive not X” (E), “I do not perceive X” (O).
Here, the A form is knowable by experience, whereas the
I form is knowable perhaps only by deductive
implication from it. On the negative side, however, the E
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form is not knowable by experience, but only by
inductive generalization from the O form (which is based
on experience).
Positive Experience Precedes Negation
Negation is a pillar of both deductive and inductive logic,
and requires careful analysis. We have to realize that
negative terms are fundamentally distinct from positive
ones, if we are to begin fathoming the nature of logic.
The following observation seems to me crucial for such
an analysis:
We can experience something positive without
having first experienced (or thought about) its
negation, but we cannot experience something
negative without first thinking about (and therefore
previously having somewhat experienced) the
corresponding positive.
a.
Cognition at its simplest is perception. Our
perceptions are always of positive particulars. The
contents of our most basic cognitions are phenomenal
sights, sounds, smells, tastes, and touch and other bodily
sensations that seemingly arise through our sense organs
interactions with matter – or mental equivalents of these
phenomena that seemingly arise through memory of
sensory experiences, or in imaginary re-combinations of
such supposed memories.
A positive particular can be experienced directly and
passively. We can just sit back, as it were, and
receptively observe whatever happens to come in our
field of vision or hearing, etc. This is what we do in
meditation. We do not have to actively think of
(remember or visualize or conceptualize) something else
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in order to have such a positive experience. Of course,
such observation may well in practice be complicated by
thoughts (preverbal or verbal) – but it is possible in some
cases to have a pure experience. This must logically be
admitted, if concepts are to be based on percepts.
b.
In the case of negative particulars, the situation is
radically different. A negative particular has no specific
phenomenal content, but is entirely defined by the
‘absence’ of the phenomenal contents that constitute
some positive particular. If I look into my material or
mental surroundings, I will always see present
phenomena. The absence of some phenomenon is only
noticeable if we first think of that positive phenomenon,
and wonder whether it is present.
It is accurate to say that our finding it absent reflects an
empirical truth or fact – but it is a fact that we simply
would not notice the negative without having first
thought of the positive. Negative knowledge is thus
necessarily (by logical necessity) more indirect and
active. It remains (at its best) perfectly grounded in
experience – but such negative experience requires a
rational process (whether verbal or otherwise).
To experience a negative, I must first imagine (remember
or invent) a certain positive experience; then I must look
out and see (or hear or whatever) whether or not this
image matches my current experience; and only then (if
it indeed happens not to) can I conclude to have
“experienced” a negative.
Thinking about X may be considered as positioning
oneself into a vantage point from which one can (in a
manner of speaking) experience not-X. If one does not
first place one’s attention on X, one cannot possibly
experience the negation of X. One may well experience
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all sorts of weird and wonderful things, but not
specifically not-X.
From this reflection, we may say that whereas
affirmatives can be experienced, negatives are inherently
rational acts (involving imagination, experience and
intention). A negative necessarily involves thought: the
thought of the corresponding positive (the imaginative
element), the testing of its presence or absence (the
experiential element) and the rational conclusion of
“negation” (the intentional element).
c.
The negation process may involve words, though
it does not have to.
Suppose I have some momentary experience of sights,
sounds, etc. and label this positive particular “X”. The
content of consciousness on which I base the term X is a
specific set of positive phenomenal experiences, i.e.
physical and/or mental percepts. Whenever I can speak
of this X, I mentally intend an object of a certain color
and shape that moves around in certain ways, emitting
certain sounds, etc.
Quite different is the negation of such a simple term, “not
X”. The latter is not definable by any specific percepts –
it
refers to no perceptible qualities. It cannot be
identified with the positive phenomena that happen to be
present in the absence of those constituting X. Thus,
strictly speaking, not-X is only definable by ‘negation’ of
X.
Note well, it would not be accurate to say (except ex post
facto) that not-X refers to all experiences other than X
(such as Y, Z, A, B, etc.), because when I look for X here
and now and fail to find it, I am only referring to present
experience within my current range and not to all
possible such experiences. We would not label a situation
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devoid of X as “not X” without thinking of X; instead, we
would label that situation in a positive manner (as “Y”,
or “Z”, or whatever).
Thus, we can name (or wordlessly think of) something
concrete “X”, after experiencing phenomena that
constitute it; but in the case of “not-X”, we necessarily
conjure the name (or a wordless thought) of it before we
experience it.
“Not-X” is thus already a concept rather than a percept,
even in cases where “X” refers to a mere percept (and all
the more so when “X” itself involves some abstraction –
as it usually does). The concept “not X” is hypothetically
constructed first and then confirmed by the attempted
and failed re-experience of X.
In short, negation – even at the most perceptual level –
involves an adductive process. It is never a mere
experience. A negative term never intends the simple
perception of some negative thing, but consists of a
hypothesis with some perceptual confirmation. Negation
is always conceptual as well as perceptual in status.
A theory cannot be refuted before it is formulated –
similarly, X cannot be found absent unless we first think
of X.
Negation is an Intention
Now, there is no specific phenomenal experience behind
the word “not”. Negation has no special color and shape,
or sound or smell or taste or feel, whether real or
illusory! What then is it? I suggest the following:
of
Negation as such refers to a ‘mental act’ – or more
precisely put, it is an act of volition (or more precisely
still,
velleity) by a Subject of consciousness.
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Specifically, negation is an intention. Note that our will
to negate is itself a positive act, even though our
intention by it is to negate something else.
Negation does express an experience – the ‘failure’ to
find something one has searched for. Some cognitive
result is willfully pursued (perception of some positive
phenomenon), but remains wanting (this experience is
qualitatively a suffering of sorts, but still a positive
intention, note) – whence we mentally (or more
precisely, by intention) mark the thing as ‘absent’, i.e. we
construct an idea of ‘negation’ of the thing sought.
Thus, negation is not a phenomenon (a physical or
mental percept), but something intuited (an event of will
within the cognizing Subject). ‘Intuition’ here, note well,
means the self-knowledge of the Subject of
consciousness and Agent of volition. This is experience
of a non-phenomenal sort. Such self-experience is
immediate: we have no distance to bridge in space or
time.
When a Subject denies the presence of a material or
mental phenomenon, having sought for it in experience
and not found it – the ‘denial’ consists of a special act of
intention. This intention is what we call ‘negation’ or
‘rejection of a hypothesis’. It occurs in the Subject,
though it is about the Object.
This intention is not however an arbitrary act. If it were,
it would be purely subjective. This act (at its best)
remains sufficiently dependent on perception to be
judged ‘objective’. The Subject must still look and see
whether X is present; if that positive experience does not
follow his empirical test, he concludes the absence of X.
Indeed, an initial negation may on closer scrutiny be
found erroneous, i.e. we sometimes think something is
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‘not there’ and then after further research find it on the
contrary ‘there’. Thus, this theory of negation should not
be construed as a claim that our negating something
makes it so. Negation is regulated by the principles of
adduction – it is based on appearance that is credible so
long as confirmed, but may later be belied.
We can ex post facto speak of an objective absence, but
we cannot fully define ‘absence’ other than as ‘non
presence’, and the ‘non-’ herein is not a phenomenon but
an intention. The ‘absence’ is indeed experienced, but it
is imperceptible without the Subject posing the prior
question ‘is X present?’
Absence, then, is not produced by the Subject, but is
made perceptible by his vain search for presence. For, to
repeat, not-X is not experienced as a specific content of
consciousness – but as a continuing failure to experience
the particular positive phenomena that define X for us.
Although we are directly only aware of apparent
existents, we can inductively infer non-apparent existents
from the experience that appearances come and go and
may change. On this basis, we consider the categories
‘existence’ and ‘appearance’ as unequal, and the former
as broader than the latter. Similarly, we inductively infer
‘objective absence’ from ‘having sought but not found’,
even though we have no direct access to former but only
indirect access by extrapolation from the latter. Such
inference is valid, with a degree of probability
proportional to our exercise of due diligence.
For these reasons, I consider the act of negation as an
important key to understanding the nature and status of
logic. Negation is so fundamental to reason, so crucial an
epistemic fact, that it cannot be reduced to something
else.
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We can describe it roughly as an intention to ‘cross-off’
(under the influence of some reason or other) the
proposed item from our mental list of existents. But this
is bound to seem like a circular definition, or a repetition
of same using synonyms. It is evident that we cannot talk
about negation without engaging in it. Thus, we had
better admit the act of negation as a primary concept for
logical science.
Note in passing: the present theory of negation
provides biology with an interesting distinction
regarding rational animals.
Sentient beings without this faculty of negation can
only respond to the present, whereas once this faculty
appears in an organism (as it did in the human
species) it can mentally go beyond the here and now.
A merely sensory animal just reacts to current events,
whereas a man can fear dangers and prepare for them.
Once the faculty of negation appears, the mind can
start
abstracting,
conceiving alternatives and
hypothesizing. Memory and imagination are required
to project a proposed positive idea, but the intent to
negate is also required to reject inadequate
projections. Without such critical ability, our
fantasies would quickly lead us into destructive
situations.
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CHAPTER SEVENTEEN
Drawn from Ruminations (2005),
Chapter 9 (sections 5-8).
ON NEGATION
Formal Consequences
Returning to logic – our insight [earlier] into the nature
of
negation can be construed to have formal
consequences. The negative term is now seen to be a
radically different kind of term, even though in common
discourse it is made to behave like any other term.
We cannot point to something as ‘negative’ except
insofar as it is the negation of something positive. This
remark is essentially logical, not experiential. The term
‘not’ has no substance per se – it is a purely relative term.
The positive must be experienced or thought of before
the negative can at all be conceived, let alone be
specifically sought for empirically. This is as true for
intuitive as for material or mental objects; and as true for
abstracts as for concretes.
One inference to draw from this realization of the
distinction of negation is: “non-existence” is not some
kind of “existence”. Non-existent things cannot be
classed under existence; they are not existent things. The
term
“non-existence”
involves
no
content
of
consciousness whatsoever – it occurs in discourse only as
the verbal repository of any and all denials of
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“existence”. Existentialist philosophers have written
volumes allegedly about “non-being”, but as Parmenides
reportedly stated:
“You cannot know not-being, nor even say it.”
This could be formally expressed and solidified by
saying that obversion (at least that of a negative – i.e.
inferring “This is nonX” from “This is not X”) is
essentially an artificial process. If so, the negative
predicate (nonX) is not always inferable from the
negative copula (is not). In other words, the form “There
is no X” does not imply “There is non-X”; or conversely,
“X does not exist” does not imply “nonX exists”.
We can grant heuristically that such eductive processes
work in most cases (i.e. lead to no illogical result), but
they may be declared invalid in certain extreme
situations (as with the term “non-existence”)! In such
cases, “nonX” is ‘just a word’; it has no conscionable
meaning – we have no specific thing in mind as we utter
it.
Logicians who have not yet grasped the important
difference of negation are hard put to explain such formal
distinctions. I know, because it is perhaps only in the last
three years or so that this insight about negation has
begun to dawn on me; and even now, I am still in the
process of digesting it.
Note that a philosophical critic of this view of
negation cannot consider himself an objective
onlooker, who can hypothesize ‘a situation where
absence exists but has not or not yet been identified’.
For that critic is himself a Subject like any other, who
must explain the whence and wherefore of his
knowledge like anyone else – including the negatives
he appeals to. No special privileges are granted.
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That is, if you wish to deny all the above, ask
yourself and tell me how you consider you go about
denying without having something to deny! Claiming
to have knowledge of a negative without first
thinking of the corresponding positive is comparable
to laying personal claim to an absolute framework in
space-time – it is an impossible exercise for us
ordinary folk.
It should also be emphasized that the above narrative
describes only the simplest kind of negation: negation of
a perceptual item. But most of the time, in practice, we
deal with far more complex situations. Even the mere act
of ‘pointing’ at some concrete thing involves not only a
positive act (“follow my finger to this”), but also the act
of negation (“I do not however mean my finger to point
at that”).
Again, a lot of our conceptual arsenal is based on
imaginary recombinations of empirical data. E.g. I have
seen “pink” things and I have seen “elephants”, and I
wonder whether “pink elephants” perhaps exist. Such
hypothetical entities are then tested empirically, and
might be rejected (or confirmed). However, note,
abstraction does not depend only on negation, but on
quantitative judgments (comparing, and experiencing
what is more or less than the other).
Abstraction starts with experiences. These are variously
grouped through comparisons and contrasts. Negation
here plays a crucial role, since to group two things
together, we must find them not only similar to each
other but also different from other things. This work
involves much trial and error.
But at this level, not only denial but also affirmation is a
rational act. For, ‘similarity’ means seemingly having
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some quality in common in some measure, although
there are bound to be other qualities not in common or
differences of measure of the common quality. The
essence of affirmation here is thus ‘measurement’.
But Nature doesn’t measure anything. Every item in it
just is, whatever it happens to be (at any given time and
place). It is only a Subject with consciousness that
measures: this against that, or this and that versus some
norm.
This weighing work of the cognizing Subject is not,
however, arbitrary (or ought not to be, if the Subject has
the right attitudes). As in the above case of mere
negation, the conclusion of it does proceed from certain
existing findings. Yet, it is also true that this work only
occurs in the framework of cognition.
Negation and the Laws of Thought
Logic cannot be properly understood without first
understanding negation. This should be obvious from the
fact that two of the laws of thought concern the relation
between positive and negative terms. Similarly, the basic
principle of adduction, that hypotheses we put forward
should be empirically tested and rejected if they make
wrong predictions – this principle depends on an
elucidation of negation.
a.
The so-called laws of thought are, in a sense, laws
of the universe or ontological laws – in that the universe
is what it is (identity), is not something other than what it
is
(non-contradiction) and is something specific
(excluded middle).
They have phenomenological aspects: appearances
appear (identity); some are in apparent contradiction to
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THE LAWS OF THOUGHT
others (a contradiction situation); in some cases, it is not
clear just what has appeared (an excluded middle
situation).
They may also be presented as epistemological laws or
laws of logic, in that they guide us in the pursuit of
knowledge. However, they are aptly named laws of
thought, because they really arise as propositions only in
the context of cognitive acts.
To understand this, one has to consider the peculiar
status of negation, as well as other (partly derivative)
major processes used in human reasoning, including
abstraction, conceiving alternative possibilities and
making hypotheses.
b.
The impact of this insight on the laws of thought
should be obvious. The law of identity enjoins us
primarily to take note of the positive particulars being
perceived. But the laws of non-contradiction and of the
excluded middle, note well, both involve negation.
Indeed, that’s what they are all about – their role is
precisely to regulate our use of negation – to keep us in
harmony with the more positive law of identity!
Their instructions concerning the subjective act of
negation, at the most perceptual level, are as follows. The
law of non-contradiction forbids negating in the
perceptible presence of the thing negated. The law of
the excluded middle forbids accepting as final an
uncertainty as to whether a thing thought of is
currently present or absent.
We are unable to cognize a negative (not-X) except by
negation of the positive (X) we have in mind; it is
therefore absurd to imagine a situation in which both X
and not-X are true (law of non-contradiction). Similarly,
if we carefully trace how our thoughts of X and not-X
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arise in our minds, it is absurd to think that there might
be some third alternative between or beyond them (law
of the excluded middle.)
Thus, these two laws are not arbitrary conventions or
happenstances that might be different in other universes,
as
some logicians contend (because they have
unfortunately remained stuck at the level of mere
symbols, “X” and “non-X”, failing to go deeper into the
cognitive issues involved). Nor are they wholly
subjective or wholly objective.
These laws of thought concern the interface of Subject
and Object, of consciousness and existence – for any
Subject graced with rational powers, i.e. cognitive
faculties that go beyond the perceptual thanks in part to
the possibility of negation.
They are for this reason applicable universally, whatever
the content of the material and mental universe faced.
They establish for us the relations between affirmation
and denial, for any and every content of consciousness.
c.
On this basis, we can better comprehend the
ontological status of the laws of thought. They have no
actual existence, since the concrete world has no use for
or need of them, but exists self-sufficiently in positive
particulars.
But the laws are a potential of the world, which is
actualized when certain inhabitants of the world, who
have the gifts of consciousness and freewill, resort to
negation, abstraction and other cognitive-volitional
activities, in order to summarize and understand the
world.
In a world devoid of humans (or similar Subject/Agents),
there are no negations and no ‘universals’. Things just
are (i.e. appear) – positively and particularly. Negation
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THE LAWS OF THOUGHT
only appears in the world in relation to beings like us
who can search for something positive and not find it.
Likewise for ‘universals’ – they proceed from acts of
comparison and contrast.
Consciousness and volition are together what gives rise
to concepts and alternative possibilities, to hypotheses
requiring testing. It is only in their context that logical
issues arise, such as existence or not, reality or illusion,
as well as consistency and exhaustiveness.
It is important to keep in mind that the laws of thought
are themselves complex abstractions implying negations – viz. the negative terms they discuss, as well as the
negation of logical utility and value in contradictory or
‘middle’ thinking. Indeed all the ‘laws’ in our sciences
are such complex abstractions involving negations.
d.
The insight that negation is essentially a
volitional act allied to cognition explains why the laws of
thought are prescriptive as well as descriptive
epistemological principles.
The laws of thought are prescriptive inasmuch as human
thought is fallible and humans have volition, and can
behave erratically or maliciously. If humans were
infallible, there would be no need for us to study and
voluntarily use such laws. There is an ethic to cognition,
as to all actions of freewill, and the laws of thought are
its top principles.
The laws of thought are descriptive, insofar as we
commonly explicitly or implicitly use them in our
thinking. But this does not mean we all always use them,
or always do so correctly. They are not ‘laws’ in the
sense of reports of universal behavior. Some people are
unaware of them, increasing probabilities of erroneous
thinking. Some people would prefer to do without them,
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and eventually suffer the existential consequences. Some
people would like to abide by these prescriptions, but do
not always succeed.
These prescriptions, as explicit principles to consciously
seek to abide by, have a history. They were to our
knowledge first formulated by a man called Aristotle in
Ancient Greece. He considered them to best describe the
cognitive behavior patterns that lead to successful
cognition. He did not invent them, but realized their
absolute importance to human thought.
the
Their justification is self-evident to anyone who goes
through
inductive
and
deductive
logical
demonstrations certain logicians have developed in this
regard. Ultimately it is based on a holistic consideration
of knowledge development.
Our insights here about the relativity of negation and
abstraction, and the realization of their role in the laws of
thought serve to further clarify the necessity and
universality of the latter.
Pure Experience
A logically prior issue that should perhaps be stressed in
this context is the existence of pure experience, as
distinct from experience somewhat tainted by acts of
thought.
Some philosophers claim that all alleged ‘experience’
falls under the latter class, and deny the possibility of the
former. But such skepticism is clearly inconsistent: if we
recognize some part of some experience as pure of
thought, this is sufficient to justify a claim to some pure
experience. Thus, the proposition “There are some pure
experiences” may be taken as an axiom of logic,
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THE LAWS OF THOUGHT
phenomenology, epistemology and ontology. This
proposition is self-evident, for to deny it is self
contradictory.
Note that this proposition is more specific than the more
obvious “There are experiences”. Denial of the latter is a
denial of the evidence before one’s eyes (and ears and
nose and tongue and hands, etc. – and before one’s
“mind’s eye”, too): it directly contravenes the law of
identity. Philosophers who engage in such denial have no
leg to stand on, anyway – since they are then hard put to
at all explain what meaning the concepts they use in their
denial might possibly have. We have to all admit some
experience – some appearance in common (however
open to debate) – to have anything to discuss (or even to
be acknowledged to be discussing).
Let us return now to the distinction between pure and
tainted experiences. This concerns the involvement of
thought processes of any kind – i.e. of ratiocinations, acts
of reason. To claim that there are pure experiences is not
to deny that some (or many or most) experiences are
indeed tainted by conceptual activity (abstraction,
classification, reasoning, etc.)
We can readily admit that all of us very often have a hard
time distinguishing pure experience from experience
mixed with rational acts. The mechanisms of human
reason are overbearing and come into play without
asking for our permission, as is evident to anyone who
tries to meditate on pure experience. It takes a lot of
training to clearly distinguish the two in practice.
But surely, any biologist would admit that lower animals,
at least, have the capacity to experience without the
interference of thought, since they have no faculty of
thought. The same has to be true to some extent for
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humans – not only in reflex actions, but also in the very
fact that reasoning of any sort is only feasible in relation
to pre-existing non-rational material. To process is to
process something.
I have already argued that what scientists call
‘experiment’ cannot be regarded as the foundation of
science, but must be understood as a mix of intellectual
(and in some cases, even physical acts) and passive
observation (if only observation of the results of
experiment displayed by the detection and measuring
instruments used). Thus, observation is cognitively more
fundamental than experiment.
Here, my purpose is to emphasize that perceptual
‘negation’ is also necessarily a mix of pure experience
and acts of the intellect. It is never pure, unlike the
perception of positive particulars (which sometimes is
pure, necessarily) – because it logically cannot be, since
to deny anything one must first have something in mind
to deny (or affirm).
Thus, negation can be regarded as one of the most
primary acts of reason – it comes before abstraction,
since the latter depends to some extent on making
distinctions, which means on negation.
Consistency is Natural
It
is
important to here reiterate the principle that
consistency is natural; whereas inconsistency is
exceptional.
Some modern logicians have come up with the notion of
“proving consistency” – but this notion is misconceived.
Consistency is the natural state of affairs in knowledge; it
requires no (deductive) proof and we are incapable of
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THE LAWS OF THOUGHT
providing such proof, since it would be ‘placing the cart
before the horse’. The only possible ‘proof’ of
consistency is that no inconsistency has been
encountered. Consistency is an inductive given, which is
very rarely overturned. All our knowledge may be and
must be assumed consistent, unless and until there is
reason to believe otherwise.
In short: harmony generally reigns unnoticed, while
conflicts erupt occasionally to our surprise. One might
well wonder now if this principle is itself consistent with
the principle herein defended that negatives are never per
se objects of cognition, but only exist by denial of the
corresponding positives. Our principle that consistency is
taken for granted seems to imply that we on occasion
have logical insights of inconsistency, something
negative!
To resolve this issue, we must again emphasize the
distinction
between pure experience and the
interpretations of experience that we, wordlessly (by
mere intention) or explicitly, habitually infuse into our
experiences. Generally, almost as soon as we experience
something, we immediately start interpreting it,
dynamically relating it to the rest of our knowledge thus
far. Every experience almost unavoidably generates in us
strings of associations, explanations, etc.
The contradictions we sometimes come across in our
knowledge do not concern our pure experiences (which
are necessarily harmonious, since they in fact exist side
by side – we might add, quite ‘happily’). Our
contradictions are necessarily contradictions between an
interpretation and a pure experience, or between two
interpretations. Contradictions do not, strictly speaking,
reveal difficulties in the raw data of knowledge, but
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merely in the hypotheses that we conceived concerning
such data.
Contradictions are thus to be blamed on reason, not on
experience. This does not mean that reason is necessarily
faulty, but only that it is fallible. Contradictions ought
not be viewed as tragic proofs of our ignorance and
stupidity – but as helpful indicators that we have
misinterpreted something somewhere, and that this needs
reinterpretation. These indicators are precisely one of the
main tools used by the faculty of reason to control the
quality of beliefs. The resolution of a contradiction is just
new interpretation.
How we know that two theories, or a theory and some
raw data, are ‘in contradiction’ with each other is a moot
question. We dismiss this query rather facilely by
referring to “logical insight”. Such insight is partly
‘experiential’, since it is based on scrutiny of the
evidence and doctrines at hand. But it is clearly not
entirely
empirical and involves abstract factors.
‘Contradiction’ is, after all, an abstraction. I believe the
answer to this question is largely given in the
psychological analysis of negation.
There is an introspective sense that conflicting intentions
are involved. Thus, the ‘logical insight’ that there is
inconsistency is not essentially insight into a negative (a
non-consistency), but into a positive (the intuitive
experience of conflict of intentions). Although the word
inconsistency involves a negative prefix, it brings to
mind something empirically positive – a felt tension
between two theses or a thesis and some data.
For this reason, to say that ‘consistency is assumable,
until if ever inconsistency be found’ is consistent with
our claim that ‘negations are not purely empirical’.
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(Notice incidentally that we did not here “prove”
consistency, but merely recovered it by clarifying the
theses involved.)
The above analysis also further clarifies how the law of
non-contradiction is expressed in practice. It does not
sort out experiences as such, but concerns more abstract
items of knowledge. To understand it fully, we must be
aware of the underlying intentions. A similar analysis
may be proposed to explain the law of the excluded
middle.
In the latter case, we would insist that (by the law of
identity) ‘things are something, what they are, whatever
that happen to be’. Things cannot be said to be neither
this
nor the negation of this, because such
characterizations are negative (and, respectively, doubly
negative) – and therefore cannot constitute or be claimed
as
positive experience. Such situations refer to
uncertainties in the knower, which he is called upon to
eventually fill-in. They cannot be proclaimed final
knowledge (as some modern sophists have tried to do),
but must be considered temporary postures in the pursuit
of knowledge.
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CHAPTER EIGHTEEN
Drawn from Logical and Spiritual Reflections (2008),
Book 1, chapter 2.
THE PRINCIPLE OF INDUCTION
The Uniformity Principle
Concerning the uniformity principle, which Hume
denies, it is admittedly an idea difficult to uphold, in the
sense that we cannot readily define uniformity or make a
generality of it. We might speak of repetition, of two or
more particular things seeming the same to us; but we are
well aware that such regularity does not go on ad
infinitum. On the contrary, we well know that sooner or
later, something is bound to be different from the
preceding things, since the world facing us is one of
multiplicity.
Therefore, this “principle” may only be regarded as a
heuristic idea, a rule of thumb, a broad but vague
practical guideline to reasoning. It makes no specific
claims in any given case. It just reminds us that there are
(or seem to us to be) ‘similarities’ in this world of matter,
mind and spirit. It is not intended to deny that there are
also (apparent) ‘dissimilarities’. It is obviously not a
claim that all is one and the same, a denial of multiplicity
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THE LAWS OF THOUGHT
and diversity (in the world of appearances, at least33). To
speak of uniformity in Nature is not to imply uniformity
of Nature.
We might also ask – can there be a world without any
‘uniformities’? A world of universal difference, with no
two things the same in any respect whatever is
unthinkable. Why? Because to so characterize the world
would itself be an appeal to uniformity. A uniformly
non-uniform world is a contradiction in terms. Therefore,
we must admit some uniformity to exist in the world. The
world need not be uniform throughout, for the principle
of uniformity to apply. It suffices that some uniformity
occurs.
Given this degree of uniformity, however small, we
logically can and must talk about generalization and
particularization.
There happens to be some
‘uniformities’; therefore, we have to take them into
consideration in our construction of knowledge. The
principle of uniformity is thus not a wacky notion, as
Hume seems to imply. It is just a first attempt by
philosophers to explain induction; a first try, but certainly
not the last. After that comes detailed formal treatment of
the topic. This proceeds with reference to specifics,
symbolized by X’s and Y’s, and to strict logic.
The uniformity principle is not a generalization of
generalization; it is not a statement guilty of circularity,
as some critics contend. So what is it? Simply this: when
we come upon some uniformity in our experience or
thought, we may readily assume that uniformity to
33
I.e. such recognition of pluralism does not at the
outset exclude monism. The former may be true at the
superficial phenomenological level, while the latter reigns at
the metaphysical level of ultimate reality.
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continue onward until and unless we find some
evidence or reason that sets a limit to it. Why? Because
in such case the assumption of uniformity already has a
basis, whereas the contrary assumption of difference has
not or not yet been found to have any. The generalization
has some justification; whereas the particularization has
none at all, it is an arbitrary assertion.
It cannot be argued that we may equally assume the
contrary assumption (i.e. the proposed particularization)
on the basis that in past events of induction other
contrary assumptions have turned out to be true (i.e. for
which experiences or reasons have indeed been adduced) – for the simple reason that such a generalization from
diverse past inductions is formally excluded by the fact
that we know of many cases that have not been found
worthy of particularization to date.
That is to say, if we have looked for something and not
found it, it seems more reasonable to assume that it does
not exist than to assume that it does nevertheless exist.
Admittedly, in many cases, the facts later belie such
assumption of continuity; but these cases are relatively
few in comparison. The probability is on the side of
caution.
In any event, such caution is not inflexible, since we do
say “until and unless” some evidence or argument to the
contrary is adduced. This cautious phrase “until and
unless” is of course essential to understanding induction.
It means: until if ever – i.e. it does not imply that the
contrary will necessarily occur, and it does not exclude
that it may well eventually occur. It is an expression of
open-mindedness, of wholesome receptiveness in the
face of reality, of ever readiness to dynamically adapt
one’s belief to facts.
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THE LAWS OF THOUGHT
In this way, our beliefs may at all times be said to be as
close to the facts as we can get them. If we follow such
sober inductive logic, devoid of irrational acts, we can be
confident to have the best available conclusions in the
present context of knowledge. We generalize when the
facts allow it, and particularize when the facts necessitate
it. We do not particularize out of context, or generalize
against the evidence or when this would give rise to
contradictions.
Hume doubted the validity of generalization because he
thought that we adopt a general proposition like All X
are Y, only on the basis of the corresponding particular
Some X are Y. But if the latter was sufficient to
(inductively) establish the former, then when we were
faced with a contingency like Some X are Y and some X
are not Y, we would be allowed to generalize both the
positive and negative particulars, and we would find
ourselves with a contradiction34 in our knowledge, viz.
with both All X are Y and No X are Y.
But since contradiction is error, according to the 2nd law
of thought, it follows that a particular is not by itself
enough to confirm a generality. To do so, we need also to
first adduce that the opposite particular is not currently
justified. Note well what we have shown here: this
criterion for generalization follows from the law of non
contradiction. Hume and his skeptical successors did not
take this additional criterion into account. They noticed
the aspect of ‘confirmation’, but ignored that of ‘non
rejection’.
34
Or more precisely a contrariety.
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The Principle of Induction
The uniformity principle ought to be viewed as an
application of a much larger and important principle,
which we may simply call the principle of induction (in
opposition to the so-called problem of induction). This
all-important principle could be formulated as follows:
given any appearance, we may take it to be real, until
and unless it is found to be illusory.35
This is the fundamental principle of inductive logic, from
which all others derive both their form and their content.
And indeed, this is the way all human beings function in
practice (with the rare exception of some people, like
Hume, who want to seem cleverer than their peers). It is,
together with Aristotle’s three laws of thought, the
supreme principle of methodology, for both ordinary and
scientific
thought,
investigation36.
whatever the domain under
35
I
have formulated and stressed this principle since I
started writing logic, although I here name it “principle of
induction” for the first time. See, for instances: Future Logic,
chapter 2, etc.; Phenomenology, chapter 1, etc.; Ruminations,
chapters 1 and 2.
36
I
stress that here, to forestall any attempt to split
ordinary and scientific thought apart. We should always stress
their continuity. The difference between them is (theoretically,
at least) only one of rigor, i.e. of effort to ensure maximal
adherence to logic and fact. This only means, at most, that
more ordinary people fail to look carefully and think straight
than do most scientists – but both groups are human. Another
important thing to stress is that this method is the same for
knowledge of matter or mind, of earthly issues or metaphysical
ones, and so forth. The principle is the same, whatever the
content.
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THE LAWS OF THOUGHT
Indeed, we could construe this principle of induction as
the fourth law of thought. Just as the three laws
proposed by Aristotle are really three facets of one and
the same law, so also this fourth law should be viewed as
implicit in the other three. Induction being the most
pragmatic aspect of logic, this principle is the most
practical of the foundations of rational discourse.
The principle of induction is a phenomenological truth,
because it does not presume at the outset that the givens
of appearance are real or illusory, material or mental, full
or empty, or what have you. It is a perfectly neutral
principle, without prejudice as to the eventual content of
experience and rational knowledge. It is not a particular
worldview, not an a priori assumption of content for
knowledge.
However, in a second phase, upon reflection, the same
principle favors the option of reality over that of illusion
as a working hypothesis. This inbuilt bias is not only
useful, but moreover (and that is very important for
skeptics to realize) logically rock solid, as the following
reasoning clearly shows:
This principle is self-evident, because its denial is self
contradictory. If someone says that all appearance is
illusory, i.e. not real, which means that all our alleged
knowledge is false, and not true, that person is laying
claim to some knowledge of reality (viz. the knowledge
that all is unreal, unknowable) – and thus contradicting
himself. It follows that we can only be consistent by
admitting that we are indeed capable of knowing some
things (which does not mean everything).
It follows that the initial logical neutrality of appearance
must be reinterpreted as in all cases an initial reality that
may be demoted to the status of illusion if (and only if)
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specific reasons justify it. Reality is the default
characterization, which is sometimes found illusory.
Knowledge is essentially realistic, though in exceptional
cases it is found to be unrealistic. Such occasional
discoveries of error are also knowledge, note well; they
are not over and above it.
If we did not adopt this position, that appearance is
biased towards reality rather than illusion, we would be
stuck in an inextricable agnosticism. Everything would
be “maybe real, maybe illusory” without a way out. But
such a problematic posture is itself a claim of knowledge,
just like the claim that all is illusory, and so self
inconsistent too. It follows that the interpretation of
appearance as reality until and unless otherwise proved is
the only plausible alternative.37
If appearance were not, ab initio at least, admitted as
reality rather than as illusion or as problematic, we would
be denying it or putting it in doubt without cause – and
yet we would be granting this causeless denial or doubt
the status of a primary truth that does not need to be
justified. This would be an arbitrary and self
contradictory posture – an imposture posing as logical
insight. All discourse must begin with some granted truth – and in that case, the most credible and consistent truth
37
Worth also stressing here is the importance of working
hypotheses as engines of active knowledge development. A
skeptical or agnostic posture is essentially static and passive;
taken seriously, it arrests all further development. Scientists
repeatedly report the crucial role played by their working
hypothesis, how it helped them to search for new data that
would either confirm or refute it, how it told them what to look
for and where and how to look (see for instance, Gould, p.
172). This is true not only of grand scientific theories, but of
ordinary everyday concepts.
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THE LAWS OF THOUGHT
is the assumption of appearance as reality unless or until
otherwise proved.
We may well later, ad terminatio (in the last analysis),
conclude that our assumption that this appearance was
real was erroneous, and reclassify it as illusory. This
happens occasionally, when we come across conflicts
between appearances (or our interpretations of them). In
such cases, we have to review in detail the basis for each
of the conflicting theses and then decide which of them is
the most credible (in accord with numerous principles of
adduction).
It should be stressed that this stage of reconciliation
between conflicting appearances is not a consequence of
adopting reality as the default value of appearances. It
would occur even if we insisted on neutral appearances
and refused all working hypotheses. Conflicts would still
appear and we would still have to solve the problem they
pose. In any case, never forget, the assumption of reality
rather than illusion only occurs when and for so long as
no contradiction results. Otherwise, contradictions would
arise very frequently.
Regarding Husserl
Note well that I do not understand appearance in quite
the same way Edmund Husserl does, as something ab
initio and intrinsically mental; such a view is closer to
Hume or even Berkeley than to me.
The ground floor of Husserl’s phenomenology and mine
differ in the primacy accorded to the concepts of
consciousness and of the subject of consciousness. My
own approach tries to be maximally neutral, in that
appearances are initially taken as just ‘what appears’,
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without immediately judging them as ‘contents of
someone’s consciousness’. Whereas, in Husserl’s
approach, the wider context of appearance is from the
start considered as part and parcel of the appearance.
For me, some content comes first, and only thereafter do
we, by a deduction or by an inductive inference, or
perhaps more precisely by an intuition (an additional,
secondary, reflexive act of consciousness), become
aware of the context of consciousness and conscious
subject. At this later stage, we go back and label the
appearance as a “content of” consciousness, i.e. as
something whose apparition (though not whose
existence) is made possible by an act of consciousness by
some subject. Content is chronologically primary, the
context is secondary.
Whereas in Husserl’s philosophy, the fact of
consciousness and its subject are present from the start,
as soon as the appearance appears. Husserl’s mistake, in
my view, is to confuse logical order and chronological
order, or ontological and epistemological. Of course,
logically
and
ontologically,
appearance implies
consciousness and someone being conscious; but
chronologically and epistemologically, they occur in
succession.
As a result of this difference, his approach has a more
subjectivist flavor than mine, and mine has a more
objectivist flavor than his. Note, however, that in his later
work Husserl tried more and more to shift from implied
subjectivism to explicit objectivism.
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THE LAWS OF THOUGHT
The Flexibility of Induction
We have seen the logic of induction in the special case of
generalization. Given the positive particular ‘Some X are
Y’ (appearance), we may generalize to the corresponding
generality ‘All X are Y’ (reality), provided we have no
evidence that ‘Some X are not Y’ (no conflicting
appearance). Without this caveat, many contradictions
would arise (by generalizing contingencies into contrary
generalities); that proves the validity of the caveat. If (as
sometimes occurs) conflicting evidence is eventually
found (i.e. it happens that Some X are not Y), then what
was previously classed as real (viz. All X are Y) becomes
classed as illusory (this is called particularization).
Induction is a flexible response to changing data, an
ongoing effort of intelligent adaptation to apparent facts.
Few logicians and philosophers realize, or take into
consideration, the fact that one of the main disciplines of
inductive logic is harmonization. They discuss
observation
and experiment, generalization and
adduction, and deduction, with varying insight and skill,
but the logic of resolving contradictions occasionally
arrived at by those other inductive means is virtually
unknown to them, or at least very little discussed or
studied. This ignorance of, or blindness to, a crucial
component of induction has led to many foolish
theories38.
Notice well, to repeat, the conditional form of the
principle of induction: it grants credibility to initial
appearances “until and unless” contrary appearances
arise, which belie such immediate assumption. Thus, in
38
For
confirmation.
example, Hempel’s so-called paradox of
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179
the case of the narrower uniformity principle, the initial
appearance is the known few cases of similarity (or
confirmation) and the fact of not having to date found
cases of dissimilarity (or conflicting data); this allows
generalization (or more broadly, theory adoption) until if
ever we have reason or evidence to reverse our judgment
and particularize (or reject, or at least modify, the
theory).
The principle of induction may likewise be used to
validate our reliance on intuition and sensory and inner
perception, as well as on conception. It may also be
applied to causality, if we loosely formulate it as: order
may be assumed to exist everywhere, until and unless
disorder appears obvious. However, the latter principle is
not really necessary to explain causality, because we can
better do that by means of regularity, i.e. with reference
to the uniformity principle, i.e. to generalization and
adduction.
In any case, the principle of induction is clearly a
phenomenological principle, before it becomes an
epistemological or ontological one. It is a logical
procedure applicable to appearance as such, free of or
prior to any pretensions to knowledge of reality devoid of
all illusion. The claims it makes are as minimal as could
be; they are purely procedural. It is for this reason as
universal and indubitable as any principle can ever be.
Moreover, the principle of induction (and likewise its
corollary the uniformity principle) applies equally to the
material, mental and spiritual realms. It is a valid method
of dealing with data, independently of the sort of data
involved, i.e. irrespective of the ‘substance’ of the data.
Many people associate induction exclusively with the
physical sciences, but this is misconceived. Inductive
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THE LAWS OF THOUGHT
logic sets standards of judgment applicable in all fields –
including in psychology and in moral and spiritual
concerns.
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181
CHAPTER NINETEEN
Drawn from Logical and Spiritual Reflections (2008),
Book 3, chapter 2.
THE PRIMACY OF THE LAWS
Briefly Put
Aristotle’s laws of thought cannot be understood with a
few clichés, but require much study to be fathomed. The
laws of thought can be briefly expressed as39:
A thing is what it is (the law of identity).
A thing cannot at once be and not-be (the law of
non-contradiction).
A thing cannot neither be nor not-be (the law of
the excluded middle).
These three principles imply that whatever is, is
something – whatever that happens to be. It is not
something other than what it is. It is not nothing
whatsoever. It is not just anything. If something exists, it
has certain features. It cannot rightly be said to have
features other than just those, or no features at all, or to
both have and lack features.
39
These are of course simple statements, which have to
be elaborated on. Note that when I speak of a ‘thing’ here, I
mean to include not only terms (percepts and concepts, or the
objects they refer to), but also propositions (which relate
percepts and/or concepts).
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THE LAWS OF THOUGHT
A thing is what it is, whether we know what it is or not,
and whether we like what it is or not. It is not our beliefs
or preferences that make a thing what it is. It is what it is
independently of them. Our beliefs can be in error, and
often are. How do we know that? By means of later
beliefs, based on better information and/or arguments.
However, a thing can have conflicting features in
different parts or aspects of its being. Notably, a thing
can change over time. So long as these differences are
separated in respect of place, time, or other relations to
other things, such as a causal relation – the contradiction
is not impossible. But if we refer to the exact same thing,
at the same place and time, and the same in all other
respects, contradiction is logically unacceptable – it is
indicative of an error of thought.
Also, we may well have no idea or no certainty what
some (indeed, many or most) features of a thing are.
Such problematic situations are indicative of our
ignorance, and should not be taken to imply that the thing
in question necessarily lacks the unknown features, or
neither has nor lacks certain features, or both has and
lacks them.
All these logical insights are evident in our ordinary
thoughts and in scientific thinking. If we look upon our
discourse clearly and honestly, we see that our conviction
in every case depends upon these criteria. Occasionally,
people try to make statements contrary to these criteria;
but
upon further analysis, they can always be
convincingly shown to be erring.
These general logical principles, and certain others
(notably the principle of induction, to name one), help us
regulate our thinking, ensuring that it sticks as close as
possible to the way things are and that we do not get
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cognitively lost in a complex maze of fantastical
nonsense.
They do not force us to be truthful, or guarantee the
success of our knowledge endeavors, but they provide us
with crucial standards by which can test our progress at
all times. (More will be said about these principles in this
volume, in addition to what has already been said in the
past.)
If the crucial epistemological and ontological roles of
Aristotle’s three laws of thought in human knowledge are
not sought out and carefully studied, there is little hope
that these little jewels of human understanding will be
treasured. It takes a lifetime of reflection on logical and
philosophical issues to fully realize their impact and
importance.
Antagonism to the Laws
I marvel at people who think they can show reason to be
unreasonable. Leaning on hip, postmodern sophists, like
Wittgenstein or Heidegger, or on more ancient ones, like
Nagarjuna, they argue confidently that the foundations of
rationality are either arbitrary, or involve circularity or
infinite regression. They do not realize that their
intellectual forebears were in fact either ignorant of logic
or intentionally illogical.
Many critics of the laws of thought simply do not
understand them; no wonder then that they are critical.
They have very narrow, shallow views about the laws of
thought; they have not studied them in any breadth or
depth. For instance, to some people, brought up under
“modern” symbolic logic, the laws of thought are simply
X=X, ~(X+~X) and ~(~X+~~X). Given such simplistic,
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THE LAWS OF THOUGHT
superficial statements, no wonder the laws seem arbitrary
and expendable to them.
The laws are not a prejudice about the world, as some
critics try to suggest. The law of identity does not tell us
about some particular identity, but only tells us to be
aware of how and what things are or even just appear to
be. The law of non-contradiction does not favor the
thesis that something is X, or the thesis that it is not X; it
allows for us sometimes facing dilemmas, only
forbidding us to settle on the implied contradictions as
final. The law of the excluded middle does not deny the
possibility of uncertainty, but only enjoins us to keep
searching for solutions to problems.
If nothing were known, or even knowable, as some
claim, this would not constitute a good reason to dump
the laws of thought – for these laws make no claims
about the specific content of the world of matter, mind or
spirit. They make no a priori demand regarding this or
that thesis. They only serve to regulate our cognitive
relation to the world, however it happens to be or seem.
They show us how to avoid and eliminate errors of
reasoning.
These laws can for a start teach us that to claim “nothing
is known or knowable” is self-contradictory, and thus
illogical and untenable.
Such a claim, about the nonexistence or impossibility of
knowledge as such, must be admitted to itself be an
allegation of knowledge (such admission being a
requirement of the law of identity). Therefore, it is
unthinkable that any Subject might attain such alleged
knowledge of its total ignorance (because such
attainment would be against the law of non
contradiction). We could not even adopt a negative
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posture of denying both knowledge and knowledge of
ignorance (in an attempted bypass of the law of the
excluded middle), for that too is an assertion, a claim to
established fact, a claim to knowledge.
All these rational insights are not open to debate.
Antagonism to the laws of thought is sure and
incontrovertible proof that one is erring in one’s thinking.
How might such antagonism be systematically justified
without appeal to those very laws? One couldn’t claim to
be generalizing or adducing it from experience, for this
would appeal to the law of generalization or the principle
of adduction, which are themselves based on the laws of
thought. One couldn’t claim to be drawing some sort of
syllogistic or other deductive conclusion, for the same
reason. Such antagonism can only be based on arbitrary
assertion, without any conceivable rational support.
Counterarguments
Arguments like this in favor of the laws of thought are
claimed by their opponents to be ‘circular’ or ‘infinitely
regressive’ – i.e. arbitrary. But to point to the fallacy of
circularity or infinite regress is to appeal to the need to
ground one’s beliefs in experience or reasoning – which
is
precisely the message of the laws of thought.
Therefore, those who accuse us of circularity or infinity
are doing worse than being circular or infinite: they are
appealing to what they seek to oppose; they are being
self-contradictory, as well as arbitrary!
It is our faculty of logical insight or rationality that
teaches us to beware of arbitrary propositions, which are
sometimes given an illusion of proof through circular or
infinite arguments. One cannot deny this very faculty of
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THE LAWS OF THOUGHT
logical insight by claiming that it can only be proven by
circular or infinite arguments. This would turn it against
itself, using it to justify its own denial. It would
constitute another fallacy – that of “concept stealing”.
The proposition “if P, then P” is not circular or infinite –
it is true of all propositions. Such a proposition does not
“prove” the truth of P, but merely acknowledges P as a
claim that may turn out to be true or false. If one
proposes “if P, then P” as a proof of P, one is then of
course engaged in circularity or infinite regression; but
otherwise no logical sin is involved in affirming it. On
the other hand, the paradoxical proposition “if P, then not
P” does imply P to be false. To affirm P as true in such
case is a logical sin, for P is definitely implied false by it.
The laws of thought are not circular or infinite – they are
just consistent with themselves. It is their opponents who
are engaged in fallacy – the failure to think reflexively,
and realize the implications of what they are saying on
what they are saying. To deny all claims to knowledge is
to deny that very claim too – it is to be self-inconsistent.
One logically must look back and check out whether one
is
self-consistent; that is not circularity, but wise
reflection.
The laws of thought are not based on any particular
argument, but the very basis of all reasoning processes.
This is not an arbitrary starting point; it is an insight
based on observation of all reasoning acts, an admission
of what evidently carries conviction for us all. These
laws cannot be disregarded or discarded, simply because
they are so universal. That these laws do not lead to any
paradox adds to their force of conviction; but that too is
just an application of their universality. They encapsulate
what we naturally find convincing in practice, provided
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we are not seeking dishonestly to pretend otherwise in
theory.
The laws of thought may be viewed as specific laws of
nature: they express the nature of rational thought, i.e. of
logical discourse. By logic is here meant simply a mass
of experiences – namely, all the ‘events having the form
expressed by the laws of thought’. That is, logic refers to
the concrete occurrences underlying the abstractions that
we name ‘laws of thought’. This is a primary given for
which no further reason is necessary. It is not arbitrary,
for it is the source of all conviction. To ask for a further
reason is to ask for a source of conviction other than the
only natural source of conviction! It is to demand the
impossible, without reason and against all reason. It is
stupid and unfair.
Our Pedestrian Path
If one examines the motives of critics of the laws of
thought, one often finds an immature and irrational
yearning for absolutes. They seek a shortcut to
omniscience, a magic formula of some sort, and think the
laws of thought are obstacles to this pipedream, and so
they abandon these laws and seek truth by less restrictive
means.
Our ordinary knowledge is very pedestrian: it progresses
step by step; it advances painstakingly by trial and error;
it is rarely quite sure, and certainly never total and final.
This relativity of common knowledge unsettles and
displeases some people. To them, such inductive efforts
are worthless – knowledge that is not omniscient is not
good enough; it is as bad as no knowledge at all. Thus,
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THE LAWS OF THOUGHT
they reject reason. This is an unhealthy attitude, a failure
of ‘realism’.
Let’s face it squarely: our knowledge as a whole has no
finality till everything about everything is known. And
how, by what sign, would we know we know everything?
Ask yourself that. There is no conceivable such sign. Our
knowledge is necessarily contextual; it depends on how
much we have experienced and how well we have
processed the data. There is no end to it.
Even so, at any given stage of the proceedings, one body
of knowledge can conceivably be considered better than
another, given experience and reasoning so far. To be
better does not necessarily mean to be the best – but it is
still better than to be worse or equal. That is a realistic
posture, and a source of sufficient security and
satisfaction.
A phenomenological approach to the problem of
knowledge is necessary, to avoid erroneous views. It
starts with mere appearance, whether of seemingly
material or mental phenomena (bodies and ideas), or of
spiritual intuitions (of self, and its cognitions, volitions
and valuations)40. The contents of one’s consciousness
40
Note well that I do not posit perception itself as the
starting point of knowledge, as some do. Perception is a
relational concept – it is perception of something by someone.
Before we become aware of our perceptual ability, we have to
exercise it – i.e. we perceive something (other than the
perceiving itself). The empirical basis of our concept of
perception is our common experience of sensory and mental
phenomenal content. When you and I were young children, we
were perceiving such phenomena – only later when we
became older did we form a concept of perception. Therefore
perception as such cannot be taken as a primary in the order
of things.
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are,
ab
initio,
appearances; this is a neutral
characterization of what we are conscious of, the raw
data and starting point of knowledge. Our first cognitive
task is to acknowledge these appearances, as apparent
and just as they appear, coolly observing them without
interference or comment before any further ado.
It is equally naïve to assume as primary given(s) matter,
or mind, or spirit; what is certainly given in experience is
the appearance of these things. Much logical work is
required before we can, ad terminatio, establish with
reasonable certainty the final status of these appearances
as matter, mind or spirit. We may indeed to begin with
assume all such appearances to be real; but in some
specific cases, due to the discovery of contradictions
between appearances or to insufficiencies in our theories
about them, we will have to admit we were wrong, and
that certain appearances are illusory.
There is an order of things in the development of
knowledge that must be respected. Everything beyond
appearances is ‘theory’ – which does not mean that it is
necessarily false, only that it must be considered more
critically. Theory involves the rational faculty in one way
or another. What is theory needs to be sorted out,
organized, kept consistent, made as complete as possible.
This is where the laws of thought are essential. But these
laws cannot make miracles; they can only help us (with
the aid of our intelligence and imaginative faculty)
formulate and select the best theory in the present context
of knowledge.41
41
Note well: the laws of thought cannot by themselves
immediately tell you whether what you have apparently
perceived is true or false – but what they can tell you is that
you should notice well what you did perceive (its configuration,
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THE LAWS OF THOUGHT
Human knowledge is thus essentially inductive and
probabilistic, depending on the scope and quality of
experience, and then on successive generalizations and
particularizations, or on competing larger hypotheses
requiring
ongoing
comparative confirmation or
refutation. The laws of thought are involved at all stages
of this process, regulating our judgments to minimize its
chances of error.
the phenomenal modalities, i.e. the sights, sounds, etc.,
apparent times, places, and so forth). Similarly for
introspective data of intuition. The question of truth and
falsehood for any single item of experience can only be solved
progressively, by holistic consideration of all other experiential
items, as well as by logical considerations (including
consistency and completeness). This is the inductive process.
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CHAPTER TWENTY
Drawn from Logical and Spiritual Reflections (2008),
Book 3, chapters 3 & 4.
Ontological Status of the Laws
Discussion of the laws of thought inevitably arrives at the
question: are these ontological or epistemological laws,
or both; and if both in what sequence? Furthermore, what
is their own ontological status – i.e. where do they
‘reside’, as it were? Are they ‘out there’ somehow, or
only ‘in our minds’?
As my thought on the issue has evolved over the years42,
I am now convinced that the traditional term “laws of
thought” is accurate, in that these statements are
primarily imperatives to us humans on how to think
about reality, i.e. how to ensure that we cognitively treat
the givens of appearance correctly, so that our ideas
remain reasonably credible possible expressions of
reality and do not degenerate into delusions.
Why? Because Nature can only posit; and so ‘negating’
depends on Man. That is to say, the world process is
always positive; negation involves a particular relation
between a conscious being and that presentation. For
negation to occur, a conscious being has to project and
look for something positive and fail to find it; otherwise,
all that occurs is positive.
42
See especially my Ruminations, chapter 9 (“About
Negation”).
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THE LAWS OF THOUGHT
Thus, when we state the laws of non-contradiction and of
the excluded middle, formally as “X and not X cannot
both be true” and “X and not X cannot both be untrue”,
we mean that such claims (i.e. ‘both true’ or ‘both
untrue’) cannot reasonably be made within discourse. We
mean that ‘X and not-X’, respectively ‘not-X and not
not-X’, cannot correctly be claimed as known or even as
reasonably opined.
Conjunctions of (positive or negative) contradictories are
thus outside the bounds of logically acceptable
discourse. These two laws of thought together and
inseparably effectively define what we naturally mean by
negation. Note well, ‘middles’ between contradictories
are as unthinkable as coexisting contradictories.
Note that the law of identity is also tacitly involved in
such definition of negation, since before we can
understand the logical act of negating, we must grasp the
fact of positive presence. So, it is not just the second and
third laws that define negation, but strictly speaking also
the first.
Such definition is, needless to say, not arbitrary or
hypothetical. Were someone to propose some other
definition of negation (e.g. using the law of non
contradiction alone, or some other statement altogether),
this would only produce an equivocation – the natural
definition with reference to the three laws of thought
would still be necessary and intended below the surface
of all discourse, however willfully suppressed.
From this it follows, by an extrapolation from logically
legitimate thought to reality beyond thought, that these
laws of thought (or, identically, of logic – ‘logic’
meaning ‘discourse’ by a thinker) are also necessarily
laws of reality.
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Words are symbols, and symbols can be made to do what
one wills, because they are per se not in fact subject to
the laws of thought. That is to say, mental gymnastics
like placing the symbol X next to the symbol not-X are
indeed feasible, but that does not mean that the things the
symbols symbolize can equally well be conjoined.
To label an observed illusion or a deliberate fantasy as
‘real’ does not make it in fact real. We can easily
verbally
imagine a ‘reality’ with non-identity,
contradictions and inclusions of the middle, but we
cannot actually conjure one.
The Need for a Subject
As for the status of the laws of thought themselves: being
products of reason, their existence depends on that of a
conscious – indeed, rational – subject. All particular acts
of
reasoning – such as negation, abstraction,
measurement, classification, predication, generalization,
etc. – depend for their existence on some such rational
subject (e.g. a man).
Take away all such subjects from the universe, and only
positive particular things or events will remain. Without
an act of negation, no mixing of or intermediate between
contradictories occurs in thought; all the more so, they
cannot occur outside thought. Similarly, with regard to
abstraction and other acts of the reasoning subject.
Concepts like similarity, difference, uniformity, variety,
continuity,
change, harmony, contradiction, and
principles like the laws of thought, being all outcomes of
such ratiocinative acts, are similarly dependent for their
existence on there being some appropriately conscious
subject(s).
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THE LAWS OF THOUGHT
These concepts and principles are, we might say,
inherent in the world in the way of a potential; but
without the involvement of such a subject, that potential
can never be actualized.
These concepts and principles depend for their existence
on there being conscious subjects to form them – but
their truth or falsehood is not a function of these
subjects. Their occurrence is dependent, but the accuracy
of their content when they occur is a different issue. It is
not subjective and relative, but on the contrary objective
and absolute.
It is important not to draw the wrong inference from the
said existential dependence, and to think it implies some
sort of relativism and subjectivism (in the most
pejorative senses of those terms) as regards issues of
truth and falsehood.
No: the ‘reasonableness’ of our basic concepts and
principles is the guarantee of their truth. To suggest some
other standard of judgment, or the equivalence of all
standards of judgment, is to tacitly claim such other
standard(s) to be somehow ‘reasonable’. A contradiction
is involved in such an attitude. Of course, you are free to
propose and accept contradictions, but you will have to
pay the cognitive and other consequences. As for me, I
prefer to stand by and rely on what is evidently
reasonable.
Fuzzy Logic
In some cases, X and notX are considered not to be
contradictory, because the term or proposition X is too
vague. If precisely what things X refers to is unclear, or
if the exact boundaries of some individual thing labeled
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195
X are uncertain, then obviously the same can be said for
the negative complement ‘not X’ (see diagram further
on). In such cases, the terms or propositions involved are
simply problematic.43
It is easy to see how such realization can lead to a
general critique of the human rational act of naming, and
to a philosophy of Nominalism. For, if we observe our
concepts carefully, we must admit that they are always in
process – they are never fully formed, never finalized.
Our ordinary knowledge is predominantly notional,
tending towards precise conception but never quite
attaining it. Thus, the meaning of words (or even of
wordless intentions) is in flux – it is becoming rather
than being.
This is not a merely epistemological critique, but one that
has ontological significance. What is being said here is
that things, the objects of our consciousness (be they
objective or subjective) are difficult, if not impossible, to
precisely pin down and delimit. This is true of concrete
individuals and of abstract classes. It is true of matter
(e.g. where does the body of a man end: if I breathe air in
or out, or swallow water or spit it out, at what stage does
the matter entering or exiting become or cease to be part
43
Note also that in some cases we face a range of
things, or different degrees of something, and we erroneously
call the extremes X and notX – whereas in fact if X is used for
one extreme, then notX must refer to all other degrees; and
vice versa, if notX is used for one extreme, then X must refer
to all other degrees; otherwise, we would be left with some
intermediate referents without name (i.e. as neither X nor
notX). It also happens that X and notX are made to overlap in
our thinking, so that X and notX are made to seem compatible.
These are simply common errors of concept formation; they
do not justify any denial of the laws of thought.
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THE LAWS OF THOUGHT
of ‘my body’?), and it is true of mind and of soul (who
knows where their respective limits are?).
Ultimately, we realize, everything is one continuum, and
the divisions we assume between things or classes are
ratiocinative and intellectual interpositions. We cannot
even truly imagine a fine line, a separation devoid of
thickness, so how can we claim to even mentally
precisely separate one thing from another? All the more
so in the physical realm, such division is impossible,
given that all is composed of continuous and endless
fields.
Another critical tack consists of saying that all our
experience (and consequently all our conceptual
knowledge) is illusory, in the way that a dream is illusory
(compared to awake experience). In a dream world, X
and not X can apparently both coexist without infringing
the law of non-contradiction. Distinctions disappear;
opposites fuse into each other.
But this is only superficially critical of our ordinary
knowledge. For what is said to coexist here are ‘the
appearance of X’ and ‘the appearance of not X’ – and not
‘X’ and ‘not X’ themselves. We have symbols, or stand
ins, or effects, instead of the objects themselves. So, this
is nothing that puts the law in doubt, but rather a
viewpoint that by its own terminology (reference to
illusion) confirms adherence in principle to that law.
Such reflections lead us to the idea of fuzzy logic, as
opposed to definite logic. The difference is illustrated in
the following diagram:
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197
Figure 20.1
Definite and Indefinite terminology.
Aristotle’s three laws of thought are aimed at a “definite
logic” model – in this model, terms and theses are in
principle clearly definable and knowable; or at least, this
is the assumption in most cases, though in a minority of
cases there might be some measure of temporary
vagueness and doubt. But this ideal is in practice rarely
met, and we should rather refer to a “fuzzy logic” model – wherein the assumption in most cases is that limits are
chronically unclear and hard to establish with certainty,
though exceptions to this rule must be acknowledged for
the sake of consistency.
Ordinarily, our reason functions in a self-confident
manner, from conviction to conviction, unfazed by the
changes in our ‘utter convictions’ that in fact occur over
time. In other words, we lay the stress on what we (think
we) know, and minimize what we consider still unknown
or the errors we made in the past. This is the approach of
definite logic, an essentially ‘deductive’ approach. The
idea of a fuzzy logic is that we ought to, on the contrary,
at the outset acknowledge our cognitive limitations and
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THE LAWS OF THOUGHT
the ongoing flux of knowing, and opt more thoroughly
for an ‘inductive’ approach.
According to this view, the logical perfection
presupposed by Aristotle is largely mythical. Our
concepts, propositions and arguments are, in practice,
usually exploratory, tentative, approximating, open
ended with regard to referents, open to change, of
uncertain pertinence and truth, and so forth. Our rational
faculty works by trial and error, constantly trying out
different overlays that might fit a momentarily glimpsed
reality, then noticing an apparent mismatch trying out
some more adjusted overlay, and so on without end.
Things are rarely quite the way we think of them, and yet
our thought of them is not entirely wrong. Hence, we
might well say that it is not correct to say that the
referents of X fit exactly what we mean by ‘X’; and it is
not correct to say that they do not all or wholly fit in.
Hence, it might be said that certain things are both X and
not X, and neither X nor not X – without really intending
to imply any contradiction, but only in the way of a
reminder to ourselves that we are functioning in shifting
sands.
Such a logical posture does not really constitute a denial
of the laws of thought. They continue to help us make
sense of things. Their precision helps us sort out the
vagueness and uncertainty we actually face in practice.
They give us an ontological and epistemological ideal we
can tend to, even if we can never hope to fully and
permanently match it.
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Stick to Logic
In the light of the aforementioned difficulties, some
logicians and philosophers are tempted to give up on all
rational knowledge, and more specifically the laws of
thought. However – and this is the point I am trying to
make here – this would be a tragic error. The error here is
to think that we humans can navigate within the sea of
phenomena and intuitions without the guiding star of the
laws of thought. Even if in particular cases these laws are
often hard to apply decisively, they help us do our best to
make sense of the world of appearances we face.
We have to stick with logic. It provides us with a
minimum of firm ground in the midst of the shifting
sands of experience and conception. Even if it is only an
ideal, a theoretical norm, its importance is crucial.
Without logic, we have no way to sort out changing
impressions and deal with the practical challenges of our
existence. Is that not the very definition of madness,
insanity?
rigid
Nevertheless, sticking to logic should not be taken to
signify
conventionality, or fearful closed
mindedness, or similar excesses of ‘rationalism’.
Sticking to logic does not exclude enlightened
consciousness, flowing with the current of life, having
faith, and similar liberating attitudes. Logic is a tool, not
an end in itself. To give up a useful tool is stupid; but it is
also stupid not to know when to put down the tool.
There is a stage in the life of the spirit when logical ifs
and buts become irrelevant, or even disturbing, and it is
wise to just be.
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THE LAWS OF THOUGHT
CHAPTER TWENTY ONE
Drawn from Logical and Spiritual Reflections (2008),
Book 3, chapter 5.
MISREPRESENTATION OF ARISTOTLE
Ups and Downs of Aristotelianism
Aristotle’s
three
laws
of
thought
are
often
misrepresented, in the service of some doctrine or other.
Often, nowadays, the motive is a desire to defend
Buddhist antinomies; some decades ago, the motive
might have been to defend Marxist contradictions; before
that, maybe Hegelian ones. Usually, the proposed
reading of Aristotle is unfair to him, a misrepresentation
of his evident intentions.
During the late Middle Ages in Europe, the authority of
Aristotelian philosophy was unmatched. The reason for
this was that before that period many of the works of
Aristotle (384-322 BCE) had been mostly lost to
Christian Europe; when they were rediscovered, the
superiority in many respects of the knowledge they
contained was such that his influence became great44.
44
The rediscovery occurred mostly by way of translation
into Latin (from Arabic, sometimes via Hebrew) of Greek
classical texts in the libraries of Moslem Spain. These
included works by Aristotle on physics, metaphysics and
ethics. Aristotle’s thought was also made known to the West
indirectly through commentators like Avicenna (Persia, 11th
century) and Averroes (Muslim Spain, 12th century). His
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201
But, as a result of that overwhelming belief in everything
Aristotelian, scientists of the Renaissance period and
after often had to struggle hard to overcome what had
become an academic bias.
It
could be argued, paradoxically, that Aristotle’s
influence on the Christian European mind was one of the
factors that led to the intellectual Renaissance;
nevertheless, just as students must rebel from teachers to
some extent to innovate and advance, an anti-Aristotelian
reaction had to occur. Many historians thus regard
Aristotelianism as the impetus of the Renaissance and
thus of modern science.
Note moreover, Aristotle himself was no rigid ideologue;
his approach was open-minded and adaptive, what we
now call ‘scientific’. Although many of his material
opinions45 have turned out to be false, they were quite
reasonable for his period of history – and for the Middle
Ages. Had he still been around in the modern era, he
would no doubt have adjusted his views.
Opposition to Aristotelianism, ranged over the special
sciences, more philosophical issues and logical aspects,
in no particular order. With regard to his logical work,
the greater emphasis Francis Bacon put on induction was
indeed a marked improvement; whereas, attempts in the
late 19th and early 20th centuries to supersede Aristotle’s
formal logic with more systematic deductive approaches
seem (to me at least) rather pretentious. The attempts,
influence reached its peak perhaps with the writings of
Thomas Aquinas (Italy, 13th century).
45
For example, his cosmological views, which led to the
Ptolemaic model that Copernicus and Galileo had to
overcome.
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THE LAWS OF THOUGHT
lately, to belittle or do away with Aristotle’s laws of
thought fall in the same category (again, in my opinion).
In many cases, criticisms of Aristotle’s thought were and
are of course justified. But in many cases, too, the critics
were and are just (I suspect) seeking a shortcut to
academic notoriety, taking an easy ride on the ongoing
wave (in some quarters) of ‘Aristotle bashing’. It is very
easy to be critical regarding someone who cannot answer
back; I daresay, if that genius were still around, they
would not dare.
Aristotle Bashing
A case in point (taken at random) is the following
presentation, drawn from an Internet site46. I quote:
The three laws of “formal logic” which Aristotle
set down in his Posterior Analytics are as
follows: (1) Law of Identity: Each existence is
identical
with itself; (2) Law of Non
contradiction: Each existence is not different
from itself; (3) Law of Excluded Middle: No
existence can be both itself and different from
itself.
Of course, nowhere in the Posterior Analytics, or
anywhere else in Aristotle’s known writings, are such
inane formulations of his laws of thought to be found.
Anyone who has read Aristotle knows this is not his
46
History and Theory of Psychology Course, by Paul F.
Ballantyne, Ph.D. “Aristotelian and Dialectical Logic”, in posted
May
2003
http://www.comnet.ca/~pballan//section1(210).htm.
at
(I
was
recently pointed to this website by a Buddhist correspondent
arguing against Aristotelian logic; that is how I came across it.)
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203
language or terminology, nor his thought or intent. He
does not speak of “existences” and is not concerned with
whether or not they are “identical with” or “different
from” themselves.
These statements are, admittedly, not presented as
verbatim quotations; but they are not, either, declared to
be mere readings or interpretations; they are made to
seem like loyal paraphrases. But they are not a fair
statement of what Aristotelian logic is about. It is not
about tautology or the lack of it, not even in an
ontological sense.
In Aristotle’s Posterior Analytics, we find the following
statements of the law of non-contradiction: “it is
impossible to affirm and deny simultaneously the same
predicate of the same subject”, and of the law of the
excluded middle: “every predicate can be either truly
affirmed or truly denied of every subject”.47
But the above author seems rather to base his
formulations on common statements of the laws of
thought, like “A is A”, “A cannot be not-A” and “Either
A or not-A”48. Such statements, however, are not meant
as
comprehensive expressions, but as shorthand
formulas; they are more like titles, stand-ins for fuller
statements that comprise all that can be said about these
laws. The simplest way to read them is as follows:
47
Both these statements are there (in Book 11) referred
to as laws, and the latter is specifically called the law of the
excluded middle. Translation by G. R. G. Mure. See
http://graduate.
gradsch.uga.edu/archive/Aristotle/Posterior_Analytics_(analyti
c).txt
48
Or at least the first two; for the third law he
misconceives altogether. See further on.
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THE LAWS OF THOUGHT
Something that is evidently A must be admitted
to be A.
Something admitted to be A cannot also be
claimed not to be A (i.e. no thing can be claimed
both to be A and not to be A).
And no thing can be claimed neither to be A nor
not to be A.
In this primary reading, note well, the term “A” is
everywhere a predicate, as Aristotle presents it, rather
than a subject, as it may seem. In all three cases, the tacit
subject of the proposition is “some thing”, an individual
thing under consideration, i.e. any apparent object of
cognition. Moreover, all three propositions are primarily
logical or epistemological statements, rather than
ontological ones. They tell us how to behave in our
discourse or cognition.
In a second phase, we can give “A” the role of subject
that it superficially has in the expressions “A is A” and
“A cannot be not-A”, and “Either A or not-A”. Such
perspective suggests a more ontological reading of these
laws, namely that every existent has a particular identity,
i.e. ‘a nature’, whatever that happen to be.
Each thing is something specific (say “A”), not just
anything whatsoever (“both A and not A”), nor nothing
at
all (“neither A nor not A”). It includes some
distinguishable aspects and excludes others: it is not
infinitely elastic in appearance. It neither includes nor
excludes
everything.
It
cannot
include
things
incompatible with it (“contradictions” of it). Its negation
may replace it, but nothing in between (no “middle”) can
replace both it and its negation.
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Note this: the law of the excluded middle could,
in analogy to the law of non-contradiction,
equally well be called the law of non-neutrality.
These laws respectively tell us that there is no
common ground and no neutral ground between
A and not-A. They ontologically together firmly
separate A and not-A, allowing of no wishy
washy togetherness or further possibility. They do
not however epistemologically exclude that we
might (occasionally, though not invariably) come
across contradiction or uncertainty in our
thinking.
Even such interpretations ought not, in any event, be
treated as the whole of the meaning of the laws of
thought, but more modestly as a beginning of
explication49. They make clear, anyway, that these laws
are not about equation or non-equation of things or
symbols with themselves, as the already mentioned
author’s formulations misleadingly suggest.
Additionally, the wording he proposes for the law of the
excluded middle “No existence can be both itself and
different from itself” – is formally wrong. This could be
construed as a statement of the law of non-contradiction,
perhaps, but the law of the excluded middle would (using
49
Many more issues arise in them, such as: what do we
mean by predicating “A” of something? What is the relation
between a label like “A” and what it intends? At what stage
may we consider “A” the exclusive label of that thing? Further:
so far, the laws have been expressed in terms of an individual
thing; but what about their application to kinds of things?
Clearly, these laws of thought are pregnant with the whole
philosophical enterprise!
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the same sort of language) have to be stated as “No
existence can be neither itself nor different from itself”.50
Clearly, Aristotle’s concern was whether the ideas we
form about the world are compatible with experiential
data and with each other. That is, one might say, an
interest in the intersection between appearance and
belief, or seeming reality and alleged knowledge. The
two components of consistency with experience and
other ideas correspond roughly to the tasks of inductive
and deductive logic, respectively.
Elsewhere on the same website51, the said author
apparently advocates, in lieu of his pseudo-Aristotelian
laws, something called “materialist dialectics,” which
“holds that the basic rules of correct thinking should
reflect a universe not in which the static and changeless
is at the core but in which change is at the core.” He goes
on to propose three questionable alternative “laws”,
which place change at the center of things.
Thus, the above quoted debatable presentation of the
laws of thought is used to convey the idea that Aristotle
had a static view of existence, and to propose instead a
50
Such a glaring formal misstatement of the law
discussed tells us much about the critic’s logical awareness,
or lack of it! When I advised him by e-mail of this formal error,
his response was at first flippant, then he made a small show
of open-mindedness, but finally he made no effort to correct
his statement. (N.B. I have just recently looked again at his
website and found out that he now seems to his credit to have
corrected this and other errors.)
51
In http://www.comnet.ca/~pballan/logic2.htm. He there
quotes statements like “What Aristotle sees as the most basic
characteristic of existence is static self-identity” by J.
Somerville, p. 45 in “The Nature of Reality: Dialectical
Materialism”, in The Philosophy of Marxism: An Exposition.
(Minneapolis: Marxist Educational Press, 1967/1983).
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more dynamic alternative set of laws. It is tendentious
rewriting of history.52
Aristotle’s Dynamism
In truth, Aristotle is throughout his work very much
concerned with dynamic becoming as well as with static
being. His laws of thought are precisely intended to help
the intellect cope with variety and change, and remain
lucid and poised in the midst of the cacophony of sense
impressions and ideas.
Consider, for instance the following statement drawn
from his Metaphysics53:
For a principle which every one must have who
understands anything that is, is not a hypothesis;
and that which every one must know who knows
anything, he must already have when he comes to
a special study. Evidently then such a principle is
the most certain of all; which principle this is, let
us proceed to say. It is, that the same attribute
cannot at the same time belong and not belong to
the same subject and in the same respect.
With characteristic intellectual accuracy, Aristotle
expresses the law of non-contradiction by saying that
nothing (i.e. no subject of a true proposition) can both be
52
For an understanding of the logic of change in
formal terms, see in my works: Future Logic, chapter 17, and
Volition and Allied Causal Concepts, chapter 14. See also,
Buddhist Illogic, chapter 6.
53
the
Book 4, Part 3. (Translated by W. D. Ross.) Posted in
Internet
Classics
website
http://classics.mit.edu/Aristotle/metaphysics.4.iv.html
at
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THE LAWS OF THOUGHT
and not-be the same thing (i.e. have and not have the
same predicate) in the same respect at the same time.
These last words are crucial to his statement, yet often
ignored by dishonest critics such as the above quoted. By
these words, Aristotle implied that something may well
be subject to both a predicate and its negation – in
different respects at the same time, or in the same respect
at different times, or in different respects at different
times.54
He is not ignoring that a given thing may have a variety
of aspects at once, or that it may change in various ways
over time. He is simply reminding us that in a given
location and time of its being, a thing cannot contradict
itself. His intent is therefore clearly not an attempt to
deny the existence of variety and change, but to affirm
the consistency that things nevertheless display at any
given place and time.
Evidently, the earlier quoted attempted reformulation of
the laws of thought as “Each existence is identical with
itself; not different from itself; and can[not] be both itself
and different from itself” is not only an inaccurate
rendition of Aristotle, but an extremely superficial one55.
54
Grass can be green and yellow, but not in exactly the
same places and times of its existence. Grass can mean what
the cows eat or what the hippies smoke, but these two same
words do not refer to the same things. If such differences of
perspective are impulsively or dogmatically ignored – well, that
does not prove that contradictions exist. To affirm
contradiction is to lack depth.
55
Due no doubt to the influence of dimwitted modern
symbolic logic, which makes every effort to reduce and limit
these complex laws to their simplest possible expression, thus
concealing most of their philosophical riches and depth. Why
do they wish to so simplify? In order to fit logic into their
simplistic “formal languages”, designed by people (like Gottlob
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Aristotle should be given the credit, respect and gratitude
due him for a timeless and irreplaceable achievement.
Frege) with hopelessly bureaucratic minds, who think that
standardizing
thought
processes makes them more
“scientific”. But science is not a deductive, Cartesian
enterprise; it is an inductive, evolutionary process. They claim
to go above common ‘intuition’; but actually, all they do is
permanently impose their own insights, and thereby inhibit
future insights in the field. Development of the science of logic
depends on alertness and flexibility, rather than on
institutionalization and rigidity.
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THE LAWS OF THOUGHT
CHAPTER TWENTY TWO
Drawn from Logical and Spiritual Reflections (2008),
Book 3, chapter 6.
NOT ON THE GEOMETRICAL MODEL
How to Validate Logic?
Since (or insofar as) the “geometrical model” of theory
justification involves arbitrary axioms, it is ultimately
conventional. If the first principles (“axioms”) of a body
of alleged knowledge cannot apparently be justified by
experience, but have to be based on mere speculation
(“arbitrary”), such principles must be admitted to be
without proof (“conventional”). If the axioms are
unproven, then logically so are all claims based on them.
This is freely admitted in the case of geometry (where for
instance Euclid’s fifth postulate may be replaced by
alternative assumptions), and similarly in other
mathematical
disciplines.
Here,
the
apparent
conventionality of certain axioms gives rise to the
possibility of alternative systems, all of which might
eventually be found useful in specific empirical contexts.
But such a liberal attitude is impossible with regard to
the science of Logic.
If we accept the geometrical model for Logic, then
Wittgenstein’s claim that “The propositions of logic are
tautologies… [and] therefore say nothing”56 is made to
56
In Tractatus, 6 (quoted in A Dictionary of Philosophy).
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211
seem true. But if we follow him, and admit that logic is
meaningless babbling, then we must regard his own
statement as meaningless – for, surely, it is itself
intended as a “proposition of logic”, indeed as the
highest principle of meta-logic! Granting that, it is as if
he has said nothing, and we can well ignore him and
move on.
Similarly, some critics have accused Aristotle of
‘begging the question’ in his defense57 of the laws of
non-contradiction and of the excluded middle, i.e. of
arguing in a circular manner using the intended
conclusion(s) as premise(s). Here again, we can more
reflexively ask: does that mean that the fallaciousness of
such petitio principii is an incontrovertible axiom of
logic? If the speaker is convinced by this rational
principle as an irreducible primary, why not also – or
even more so – by the second and third laws of thought?
Can he justify his antipathy to circularity without
committing circularity?
If Logic is not solidly anchored in reality through some
more rigorous process of validation, then all knowledge
is put in doubt and thus effectively invalidated. If all
knowledge is without validity, then even this very claim
to invalidity is without validity. The latter insight implies
that this skeptical claim is itself invalid, like all others,
note well. Therefore, since this skeptical claim is
paradoxical, i.e. self-denying, the opposite claim (which
is
not inherently paradoxical) must be admitted as
necessarily true. That is to say, we must admit that Logic
has undeniable validity. Only given this minimal
57
For instance, in Chapter IV of his Metaphysics,
Gamma.
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admission, does it become possible to admit anything
else as true or false.
I have said all this before again and again, but must keep
repeating it in view of the ubiquity of statements I
encounter these days in debates to the effect that
Aristotle’s three Laws of Thought are mere conventions.
To make such a statement is to imply one has some
privileged knowledge of reality – and yet at the same
time to explicitly suggest no such knowledge is even
conceivable. Thus, any such statement is self
contradictory, and those who utter it are either fools or
knaves, kidding themselves and/or others.
The said laws of thought must not be viewed as axioms
of knowledge within a geometrical model. The very idea
of such a model is itself an offshoot of Aristotle’s logic –
notably his first-figure syllogism, where a broad
principle or general proposition (the major premise) is
used to derive a narrower principle or particular
proposition (the conclusion). It follows that such a model
cannot be used to justify Logic, for in such case we
would be reasoning in circles and obviously failing to
anchor our truths in reality.
The Inductive Nature of Knowledge
The only way out of this quandary is to notice and
understand the inductive nature of all knowledge,
including deductive knowledge. The ground of all
knowledge is experience, i.e. knowledge of appearances
(material, mental and spiritual appearances of all sorts).
Without cognition of such data, without some sort of
given data whatever its ultimate status (as reality or
illusion), no knowledge true or false even arises.
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There is no such thing as “purely theoretical” knowledge:
at best, that would consist of words without content; but
upon reflection, to speak even of words would be to
admit them as experienced phenomena. To attempt to
refer, instead, to wordless intentions does not resolve the
paradox, either – for intentions that do not intend
anything are not. There has to be some experiential basis
to any knowledge claim. Whether the knowledge so
based is indeed true, or the opposite of it is true, is
another issue, to be sorted out next.
Logic comes into play at this stage, when we need to
discriminate between true and false theoretical
knowledge. We are always trying to go beyond
appearances – and that is where we can go wrong (which
does not mean we cannot sometimes be right). If we
stayed at the level of pure appearance – the
phenomenological level – we would never be in error.
But because we existentially need to surpass that stage,
and enter the rational level of consciousness, we are
occasionally evidently subject to error.
Moreover, it is very difficult for us to remain at the
purely phenomenological level: we seem to be
biologically programmed to ratiocinate, conceptualize
and argue; so we have little choice but to confront logical
issues head on. The principles of Logic, meaning the
laws of thought and the specific logical techniques
derived from them, are our tools for sorting out what is
true and what is false. We do not infer truth from these
principles, as if they were axioms containing all truth in
advance. Rather, these principles help us to discern truth
from falsehood in the mass of appearances. Without
some appearance to work with, logic would yield no
conclusion – it would not even arise.
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The validity of Logic is, thus, itself an inductive truth,
not some arbitrary axiom. Logic is credible, because it
describes how we actually proceed to distinguish truth
from falsehood in knowledge derived from experience.
No other logic than the standard logic of the three laws of
thought is possible, because any attempt to fancifully
propose any other logic inevitably gets judged through
standard logic. The three laws of thought are always our
ultimate norms of discursive conduct and judgment.
They point us to an ideal of knowledge we constantly try
to emulate.
The Crucial Role of Negation
This logical compulsion is not some deterministic force
that controls our brain or mind. It is based on the very
nature of the ratiocination that drives our derivation of
abstract knowledge from concrete appearances. The
primary act of ratiocination is negation: thinking “not
this” next to the “this” of empirical data. That act is the
beginning of all knowledge over and above experience,
and in this very act is the secret of the laws of thought,
i.e. the explanation as to why they are what they are and
not other than they are.
For, whereas the law of identity (A is A) is an
acknowledgment of experience as it presents itself, the
law of non-contradiction (nothing can be both A and not
A) and the law of the excluded middle (nothing can be
neither A nor not A) both relate to things as they do not
present themselves. These two laws define for us what
denial of A means – they set the standard for our
imagination of something not presented in experience at
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the time concerned. Note this well, for no one before has
noticed it that clearly.
Negation is the beginning of the “bigbang” of conceptual
and argumentative knowledge, the way we pass from
mere experience to concepts and principles; and the only
way to test and ensure that our rational framework
remains in reasonable accord with the givens of
experience is to apply the laws of thought. Negations are
never directly positively experienced: they are only
expressions that we have not experienced something we
previously imagined possible. There is no bipolarity in
concrete existence; bipolarity is a rational construct.58
The concept or term ‘not X’ can be interpreted to mean
‘anything except X’ (whether X here intends an
individual thing or a group of things). To deny the law of
non-contradiction is to say that this “except” is not really
meant to be exclusive – i.e. that ‘not X’ can sometimes
be included in ‘X’. Again, to deny the law of the
excluded middle is to say that this “anything” is not
really meant to be general – i.e. that besides ‘X’ there
might yet be other things excluded from ‘not X’. Thus, to
deny these laws of thought is to say: “I do not mean what
I say; do not take my words seriously; I am willing to
lie”.
58
This is made clear if we consider what we mean when
we say, for example, neither the dog nor the cat is in the room
we are in. The absence of the dog and the absence of the cat
look no different to us; what we actually see are the positive
phenomena only, i.e. the carpet, the desk, the chairs, etc. We
do not see a non-dog and a non-cat, or anything else that “is
absent” from this room, as if this is some other kind of
“presence”. (However, it does not follow that non-dog and
non-cat are equivalent concepts – for the cat may be present
when the dog is absent and vice versa.)
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CHAPTER TWENTY THREE
Drawn from Logical and Spiritual Reflections (2008),
Book 3, chapter 7.
A POISONOUS BREW
Truth vs. Proof
Despite its name, the modern theory of knowledge called
Intuitionism, developed by L.E.J. Brouwer59, can be
classed as an excessively deductive approach. It was,
significantly, originally intended and designed for
mathematics, and was thereafter by extrapolation applied
to all knowledge60. Equating for all intents and purposes
the logical modality of proof with that of fact,
“Intuitionist logic” rejects the law of the excluded middle
(and hence the inference of a positive statement from a
double negation).
Arguing that nothing can be claimed to be true if it is not
proved to be true, Intuitionism claims to accept the law
59
60
Holland, 1881-1966.
Such
extrapolations
are
unfortunate:
since
mathematics deals with special classes of concepts (notably
numerical and geometrical ones), insights concerning it cannot
always be generalized to all other concepts. Inversely,
comments concerning logic in general like the ones made
here do not exclude the possibility of specific principles for the
mathematical field. I am not a mathematician and do not here
intend to discuss that subject.
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of non-contradiction (since we cannot both prove A and
prove not-A), but denies the law of the excluded middle
(since we can both fail to prove A and fail to prove not
A). Thus, whereas Aristotle originally formulated these
laws with reference to facts (as nothing can be A and not
A, and nothing can be neither A nor not-A), Brouwer
focused on proof alone.
Many errors are involved in this change of perspective.
For a start, one can refute it on formal grounds: just as
we cannot both prove A and prove not-A, we cannot both
disprove A (= prove not-A) and disprove not-A (= prove
A). The fact that we can be in ignorance of both A and
not-A, i.e. uncertain as to which is true and which is
false, does not change the fact that A and not-A cannot
be both true or, equally, be both false. The two laws are
symmetrical and cannot be taken separately.
Note that Aristotle’s approach was to set ontological
standards that would serve as epistemological guides,
whereas Brouwer tried to place epistemology squarely
before ontology. The former implicitly allowed for
knowledge not at all dependent on rational processes,
viz. knowledge from experience, whereas the latter
considered all knowledge as dependent on reasoning, i.e.
as purely mental construction.
For classical logic, proof is a conflation of empirical
givens and conceptual constructs. To anchor concepts in
experience involves deductive methods, but the result is
always inductive. If we precisely trace the development
of our knowledge, we always find ultimate dependence
on empirical givens, generalization and adduction. There
is
no purely deductive truth corresponding to the
Intuitionist’s notion of “proved” knowledge. The
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THE LAWS OF THOUGHT
Intuitionist’s idea of proof is misconceived; it is not
proof.
Even an allegedly “purely deductive system” would need
to rely on our experience of its symbols, axioms and
rules. Thus, it cannot logically claim to be purely
deductive (or a priori or analytic, in Kantian terms), i.e.
wholly independent of any experience. Moreover, our
understanding of the system’s significance is crucial. A
machine may perform operations we program into it, but
these are meaningless without an intelligent human being
to consume the results. Brouwer’s assumptions are rife
with ignored or hidden issues.
Note too that Brouwer effectively regards “proved” and
“not proved” to be exhaustive as well as mutually
exclusive. This shows that he implicitly mentally relies
on the law of the excluded middle (and on double
negation), even while explicitly denying it. Certainly, we
have to understand him this way – otherwise, if the terms
proved and unproved (N.B. not to confuse with
disproved) allow for a third possibility, his theory loses
all its force. That is, something in between proved and
not proved (N.B. again, not to confuse with proved not)
would have to somehow be taken into consideration and
given meaning!
Brouwer’s denial of the law of the excluded middle is in
effect nothing more than a recognition that some
knowledge has to be classed as problematic. That was
known all along, and we did not need to wait for Mr.
Brouwer to realize it. The law of the excluded middle
does not exclude the possibility of problemacy, i.e. that
humans may sometimes not know for sure whether to
class something as A or not-A. On the contrary, the law
of the excluded middle is formulated on that very
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219
assumption, to tell us that when such problemacy occurs
(as it often does), we should keep looking for a solution
to the problem one way or the other.
The law of non-contradiction is similarly based on
human shortcoming, viz. the fact that contradictions do
occur occasionally in human knowledge; and its function
is similarly to remind us to try and find some resolution
to the apparent conflict. Note here the empirical fact that
we do sometimes both seem to prove A from one angle
and seem to prove not-A from another tack. In other
words, if we follow Brouwer’s formulation of the law of
non-contradiction, that law of thought should also be
denied!
The fact of the matter is that what we commonly call
proof is something tentative, which may turn out to be
wrong. The genius of classical logic is its ability to take
even such errors of proof in stride, and lead us to a
possible resolution. It is a logic of realism and
adaptation, not one of rigid dogmas.
Indeed, if there is anything approaching purely deductive
truth in human knowledge, it is the truth of the laws of
thought. So much so, that we can say in advance of any
theory of knowledge that if it postulates or concludes that
any law of thought is untrue – it is the theory that must
be doubted and not these laws. Such antinomy is sure
proof that the theory is mixed-up in some way (just as
when a theory is in disagreement with empirical facts, it
is put in doubt by those facts).
In the case of Intuitionism, the confusion involved is a
misrepresentation of what constitutes “proof”. Only
people ignorant of logic are misled by such trickery.
Why on earth would we be tempted to accept Brouwer’s
idea of “proof” in preference to the law of the excluded
220
THE LAWS OF THOUGHT
middle (which this idea denies)? Has he somehow
“proved” his idea, or even just made it seem less
arbitrary, more credible or more logically powerful than
the idea of the law of the excluded middle? His view of
proof is not even “proved” according his own standards –
and it is certainly not proved (indeed it is disproved) by
true logic.
Consider the implications of denials of the second and
third laws of thought on a formal level. To deny the law
of non-contradiction only is to wish to logically treat X
and not-X as subcontraries instead of as contradictories.
To deny the law of the excluded middle only is to wish to
logically treat X and not-X as contraries instead of as
contradictories. To deny both these laws is to say that
there is no such thing as negation. All the while, the
proponent of such ideas unselfconsciously affirms some
things and denies others.
Reflect and ask yourself. If X and not-X cannot be
contradictories, why should they be contraries or
subcontraries? On what conceivable basis could we say
that incompatibility (as that between X and not-X) is
possible, but exhaustiveness is not; or vice versa? And if
nothing can be incompatible and nothing can be
exhaustive – what might negation refer to? It is clear that
all such proposed antinomial discourse is absurd, devoid
of any sort of coherence or intelligence. It is just a
manipulation of symbols emptied of meaning.
Double Negation
The deeper root of Intuitionist logic is of course a failure
to understand the nature of negation. What does ‘not’
mean, really? How do we get to know negative terms,
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221
and what do they tell us? How does negation fit in the
laws of thought? I will not go far into this very important
field here, having already dealt with it in detail in the
past61; but the following comments need be added.
Another, related weakness of Intuitionist is ignorance of
inductive logic. As already stated, Brouwer functioned on
an essentially deductive plane; he did not sufficiently
take induction into consideration when formulating his
ideas. In a way, these were an attempt to get beyond
deductive logic; but his analysis did not get broad
enough.
This can be illustrated with reference to double negation.
On a deductive plane, negation of negation is equivalent
to affirmation. This is an implication and requirement of
the laws of thought. However, on an inductive plane, the
matter is not so simple, because negation is always a
product of generalization or adduction. That is to say,
‘not’ always means: ‘so far, not’; i.e. it is always relative
to the current context of knowledge.
What distinguishes deductive from inductive logic is that
in the former the premises are taken for granted when
drawing the conclusion, whereas in the latter the
uncertainty of the premises and therefore of the
conclusion are kept in mind. Thus, deductively: ‘not not
X’ means exactly the same as, and is interchangeable
with, ‘X’; but inductively: the premise ‘not not X’ tends
towards an ‘X’ conclusion, but does not guarantee it.
Since ‘not X’ really means ‘we have looked for X but not
found it so far’, it always (with certain notable
exceptions) remains somewhat uncertain. On the other
hand, a positive, namely ‘X’ here, can be certain insofar
61
In Chapter 9 of my book Ruminations. I strongly
recommend the reader to read that crucial essay.
222
THE LAWS OF THOUGHT
as it can be directly perceived or intuited (and in this
context, the experience ‘not found’ must be considered as
a positive, to ensure theoretical consistency).
If ‘not X’ is always uncertain to some degree, it follows
that ‘not not X’ is even more uncertain and cannot be
equated in status to the certainty inherent in ‘X’ (if the
latter is experienced, and not merely a conceptual
product). Double negation involves two generalizations
or adductions, and is therefore essentially an abstraction
and not a pure experience.
Moreover, the expression ‘not (not X)’ inductively
means ‘we have looked for the negation of X and not
found it’. But since ‘not X’ already means ‘we have
looked for X and not found it’, we may reasonably ask
the question: is the path of ‘not not X’ the way to find
‘X’ in experience? Obviously not! If we seek for X, we
would directly look for it– and not indirectly look for it
through the negation of its negation.
Note, too, that having found ‘X’ in experience we would
consider ‘not not X’ to follow with deductive force, even
though the reverse relation is (as already mentioned)
much weaker.
Thus, the problem of double negation posed by Brouwer
is a very artificial one, that has little or nothing to do with
actual cognitive practice. Not only are the laws of
thought nowhere put in doubt by this problem – if we are
careful to distinguish induction from deduction – but it is
not a problem that would actually arise in the normal
course of thought. It is a modern sophistical teaser.
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223
CHAPTER TWENTY FOUR
Drawn from Logical and Spiritual Reflections (2008),
Book 3, chapter 8.
THE GAME OF ONE-UPMANSHIP
Misleading Symbolism
People who think the law of non-contradiction and/or the
law of the excluded middle is/are expendable have
simply not sufficiently observed and analyzed the
formation of knowledge within themselves. They think it
is just a matter of playing with words, and they are free
to assert that some things might be “both A and not A”
and/or “neither A nor not A”. But they do not pay
attention to how that judgment arises and is itself judged.
They view “A is A”, etc.62, as verbal statements like any
other, and think they can negate such statements like all
others, saying “A is not A”, etc. But in fact, negation is
not possible as a rational act without acceptance of the
significance of negation inherent in the second and third
laws of thought, in comparison to the first law of thought.
To say “not” at all meaningfully, I must first accept that
62
Incidentally, I notice people on the Internet nowadays
labeling the three laws of thought (LOT): LOI, LNC and LEM,
for brevity’s sake. Sure, why not?
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THE LAWS OF THOUGHT
“A cannot be not A” and that “there’s no third alternative
to A and not A”.63
To try to introduce some other (less demanding)
definition of negation is impossible, for true negation
would still have to be thought of (in a hidden manner or
using other words). Inventing a “many-valued logic” or a
“fuzzy logic” cannot to do away with standard two
valued logic – the latter still remains operative, even if
without words, on a subconscious level. We have no way
to think conceptually without affirmation and denial; we
can only pretend to do so.
Many “modern” logicians are so imprisoned by symbolic
logic that they have lost contact with the intended
meanings of their symbols. For this reason, the symbols
‘X’ and ‘not X’ seem equivalent to them, like ‘X’ and
‘Y’. But for classical logicians, a term and its negation
have a special relationship. The negation of X refers to
all but X, i.e. everything that is or might be in the whole
universe other than X.64
63
Some logicians accept the law of non-contradiction as
unavoidable, but consider the law of the excluded middle as
expendable: this modern notion is quite foolish. Both laws are
needed and appealed to in both deductive logic and in
inductive logic. They do not only serve for validation (e.g. of
syllogisms or of factorial inductions), but they generate
questions and research (e.g. what does this imply? or what
causative relation can be induced from that?). Moreover, they
are mirror images of each other, meant to complement each
other so as to exhaust all possibilities, and they ultimately
imply each other, and both imply and are implied by the law of
identity.
64
Note that difference does not imply incompatibility.
Two things, say X and Y, may be different, yet compatible – or
even imply each other. We are well able to distinguish two
things (or characteristics of some thing(s)), even if they always
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Figure 24.1
Visualizations of Negation.
The diagram above illustrates how differently these
people effectively visualize negation:
Obviously, if a person mentally regards ‘X’ and ‘not X’
as commensurate, he will not understand why they
cannot both be affirmed or both be denied at once; the
second and third laws of thought will seem to him
prejudicial and conventional. To return to a rational
viewpoint, that person has to become conscious of the
radical intent of the act of negation; it leaves no space for
mixtures or for additional concoctions.
Bipolar logic is not a mere “convention”, for the simple
reason that making a convention presupposes we have a
choice of two or more alternatives, whereas bipolarity is
the only way rational thought can at all proceed. We do
occur in tandem and are never found elsewhere. Their
invariable co-incidence does not prevent their having some
empirical or intellectual difference that allows and incites us to
name them differently, and say that X is not the same thing as
Y. In such case, X as such will exclude Y, and not X as such
will include Y, even though we can say that X implies Y, and
not X implies not Y.
226
THE LAWS OF THOUGHT
not arbitrarily agree bipolarity, because it is inherent in
the very asking of the question. To claim something to be
conventional is already to acknowledge the conflict
between it and the negation of it, and the lack of anything
intelligible in between the two.
The motive behind the attempts of some thinkers to deny
the laws of thought (i.e. the laws of proper affirmation
and denial) is simply an ego ambition to “beat the
system”, or more specifically (in the case of Western
philosophers) to surpass Aristotle (the one who first
made these laws explicit objects of study). “You say X? I
will ‘up the ante’ and say Not X (etc.) – and thus show I
am the greatest!”
Upping the Ante
This is not mere perversity – but a sort of natural denial
instinct gone mad. For, funnily enough, to deny some
suggestion (including the suggestion there are three laws
of thought) is in the very nature of conceptual knowing, a
protective mechanism to make sure all alternative
interpretations of fact are taken into consideration. This
is precisely the faculty of negation – the very one which
gives rise to the need for the laws of thought! The
problem here is that it is being turned on itself – it is
being over-applied, applied in an absurd way.
This can go on and on ad infinitum. Suppose I say “A”
(meaning “A but not notA”), you answer “not A”
(meaning “notA but not A”)65; I reply “both A and
65
Note that if we start admitting the logical possibility of
“A and notA” (or of “not A and not notA”), then we can no
longer mention “A” (or “notA”) alone, for then it is not clear
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notA”, you oppose “neither A nor notA”; what have we
said or achieved? Perhaps I will now say: “all of these
four alternatives”; and you will reply: “none of these four
alternatives”. Then I trump you, asserting: “both these
last two alternatives” and you answer: “neither of them”.
And so forth. Whither and what for?
A more complex version of the same game of one
upmanship can be played with reference to the laws of
thought:
A is A (affirming the law of identity).
A is not A (denying the law of identity).
Both (1) and (2). A is A, and A is not A.
(disregarding the law of non-contradiction).
Neither (1) nor (2). A is not A, and A is not not A
(disregarding the law of the excluded middle).
Both (3) and (4).
Neither (3) nor (4).
Both (5) and (6).
Neither (5) nor (6).
And so on and so forth.
Thus for the first law of thought; and similarly for the
other two. We do not merely have a choice of four
alternatives (the first four in the above list), a so-called
‘tetralemma’, but an infinite choice of denials of denials
of denials… How would we even evaluate the meaning of
all these alternatives without using the laws of thought?
They would all be meaningless, because every proposed
interpretation would be in turn deniable.
whether we mean “A with notA” or “A without notA” (etc.). This
just goes to show that normally, when we think “A” we mean
“as against notA” – we do not consider contradictory terms as
compatible.
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THE LAWS OF THOUGHT
Thus, the attempt to propose a radically “alternative
logic”, instead of the standard (Aristotelian) logic, is
really the end of all intelligible logic, the dissolution of
all rationality. It is not a meaningful option but a useless
manipulation of meaningless symbols. None of it makes
any sense; it is just piling up words to give an optical
illusion of depth. People who engage in such moronic
games should clearly not be granted the status of
“logicians”.
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CHAPTER TWENTY FIVE
Drawn from Logical and Spiritual Reflections (2008),
Book 3, chapter 9.
IN BUDDHIST DISCOURSE
Mitigating Clarifications
Opposition by some Western logicians to (one or more
of) the laws of thought is mostly naïve symbolic games,
without any profound epistemological or ontological
reflection; of quite another caliber is the opposition to
these laws found in some Buddhist literature66. But we
can, with a bit of effort of reflection, explain away the
apparent antinomies in their discourse.
When Buddhist philosophers make statements of the
form “not X and not notX”, they should not (or not
always) be viewed as engaging in antinomy, or in
rejection of the laws of thought. Rather, such statements
are abridged expressions intending: “don’t look for X
and don’t look for not X”, or “don’t think X and don’t
think not X”, or “don’t say X and don’t say not X”, or
66
I
am of course over-generalizing a bit here, for
emphasis. There are of course more savvy Western logicians
and less savvy Oriental (including Buddhist) logicians. A case
of the latter I have treated in some detail in past works is
Nagarjuna.
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THE LAWS OF THOUGHT
“don’t attach to X and don’t attach to not X”, or the
like.67
When thus clarified, statements superficially of the form
“neither X nor not X” (or similarly, in some cases, “both
X and not X”) are seen to be quite in accord with logic.
For the laws of thought do not deny that you cannot look
for ‘X’ and for ‘not X’, or for that matter for ‘both X and
not X’, or even ‘neither X nor not X’. Similarly, with
regard to thinking this or that, or to claiming this or that,
or to attaching to this or that, etc.
The laws of logic would only say that you cannot at once
‘look for X’ and ‘not look for X’, and so forth. It does
not say you cannot at once ‘look for X’ and ‘look for not
X’, and so forth. The latter situation merely asserts that
the issue of X or not X ought to be left problematic. An
unsolved problem is not an antinomy. The most we can
say is that whereas Buddhism might be deemed to enjoin
us to accept such uncertainty as final, Western logic
would recommend pressing on to find a solution of sorts.
Thus, in some cases, the apparent contradictions and
inclusions of middle terms in Buddhist philosophy (and
similarly in some other texts) are merely verbal. They are
due to inaccuracy in verbal expression, omitting
significant implicit aspects of what is really meant. The
reason for such verbal brevity is that the focus of such
statements is heuristic, rather than existential. They are
merely meant as “skillful means” (to the end of
Realization), not as factual descriptions. That is to say,
67
For example, the following is a recommendation to
avoid making claims of truth or falsehood: “Neither affirm nor
deny… and you are as good as a enlightened already.” Sutra
of Supreme Wisdom, v. 30 – in Jean Eracle (my translation
from French).
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they are statements telling the subject how to proceed
(cognitively, volitionally or in valuation), rather than
telling him/her how things are.
Examples
To give an actual example from Buddhist literature, I
quote the following passage from the Wake-up Sermon
attributed to Bodhidharma:
“Mortals keep creating the mind, claiming it
exists. And arhats keep negating the mind,
claiming it doesn’t exist. But bodhisattvas and
buddhas neither create nor negate the mind. This
is what’s meant by the mind that neither exists
nor doesn’t exist… called the Middle Way.”68
When we face an unresolved contradiction or an
unsolved problem of any sort, we are from the point of
view of knowledge in front of a void. This ‘emptiness’
can be looked upon with anxiety, as a precipice, as a
deficiency of means to deal with the challenges of life.
Or it may be viewed as something pregnant with
meaning, a welcome opportunity to dive fearlessly into
infinity. The former attitude gives rise to Western
science, the latter to Zen meditation.
Or again, consider the following quotation from Huang
Po’s teaching:
68
P. 53. This passage is particularly clear in its
explanation of “neither exists nor does not” as more precisely
“is neither created nor negated”. Whereas the former is
logically contradictory, the latter is in fact not so. What is
advocated here is, simply put, non-interference.
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THE LAWS OF THOUGHT
“If only you will avoid concepts of existence and
non-existence in regard to absolutely everything,
you will then perceive the Dharma.” (P. 43.)
Here again, the meaning is clear. The Zen master is not
here denying existence or non-existence or both; he is
just telling us not to engage in judgments like ‘this
exists’ or ‘this does not exist’ that are inherent to all
conceptualization. He refers to such judgments as
“dualism”, because they require a decision between two
alternatives. Clearly, Huang Po’s statement is not a
formally contradictory ontological proposition, but a
prima facie coherent epistemological injunction not to be
concerned with judging whether what one experiences is
real or unreal.
Admittedly, some Buddhists69 do take such a statement
as implying that existence does not exist, or that it both
exists and does not exists, or neither exists nor does not
exist. But as far as commonsense logic is concerned,
existence does exist – i.e. whatever is, is (Aristotle’s law
of identity). Any clear denial of this fundamental truth
would just be self-contradictory – it would deliberately
ignore the fact and implications of its own utterance (i.e.
that a statement has been made, alleging a truth, by
someone to someone, etc.)
69
In truth, Huang Po is among them, since elsewhere he
piously states: “from first to last not even the smallest grain of
anything perceptible has ever existed or ever will exist” (p.
127). This is a denial of all appearance, even as such. Of
course, such a position is untenable, for the existence of mere
appearance is logically undeniable – else, what is he
discussing? Before one can at all deny anything, one must be
able to affirm something. Also, the act of denial is itself an
existent.
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Doing Rather than Talking
More precisely, in the present context, we must
acknowledge that whatever but appears, certainly exists – whether it is eventually judged to be real or illusory.
On this basis, we can reasonably interpret Huang Po (at
least in the citation above) as simply saying “do not ask
whether some particular (or general) thing exists or not,
or whether it is real or not, because such questioning
diverts your attention from a much more important
insight into the nature of being”.
It should be added that, even though I above admit that
Huang Po’s position is prima facie coherent, it is not so
coherent upon further scrutiny. He cannot strictly
speaking utter a statement without using concepts and he
cannot be understood by us without use of our conceptual
faculty. All discourse is conceptual, even anti-conceptual
discourse. That is, in the very act of preaching abstinence
from concepts, he is in fact not practicing what he
preaches.
This shows that even persons presumed to be enlightened
need concepts to communicate, and also that such
conceptuality does not apparently (judging by the claims
of those who practice it) affect their being enlightened.
So concepts cannot be intrinsically harmful to
enlightenment, and the claim that they must be eschewed
is internally inconsistent! This is not a game of words (as
some might argue) – it is a logical insight that cannot be
waved off. One can only at best argue against excessive
conceptualization.
In any event, it must be understood that Buddhist anti
conceptual philosophy is aimed at psychological
development: it is primarily a “way” or “path”. Its focus
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THE LAWS OF THOUGHT
is how to react to ordinary experiences, so as to get to see
the ultimate reality beyond them. It refers to the object
(X or not X), not independently (as in most Western
logic), but as an object of the Subject (i.e. sought out,
thought of, claimed, or attached to by the subject-agent).
The latter ‘subjectivity’ (i.e. dependence on the subject
agent) is very often left implicit, simply because it is so
pervasive. Notwithstanding, there are contexts in which
the intent is more ‘objective’ than that70.
Imprecise Language
It should also be noticed that many of the contradictions
or paradoxes that Buddhist philosophers produce in their
discourse are due to their tendency to make apparently
general statements that in the last analysis turn out to be
less than all-inclusive. Even while believing that there is
more to the world as a whole than what is commonly
evident, they formulate their ideas about the phenomenal
world as unqualified universal propositions. There are
many examples of this tendency.
“All is unreal”, says the Dhammapada (v. 279). Calling
all unreal or illusory is of course possible in imagination,
i.e. verbally – by taking the predicate ‘unreal’ or
‘illusory’ from its original legitimate subjects of
application and applying it to ‘all’ subjects. Implicit in
this manipulation is an analogy – i.e. a statement that just
as within the realm of appearance some items are found
not real and labeled illusory, so we can project a larger
70
For a start, to claim a means as skillful is a kind of
factual description.
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realm in which the whole current realm of appearance
would seem unreal.
This explains how people assimilate that oft-repeated
Buddhist statement, i.e. why it seems thinkable and
potentially plausible. But it does not constitute logical
justification for it. The only possible justification would
be to personally experience a realm beyond that of
ordinary experience. Even then, the logically consistent
way to make the statement would be “all ordinary
experience is unreal” (because saying just “all” would of
course logically have to include the extraordinary
experience).
Another frequently found example is “existence is
suffering71.” This statement is true, all too true, about the
world we commonly experience, i.e. the world of
material and mental phenomena. If one is observant, one
discerns that we are always feeling some unpleasantness
in the background of our existence. No earthly happiness
is ever complete, if only because it is tenuous. Even
sexual pleasure or orgasm – which more and more of my
contemporaries seem to regard as the ultimate ecstasy
and goal of existence – is a pain of sorts72.
Buddhism has displayed extreme wisdom in emphasizing
the fact of suffering, because once we realize it we are by
this very simple realization already well on the way to
71
This is the usual translation of the Sanskrit term is
dukkha. This connotes not only physical and emotional pain,
but more broadly mental deficiencies and disturbances, lack of
full satisfaction and contentment, unhappiness, absence of
perfect peace of mind.
72
If
we are sufficiently attentive, we notice the pain
involved in sexual feelings. Not just a pain due to frustration,
but a component of physical pain in the very midst of the
apparent pleasure.
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THE LAWS OF THOUGHT
being freed of suffering. If one were visiting hell, one
would not expect to experience heaven there; likewise, it
is natural in this halfway world to experience some
suffering. I used to suffer a lot at the sight of people
getting away with injustices or other ugly acts; but lately
I just tell myself: “well, I am in samsara and this is
normal behavior in samsara73 – so long as I am here, I
have to expect this kind of unpleasant experience and
take it in stride!”
But the statement “existence is suffering” is wrongly
formulated from the logical point of view, and for that
reason it is bound to lead to paradoxes. For if we believe
(as Buddhists do) that suffering can eventually be
overcome (specifically, when nirvana is attained), then
the truth of suffering must be formulated less universally
as:
“mundane existence is suffering”. The usual
formulation of the first Noble Truth, “existence is
suffering,” is not intended to be as all-inclusive as it
seems – for suffering disappears according to the third
Noble Truth when we become enlightened. Therefore, to
make the former consistent with the latter, it has to be
rephrased more restrictively.
Another example of the tendency to artificially refuse to
count the experience of enlightenment as part of the
world as a whole is the idea that enlightenment takes us
“beyond good and evil”. This is logically incorrect – if
we regard enlightenment as the summum bonum, the
ultimate good (which we do, if we enjoin people to prefer
it to all other pursuits).
73
Or, using Jewish terminology: “I am in galut (exile, in
Hebrew), and such unpleasantness is to be expected here”.
Note in passing, the close analogy between the Buddhist
concept of samsara and the kabbala concept of galut.
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The phrase “beyond good and bad” is intended to stress
the practical problem that pursuing good is as much a
form of attachment as avoiding evil. The pursuit of
worldly good things is ultimately bad, because it just ties
us to this world and subjects us to the bad in it. And
indeed, even the pursuit of liberation from this world, i.e.
of an otherworldly good, is problematic, in that it
involves the wrong attitude, a grasping or clinging
attitude that is not conducive to success. All this is true,
but tends towards paradox.
To avoid confusion, we must simply rephrase our goal as
“beyond pursuit of good and avoidance of evil”. That is
to say, we must admit that nirvana is ‘good’ in the most
accurate sense of the term, while what we call ‘good’ in
the world of samsara (i.e. wealth position, power, sensual
pleasure, etc.) is really not much better than what we call
‘bad’. Alternatively, we should distinguish good in an
absolute sense (the good of nirvana) and good in a
relative sense (the goods within samsara). Relative goods
would then to be classified as not so good from the
absolute point of view.
The result of this change of perspective is that, rather
than view existence as fundamentally bad (due to
suffering), we may now view it as fundamentally good
(since nirvana underlies all samsaric existence). Our
common view and manner of existence is just an error of
sorts, causing us much suffering; if we but return to
correct cognition and behavior, we will experience the
natural good at the core of all things. Here, the illusory
good and evil of the mundane are irrelevant, and we are
fully immersed in the real good.74
74
We could read S. Suzuki as saying much the same
thing, when he says: “Because we are not good right now, we
238
THE LAWS OF THOUGHT
To conclude – Buddhist discourse often leads to paradox
or contradiction because it insists on using terms in
conventional ways and uttering generalities that apply to
only part of the totality of experience (namely, the
mundane part, to the exclusion of the supramundane
part). To avoid the doctrinal problems such discursive
practices cause, we must either clearly specify the terms
used as having such and such conventional senses, or
particularize statements that were formulated too
generally (i.e. which did not explicitly take into
consideration the data of enlightenment).
want to be better, but when we attain the transcendental mind,
we go beyond things as they are and as they should be. In the
emptiness of our original mind they are one, and there we find
perfect composure” (p.130).
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CHAPTER TWENTY SIX
Drawn from Logical and Spiritual Reflections (2008),
Book 3, chapter 20.
THE LAWS OF THOUGHT IN MEDITATION
Cognitive Virtues
The three laws of thought are commonly considered by
many current commentators75 to be (at best) only
relevant to rational discourse, and not relevant at all or
even antithetical to meditation and all the more so to its
finale of enlightenment. Nothing could be further from
the truth, as will now be explicated.
The laws of thought are principally ‘moral’ imperatives
to the thinker, enjoining him or her to have certain
cognitive attitudes in all processes of thought. They call
upon the thinker to make an effort, so as to guarantee
maximum efficiency and accuracy of his or her thoughts.
The ‘metaphysical’ aspect of the laws of thought is a
substratum and outcome of this practical aspect.76
The law of identity is a general stance of ‘realism’.
In discursive thought, this means: to face facts; to
observe and think about them; to admit the factuality
75
76
Judging by Internet postings and debate on this topic.
It could also be said that the two aspects are ‘co
emergent’, mutually significant and equally important. But here
I wish to stress the psychological side of the issue.
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THE LAWS OF THOUGHT
of appearances as such and that of logical arguments
relating to them; to accept the way things are (or at
least the way they seem to be for now), that things
are as they are, i.e. whatever they happen to be; and
so on.
Clearly, these same cognitive virtues are equally
applicable to meditation practice, which requires
awareness, receptivity and lucidity. The antitheses of
these attitudes are evasiveness, prejudice and
obscurantism, resulting in “sloth and torpor”77.
At the apogee of meditation, in the enlightenment
experience, this is expressed as (reportedly)
consciousness of the “thus-ness” (or “such-ness”) of
“ultimate reality”.
The law of non-contradiction is a general stance of
‘coherence’ (which is an aspect of ‘realism’).
In discursive thought, this means: while giving initial
credence to all appearances taken singly, not to
accept two conflicting appearances as both true (or
real), but to place one or both of them in the category
of falsehood (or illusion); to seek to resolve or
transcend all apparent contradictions; to pursue
consistency in one’s concepts and theories; to reject
inconsistent ideas as absurd and self-contradictions as
untenable nonsense; and so on.
Clearly, these same cognitive virtues are equally
applicable to meditation practice, which requires
harmony, balance and peace of mind. The antitheses
of these attitudes are conflict, confusion and neurosis
77
See Kamalashila, p. 253.
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(or madness), resulting in “restlessness and
anxiety”78.
At the peak of meditation, in the enlightenment
experience, this is expressed as (reportedly) the “one
ness” (monism or monotheism) of “ultimate reality”.
The law of the excluded middle is a general stance
of ‘curiosity’ (which is also an aspect of ‘realism’).
In discursive thought, this means: engaging in
research and study, so as to fill gaps in one’s
knowledge and extend its frontier; engaging in
speculation and theorizing, but always under the
supervision and guidance of rationality; avoiding
fanciful escapes from reality, distorting facts and
lying to oneself and/or others; accepting the need to
eventually make definite choices and firm decisions;
and so on.
Clearly, these same cognitive virtues are equally
applicable to meditation practice, which requires
clarity, judgment and understanding. The antitheses
of these attitudes are ignorance, uncertainty and
delusion, resulting in “doubt and indecision”79.
At the pinnacle of meditation, in the enlightenment
experience, this is expressed as (reportedly) the
“omniscience” of “ultimate reality”.
Thus, I submit, rather than abandon the laws of thought
when we step up from ordinary thinking to meditation,
and from that to enlightenment, we should stick to them,
while allowing that they are expressed somewhat
differently at each spiritual stage. Whereas in discursive
78
79
See Kamalashila, p. 249.
See Kamalashila, p. 258.
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THE LAWS OF THOUGHT
thought awareness is expressed by intellectual activity, in
meditation the approach is gentler and subtler, and in
enlightenment we attain pure contemplation.
When such final realization is reached80, the laws of
thought are not breached, but made most evident. “Thus
ness” is the essence of existence; it is the deepest stratum
of identity, not an absence of all identity. “One-ness” is
not coexistence or merging of opposites, but where all
oppositions are dissolved or transcended. “Omniscience”
is not in denial of ordinary experience and knowledge,
but their fullest expression and understanding. What in
lower planes of being and knowing seems obscure,
divergent and uncertain, becomes perfect at the highest
level.81
80
I submit, on the basis of my own limited experience,
but also out of logical expectation of consistency between all
levels of being. I think many people more knowledgeable than
me would agree with the descriptions here given of the higher
realms.
81
Buddhist, and especially Mahayana, philosophers
often stress that nirvana (the common ground of all being) and
samsara (the multiplicity of changing appearances) are
ultimately one and the same. Even while admitting this, we
must remain aware of their apparent difference. The whole
point of the philosophical idea of monism (“nirvana”) is of
course to resolve the contradictions and gaps inherent in the
experience of plurality (“samsara”). At the same time, the one
ness of nirvana is in a sort of conflict with the multiplicity of
samsara. We must somehow both admit and ignore this
tension. In truth, all this remains an unsolved problem at some
level.
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The Absurdity of the Antitheses
Those teachers or commentators who claim that the laws
of thought are abrogated once we transcend ordinary
discourse are simply misinterpreting their experiences.
Either their experience is not true “realization”, or their
particular interpretation of their realization experience is
just an erroneous afterthought that should not be viewed
as part of the experience itself.
Instead of the laws of identity, non-contradiction and
exclusion of any middle, they propose a law of non
identity, a law of contradiction, and a law of the included
middles! According to them, the ultimate reality is that
nothing has an identity, all contradictories coexist quite
harmoniously, and there may be other alternatives
besides a thing and its negation!
They adduce as proofs the Buddhist principles of non
selfhood, impermanence and interdependence.
But they cannot claim that something has no “nature”
whatsoever, for then what is that “something” that they
are talking about? If it is truly non-existent, why and how
are we at all discussing it and who are we? Surely these
same people admit the existence of an “ultimate reality”
of some sort – if only a single, infinite, universal
substratum82. They call it “void” or “empty”, but surely
82
The “great self” or “ocean of permanence”, to use the
words of Dogen (p. 267. Note that Dogen is not here saying
there is no such thing, but is stressing that we do not – as
some people claim – automatically all return there after death,
but rather are subject to various rebirths according to our
respective karmas; he is implying that to get there is hard-won
realization, not something given gratis to all comers). Some
identify this underlying ultimate reality with the “Deus sive
Natura” of Baruch Spinoza (Holland, 1632-77). But I hasten to
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THE LAWS OF THOUGHT
such a negation is not logically tenable without the
admission that something positive is being negated; a
negation can never be a primary given.
Similarly, we might argue, “impermanence” means the
impermanence of something and “interdependence”
means the interdependence of two or more things. They
cannot claim infinite impermanence, without admitting
the extended existence in time of something however
temporary; and they cannot claim a universal
interdependence, without admitting causal connections
between actual facts.
There is an unfortunate tendency here to use words
without paying attention to their relational implications.
Another example of this practice is to speak of
“consciousness” (or perception or thought or some such
cognitive act), without admitting that this implies
consciousness of something (called an object) by
something (called the Subject).
This is done deliberately, to conform with the ideological
prejudice that there is no cognizing self and nothing to
cognize. Similarly, so as not to have to mention the
Agent willing an action, volition is concealed and the
action is made to appear spontaneous or mechanical.
They refuse to admit that someone is suffering, thinking,
meditating or becoming enlightened.
Another claim often made is that our common experience
of the world is like a dream compared to ultimate reality.
The implication being that the laws of thought are not
add that I do not subscribe to Spinoza’s equation of God and
Nature, which implies that God is like Nature subject to
determinism. For me, as in normative Judaism, God is the
free, volitional creator of Nature. He underlies and includes it.
It is a mere product His and but a tiny part or aspect of Him.
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obeyed in a dream. But in truth, even in a dream, though
images and sound come and go and seem to intertwine,
actually there is no contradiction if we observe carefully.
As for the difference between dream and awake
experience, it is not strictly a contradiction since they are
experienced as distinct domains of being.
Contradiction is not even thinkable, except in words (or
intentions). We cannot even actually imagine a
contradiction, in the sense defined by Aristotle (is and is
not at once in every respect). We can only say (or
vaguely believe) there is one. We of course commonly
encounter apparent contradiction, but that does not prove
that contradiction exists in fact. It is an illusion, a conflict
between verbal interpretations or their non-verbal
equivalents.
We formulate theories; they yield contradictions; we
correct the theories so that they no longer yield these
contradictions. We tailor our rational constructs to
experience. We do not infer contradiction to exist from
contradictions in our knowledge. We question and fix our
knowledge, rather than impose our beliefs on reality.
That is sanity, mental health. That is the way knowledge
progresses, through this dialectic of thesis-antithesis
synthesis.
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CHAPTER TWENTY SEVEN
Drawn from A Fortiori Logic (2013),
Chapters 3.1, 7.2.
UNDERSTANDING THE LAWS OF
THOUGHT
Adapting the Laws of Thought
Many people regard Aristotle’s three ‘laws of thought’ –
the laws of identity, of non-contradiction and of the
excluded middle83 – as rigid prejudices. They think these
are just conventions, that some moronic old fellow called
Aristotle had the bad grace to impose on the rest of us,
and that we can just chuck ‘em out at will. In each of my
past works, I have tried to explain why these are
fundamental human insights that cannot under any
pretext be discarded. I would like to add a few more
explanations in the present work.
83
Aristotle states the laws of non-contradiction and of
the excluded middle in his Metaphysics, B, 2 (996b26-30), Γ, 3
(1005b19-23), Γ, 7 (1011b23-24). Metaph. Γ, 7 (1011b26-27)
may be viewed as one statement by Aristotle of the law of
identity: “It is false to say of that which is that it is not or of that
which is not that it is, and it is true to say of that which is that it
is or of that which is not that it is not.” These references are
found in the Kneales, p. 46 (although they interpret the latter
statement as somewhat defining truth and falsehood, rather
than as expressing the law of identity).
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The laws of thought must not be thought of as
mechanical rules, but as repeated insights of our
intelligence. Every ‘application’ of these laws in a new
context demands a smart new insight from us. We must
in each new context reaffirm these laws, and use them
creatively to deal with the complexities of the case at
hand.
In a fortiori logic, where new forms are encountered, and
new problems need solutions, we can expect our
intelligence and creativity to be called upon. We have
already come across many contexts where subtlety was
required. The distinction between a proposition like ‘X is
Y’ and ‘X is R enough to be Y’ was one such context.
Another was our development of a distinction between
absolute terms (R and notR) and relative terms (R and
notR). The laws of thought are ever present in logical
discourse, but they must always be understood and
adapted in ways that are appropriate to the context at
hand – so they are not mechanical laws, but ‘smart laws’.
The laws of thought have to repeatedly be adapted to the
increasing complexity of discourse. Originally, no
doubt, Aristotle thought of the laws with reference to
indefinite propositions, saying that ‘A is B’ and ‘A is not
B’ were incompatible (law of non-contradiction) and
exhaustive (law of the excluded middle). In this simplest
of contexts, these laws implied only two alternatives.
However, when Aristotle considered quantified
propositions, ‘All A are B’ and ‘Some A are B and some
A are not B’ and ‘No A is B’ – he realized that the
application in this new context of the very same laws
implied three alternatives. From this example, we see
that the subtleties of each situation must be taken into
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THE LAWS OF THOUGHT
consideration to properly ‘apply’ the laws. They are not
really ‘applied’; they are intelligently formulated anew as
befits the propositional forms under consideration.
We could say that the disjunction “Either ‘A is B’ or ‘A
is not B’” refers to an individual subject A, whereas the
disjunction “Either ‘All A are B’ or ‘Some A are B and
some A are not B’ or ‘No A is B’” refers to a set of
things labeled A. But then the question arises: what do
we mean when we say that an individual A ‘is B’? Do we
mean that A is ‘entirely B’, ‘partly B and partly not B’?
Obviously, the mutually exclusive and exhaustive
alternatives here would be: “Either ‘A is wholly B’ or ‘A
is partly B and partly not B’ or ‘A is not at all B’.” It
seems obvious that in most cases ‘A is B’ only intends
‘A is partly B and partly not B’ – for if ‘A is wholly B’,
i.e. ‘A is nothing but B’ were intended, why would we
bother verbally distinguishing A from B? Well, such
tautologies do occur in practice, since we may first think
of something as A and then of it as B, and belatedly
realize that the two names in fact refer to one and the
same thing. But generally we consider that only B is
‘wholly B’, so that if something labeled A is said to have
some property labeled B, A may be assumed to be
intended as ‘only partly B’.
To give a concrete example: my teacup is white. This is
true, even though my teacup is not only colored white,
but also has such and such a shape and is made of such
and such a material and is usually used to drink tea.
Thus, though being this teacup intersects with being
white, it does not follow that the identity of this
individual teacup is entirely revealed by its white color
(which, moreover, could be changed). With regard to
classes, even though we may choose to define the class
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of all A by the attribute B, because B is constant,
universal and exclusive to A, it does not follow that A is
thenceforth limited to B. B remains one attribute among
the many attributes that are observed to occur in things
labeled A. Indeed, the class A may have other attributes
that are constant, universal and exclusive to it (say C, D,
etc.), and yet B alone serves as the definition, perhaps
because it intuitively seems most relevant. Thus, to
define concept A by predicate B is not intended to limit
A to B. If A were indeed limited to B, we would not
name them differently.
These thoughts give rise to the logical distinction
between ‘difference’ and ‘contradiction’, which calls
forth some further use of ad hoc intelligence. When we
say that ‘A and B are different’, we mean that these
labels refer to two distinct phenomena. We mean that to
be A is not the same as to be B, i.e. that B-ness is
different from A-ness. It does not follow from this that
No A is B. That is to say, even though A is not the same
thing as B, it is conceivable that some or all things that
are A may yet be B in some way. To say the latter
involves no contradiction, note well. Therefore, the laws
of non-contradiction and of the excluded middle cannot
in this issue be applied naïvely, but only with due regard
for the subtleties involved. We must realize that
‘difference’ is not the same as ‘contradiction’. Difference
refers to a distinction, whereas contradiction refers to an
opposition. Two propositions, say X and Y, may have
different forms and yet imply each other. It is also
possible, of course, that two propositions may be both
different and contradictory.
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THE LAWS OF THOUGHT
Another subtlety in the application of the laws of thought
is the consideration of tense. Just as ‘A is B’ and ‘A is
not-B’ are compatible if they tacitly refer to different
places, e.g. if they mean ‘A is B here’ and ‘A is not-B
there’, so they are compatible if they tacitly refer to
different times, e.g. if they mean ‘A is B now’ and ‘A is
not-B then’. Thus, if a proposition is in the past tense and
its negation is in the present or future tense, there is no
contradiction between them and no exclusion of further
alternatives. Likewise, if the two propositions are true at
different moments of the past or at different moments of
the
future,
they
are logically compatible and
inexhaustive.
These matters are further complicated when we take into
consideration the various modalities (necessity, actuality,
possibility), and still further complicated when we take
into consideration the various modes of modality
(natural, extensional, logical). I have dealt with these
issues in great detail in past works and need not repeat
myself here. In the light of considerations of the
categories and types of modality, we learn to distinguish
factual propositions from epistemic propositions, which
qualify our knowledge of fact. In this context, for
instances, ‘A is B’ and ‘A seems not provable to be B’,
or even ‘A is B’ and ‘A seems provable not to be B’,
might be both true.
One of the questions Aristotle made a great effort to
answer, and had some difficulty doing, was how to
interpret the disjunction: “Either there will be a sea battle
tomorrow or there will not be a sea battle tomorrow”84.
But the solution to the problem is simple enough: if we
can truly predict today what will (or will not) happen
84
See De Interpretatione, 9 (19a30).
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tomorrow, it implies that tomorrow is already
determined at this earlier point in time and that we are
able to know the fact; thus, in cases where the fact is not
already determined (so that we cannot predict it no
matter what), or in cases where it is already inevitable
but we have no way to predict the fact, the disjunction
obviously cannot be bipolar, and this in no way
contravenes the laws of thought. Nothing in the laws of
thought allows us to foretell whether or not
indeterminism is possible in this world.
As a matter of fact, either now there will be the sea battle
tomorrow or now there won’t be one or the issue is still
undetermined (three alternatives). As regards our
knowledge of it, either now there will be the battle
tomorrow and we know it, or now there won’t be and we
know it, or now there will be and we don’t know it, or
now there won’t be and we don’t know it, or it is still
undetermined and so we cannot yet know which way it
will go (five alternatives). We could partially formalize
this matter by making a distinction between affirming
that some event definitely, inevitably ‘will’ happen, and
affirming only that it just possibly or even very likely
‘will’; the former is intended in deterministic contexts,
whereas the latter is meant when human volition is
involved or eventually when natural spontaneity is
involved. These alternatives can of course be further
multiplied, e.g. by being more specific as regards the
predicted time and place tomorrow.
What all this teaches us is that propositions like ‘A is B’
and ‘A is not B’ may contain many tacit elements, which
when made explicit may render them compatible and
inexhaustive. The existence of more than two alternatives
is not evidence against the laws of thought. The laws of
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THE LAWS OF THOUGHT
thought must always be adapted to the particulars of the
case under consideration. Moreover, human insight is
required to properly formalize material relations, in a
way that keeps our reading in accord with the laws of
thought. This is not a mechanical matter and not
everyone has the necessary skill.
Another illustration of the need for intelligence and
creativity when ‘applying’ the laws of thought is the
handling of double paradoxes. A proposition that
implies its contradictory is characterized as paradoxical.
This is a logical possibility, in that there is a quick way
out of such single paradox – we can say that the
proposition that implies its contradictory is false, because
it leads to a contradiction in knowledge, whereas the
proposition that is implied by its contradictory is true,
because it does not lead to a contradiction in knowledge.
A double paradox, on the other hand, is a logical
impossibility; it is something unacceptable to logic,
because in such event the proposition and its
contradictory both lead to contradiction, and there is no
apparent way out of the difficulty. The known double
paradoxes are not immediately apparent, and not
immediately resolvable. Insights are needed to realize
each unsettling paradox, and further reflections and
insights to put our minds at rest in relation to it. Such
paradoxes are, of course, never real, but always illusory.
Double paradox is very often simply caused by
equivocation, i.e. using the same word in two partly or
wholly different senses. The way to avoid equivocation is
to practice precision and clarity. Consider, for instance,
the word “things.” In its primary sense, it refers to
objects of thought which are thought to exist; but in its
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expanded sense, it refers to any objects of thought,
including those which are not thought to exist. We need
both senses of the term, but clearly the first sense is a
species and the second sense is a genus. Thus, when we
say “non-things are things” we are not committing a
contradiction, because the word “things” means one
thing (the narrow sense) in the subject and something
else (the wider sense) in the predicate. The narrow sense
allows of a contradictory term “non-things;” but the
wider sense is exceptional, in that it does not allow of a
contradictory term – in this sense, everything is a “thing”
and nothing is a “non-thing,” i.e. there is no “non-thing.”
The same can be said regarding the word “existents.” In
its primary sense, it refers to actually existing things, as
against non-existing things; but in its enlarged sense, it
includes non-existing things (i.e. things not existing in
the primary sense, but only thought by someone to exist)
and it has no contradictory. Such very large terms are, of
course, exceptional; the problems they involve do not
concern most other terms. Of famous double paradoxes,
we can perhaps cite the Barber paradox as one due to
equivocation85. Many of the famous double paradoxes
have more complex causes. See for examples my latest
analyses [in chapter 30, below] of the Liar and Russell
paradoxes. Such paradoxes often require a lot of
ingenuity and logical skill to resolve.
85
I deal with this one in my Future Logic, chapter 32.3.
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THE LAWS OF THOUGHT
Two More Laws of Thought86
The first three laws of thought, which were formulated
by Aristotle, are that we admit facts as they are (the law
of identity), in a consistent manner (the law of non
contradiction) and without leaving out relevant data pro
or con (the law of the excluded middle). To complete
these axioms of logic, and make them fully effective in
practice, we must add two more. The fourth, which I
have called the principle of induction; and a fifth,
which I call the principle of deduction.
These five laws are nothing new, being used in practice
by mankind since time immemorial; only our naming
them in order to spotlight them and discuss them is a
novelty. They qualify as ‘laws of thought’ because they
are self-evident, and necessary to and implied in all
rational thought.
The principle of deduction is a law of logic that no
information may be claimed as a deductive conclusion
which is not already given, explicitly or implicitly,
verbally or tacitly, in the premise(s). The premises must
obviously fully justify the conclusion, if it is to be
characterized as deduced. This fundamental rule is true
for all forms of deductive (as against inductive)
arguments, which helps us avoid fallacious reasoning. It
may be viewed as an aspect of the law of identity, since it
enjoins us to acknowledge the information we have, as it
is, without fanciful additions.
It may also be considered as the fifth law of thought, to
underscore the contrast between it and the principle of
86
This essay was originally written for A Fortiori Logic,
and may still be found there in a scattered way, notably in
chapter 7.2.
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induction, which is the fourth law of thought. The
principle of induction may, in its most general form, be
stated as: what in a given context of information appears
to be true, may be taken to be effectively true, unless or
until new information is found that puts in doubt the
initial appearance. In the latter event, the changed
context of information may generate a new appearance as
to what is true; or it may result in some uncertainty until
additional data comes into play.
Deduction must never be confused with induction.
Although deduction is one of the tools of induction in a
broad sense, it is a much more restrictive tool than
others. Deduction refers specifically to inferences with
100% probability; whereas induction in a narrow sense
refers to inferences with less than 100% probability.
Inductive reasoning is not subject to the same degree of
restriction as deduction. Induction is precisely the effort
to extrapolate from given information and predict things
not deductively implied in it. In inductive reasoning, the
conclusion can indeed contain more information than the
premises make available; for instance, when we
generalize from some cases to all cases, the conclusion is
inductively valid provided and so long as no cases are
found that belie it. In deductive reasoning, on the other
hand, the conclusion must be formally implied by the
given premise(s), and no extrapolation from the given
data is logically permitted. In induction, the conclusion is
tentative, subject to change if additional information is
found, even if such new data does not contradict the
initial premise(s)87. In deduction, on the other hand, the
87
For example, having generalized from “some X are Y”
to “all X are Y” – if it is thereafter discovered that “some X are
not Y,” the premise “some X are Y” is not contradicted, but the
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THE LAWS OF THOUGHT
conclusion is sure and immutable, so long as no new data
contradicts the initial premise(s).
As regards the terms, whereas in induction the
conclusion
may contain terms, denotations or
connotations that are not manifest in the premise(s), in
deduction the terms, denotations and connotations in the
conclusion must be uniform with those given in the
premise(s). If a term used in the conclusion of a
deductive argument (such as syllogism or a fortiori)
differs however slightly in meaning or in scope from its
meaning or scope in a premise, the conclusion is
deductively invalid. No equivocation or ambiguity is
allowed. No creativity or extrapolation is allowed. If the
terms are not exactly identical throughout the argument,
it might still have some inductive value, but as regards its
deductive value it has none.
Any deductive argument whose conclusion can be
formally validated is necessarily in accord with the
principle of deduction. In truth, there is no need to refer
to the principle of deduction in order to validate the
conclusion – the conclusion is validated by formal
means, and the principle of deduction is just an ex post
facto observation, a statement of something found in
common to all valid arguments. Although useful as a
philosophical abstraction and as a teaching tool, it is not
necessary for validation purposes.
Nevertheless, if a conclusion was found not to be in
accord with the principle of deduction, it could of course
be forthwith declared invalid. For the principle of
deduction is also reasonable by itself: we obviously
cannot produce new information by purely rational
conclusion “all X are Y” is indeed contradicted and must be
abandoned.
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means; we must needs get that information from
somewhere else, either by deduction from some already
established premise(s) or by induction from some
empirical data or, perhaps, by more mystical means like
revelation, prophecy or meditative insight. So obvious is
this caveat that we do not really need to express it as a
maxim, though there is no harm in doing so.
The principle of deduction is that the putative conclusion
of any deductive argument whatsoever must in its
entirety follow necessarily from (i.e. be logically implied
by) the given premise(s), and therefore cannot contain
any claims not supported in the said premise(s). If a
putative conclusion contains additional information and
yet seems true, that information must be proved or
corroborated from some other deductive or inductive
source(s). Inference in accord with this principle is truly
deductive. Inference not in accord with this principle
may still be inductively valid, but is certainly not
deductively valid.
In truth, the principle of deduction is a redundancy. That
the conclusion cannot go beyond what is given in the
premises is true of all deductive argument, without any
need to state it as a special principle; it is the very
definition of deduction, as against induction or fallacious
thought, and so the subtext of any deductive act. Clearly,
the principle of deduction is not an artificial, arbitrary or
conventional limitation, but a natural, rational one.
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THE LAWS OF THOUGHT
CHAPTER TWENTY EIGHT
Drawn from A Fortiori Logic (2013),
Chapter 12.1 and Appendix 7.3.
ASSAULTS ON LOGIC
Zen’s Anti-logic
Zen logic, as is well known, is no logic, but a sort of
anti-logic, an antithesis of logic88. It thrives on paradox
and even contradiction, at least apparent if not real. A
major feature of Zen logic, though this may not be
distinctive to Buddhist or even to Indian or Chinese
logic, is its belief in the ‘tetralemma’ (or catuskoti).
According to this viewpoint, not only a thesis alone (A
and not not-A) or alternatively its antithesis alone (not-A
and not A) may in fact be true, but there is a real
possibility that both the thesis and its contradictory (A
and not-A) are true, or neither the thesis nor its
contradictory (not A and not not-A) are true – or even,
eventually, that two or more of these four compounds are
true together or all false together.
For example, the “two truths” doctrine, formulated by the
Buddhist philosopher Nagarjuna (India, ca. 150-250 CE),
88
I should reiterate here that though I repeatedly
criticize Buddhism for its illogic, my purpose is not to totally
discredit it. I greatly respect this philosophy of life, and am
myself positively influenced by it daily. However, there is much
in its philosophical discourse that needs to be revised. Its
cavalier attitude to logic is simply untenable.
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which distinguishes the “relative truth” of conventional
minds and the “absolute truth” of enlightened minds may
be classified under the tetralemma category of “neither A
nor not-A,” since relative truth is neither absolutely true
nor untrue, but something in between. Again, the
doctrine that “Nirvana and Samsara are one” may be
classified under the tetralemma category of “both A and
not-A,” since it proposes a mixture of opposites. These
two doctrines are paradoxically considered as mutually
supportive, but of course that is quite illogical: if truth is
twofold, its two aspects cannot be one; you can’t have it
both ways. In scientific Western thought89, truth is one; if
it is merely ‘relative’, it is simply untruth. Again, if two
things are opposites, they cannot overlap.
Moreover, Buddhists argue that existents have no
identity of their own, being merely aggregates, constantly
in
flux, and thoroughly dependent on causes and
conditions. They apply this idea to mind as well as
matter, and deny existence of the self or soul. Such
claims are logically patently absurd. To deny the self or
soul is to deny the existence of someone doing the
denying90. If literally everything is aggregated, then the
89
The 12th century CE Islamic philosopher Averroës (or
Ibn Rushd) tried to introduce a similar notion of “double truth”
(one for common people and one for the élite). Some Christian
philosophers, possibly including Boethius, tried to follow suit.
But such tendencies were ultimately rejected in both cultures,
as it was realized that if religion was cut off from reason, it
ultimately implied that religion is irrational and therefore
untrue. More recently, most Christians have gradually adapted
their beliefs to empirical science and history (though many still
resist, e.g. with regard to Darwinism). Islam, on the other
hand, is still firmly marooned in the Middle Ages.
90
This goes against Descartes’ phenomenological
dictum: “I think, therefore I am,” which means that as of the
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THE LAWS OF THOUGHT
elements that are aggregated must also be aggregated, ad
infinitum91, in which case there is ultimately nothing to
aggregate. There cannot be such a thing as aggregation
without something non-aggregated to aggregate; the buck
has to stop somewhere. Similarly, to the Buddhist
doctrine of impermanence, which claims that literally
everything is constantly in flux, we must ask: flux of
what? Change must be change of something to
something, with an least momentary stationary existence
before and after the change. There cannot be such a thing
moment one acknowledges the phenomenon of thought by
venturing some proposition, one logically must acknowledge
the existence of someone having that thought.
‘Consciousness’ presupposes some sort of subject and some
sort of object, being a special relation between two things, the
conscious one being called ‘subject’ and the one the subject is
conscious of being called ‘object’. The difficulty of fathoming
this relation, due to its ontological distinctiveness and
therefore primacy, does not make it any the less real; there
are a great many things we cannot fathom, but must take for
granted. Knowledge must start with some irreducible
primaries; it cannot be grounded in an infinity of definitions
and proofs. To make a demand for endless grounding is to
claim that demand as an irreducible primary; it is self
contradictory. Buddhists consider that what we call the self is
simply the totality of our sensory and mental experiences at
any given moment of time: for them, there is no one having
those experiences – they just are, forming a changing bubble
of manifest being (which they call ‘consciousness’); this
bubble being particular gives the illusion of selfhood. But the
question remains: who has this illusory idea of being a self?
How can a non-self imagine that it is a self? They have no
answer to such questions, and avoid to ask them, being
dogmatically attached to the idea of no-self.
91
Moreover, how can a human mind go all the way to
infinity and observe that aggregation continues there, before
making such a bold claim?
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as change without something static that changes or
emerges from the change. The same applies to the
Buddhist idea of interdependence, or co-dependence of
everything92: one thing cannot depend on another if that
other thing is as devoid of independent existence as
itself. Dependence presupposes something more firmly
rooted in being, which can be depended on.
Simultaneous mutual dependence is unconscionable.
Thus, Buddhist discourse is built on stolen concepts,
ignoring their conceptual basis.93
Such Buddhist beliefs are contrary to the laws of thought
discovered by Aristotle, namely the laws of identity, of
non-contradiction and of the excluded middle. For
Buddhists, all existents are ultimately “empty” of any
nature. But the law of identity is that every existing thing
has an identity, a specific nature (whatever that happens
92
Here again, how can a human mind know the
dependence of literally all things on each other? To have such
knowledge, of all things past, present and future throughout
the universe and their exact relations to each other, is
conceivable for God – but how can a mere mortal obtain it?
93
Although Buddhists claim that enlightenment brings
about omniscience, such a claim is not empirically justified.
For a start, Buddhism has made and still makes many claims
about the physical world and the history of life and men that
are rejected by modern science; e.g. that the world and life
have existed forever. More specifically, consider the following
blooper: Zen master Dogen, after attaining enlightenment in
1227-8, wrote in an essay dated 1231 that the Buddha was
active about 2000 years before, whereas we know that he
lived in circa the 6th-5th centuries BCE, i.e. some two to four
hundred years later than Dogen thought. See: Beyond
Thinking: A Guide to Zen Meditation, Ed. Kazuaki Tanahashi
(Boston, MA: Shambhala, 2004), p. 31. Dogen claims having
attained enlightenment in another essay (p. 13).
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to in fact be94): it is not ‘just anything’ and it is not
‘nothing’. Every existent is something in particular, with
features and behavior peculiar to it. Moreover, a fact is a
fact: while it occurs, its constituents, its history and its
causes and conditions, if any, are irrelevant to the fact of
its existence: it just is. Moreover, identities, facts, are
mostly objective givens, not products of mind; to claim
otherwise is to affirm one’s claim itself to be imaginary
and thus untrue. The law of contradiction is that an
existent cannot at once have and not-have a particular
identity; presence and absence are incompatible. The law
of the excluded middle is that an existent cannot at once
neither have nor not-have a particular identity; there is
nothing besides presence or absence. These laws,
properly understood, are absolute; they are not subject to
any exceptions, under any circumstances whatsoever.
These laws – which have been foundational for Western
logical thought and the source of its successes (although
in today’s atmosphere of willful unreason many people
do take a perverse pleasure in disowning them) – were
never, it seems, very influential further East. The
tetralemma was evidently freely used very early on in
East Asia and the Far East, since it is so pervasive in later
literature. The “reasoning” behind this irrational belief is
that
all ordinary human cognition is necessarily
“dualistic.” According to its proponents, as soon as
anything comes to mind, through perception or
94
How the identity of things is to be known is the
question the science of logic seeks to answer. The short
answer is, of course, by means of our senses and our reason.
That is, empirically and logically, inductively and deductively.
Not all identities are necessarily knowable; but we must admit
that some are, for otherwise we would be involved in self
contradiction (claiming knowledge and denying it at once).
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conception, its negation must also be considered, even if
we tend to either ignore it or arbitrarily reject it. Thus, a
positive is unthinkable without a corresponding negative.
In one author’s words:
“In Buddhist logic, it is said that all concepts are
based upon exclusion. As soon as we affirm
something by saying ‘It is this,’ we automatically
exclude so many other things it might have been.
By imposing a conceptual limitation we fabricate
an idea. The suggestion here is that it is just an
idea – it is not an open experience.”95
But of course this ‘reasoning’ is quite fallacious.
Knowledge starts with pure perception of positive
phenomena; negatives are never pure percepts but are
necessarily products of conception. Note well: positives
come before negatives; and negatives are inconceivable
without positives. I can cognize a positive through
perception and therefore without any reference to its
contradictory; but I cannot do the same with a negative.
95
See Traleg Kyabgon Rinpoche, “The Path of
Mahamudra,” in The Best Buddhist Writing 2005, Ed. Melvin
McLeod and the editors of the Shambhala Sun (Boston, MA:
Shambhala, 2005), p. 98. Although he refers specifically to
conception, the implication of such statements is usually taken
to be that all affirmation implies negation, i.e. even affirmation
based exclusively on perception. Note however, the contrary
statement by Eleanor Rosch, in the same collection of essays,
p. 114: “According to Buddhist teachings, while all of the
interdependent past can be causally gathered into the
microcosm of the moment of present experience, that does
not mean that the basic mode of apprehending the present
moment is somehow filtered or distorted or abstractly
representational.” In other words, Buddhists do ultimately
admit of unadulterated percepts (if only in the context of the
enlightenment experience).
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In the latter case, I must first have some idea however
vague or hypothetical of the positive, before I can even
think of, let alone check out, the negative. Thus, though
exclusion is indeed eventually part of the knowing
process, it is certainly not a primary act: it is only
possible after the pure perception of some things and the
subsequent imagination of their possible negation.
Some Buddhist philosophers go still further and,
appealing to the notion of “emptiness” (shunyata), claim
the ontological primacy of negation over affirmation. But
here again the question they do not ask is: “negation of
what?” If as they suggest there is nothing there at all,
then even negatives have no foot to stand on. The
negation of a nothing does not produce a something.
What needs to be understood by such people is that the
word ‘not’ is more akin to a verb than to a noun. It
expresses the Subject’s mental act of rejection of a
proposed object. It is therefore necessarily conceptual,
and never perceptual. Moreover, such claims invariably
ignore the positive existence of the claim, and of
someone doing the claiming, and of someone receiving
the claim. Such people imagine they can speak in a
vacuum, without acknowledging the existential context
of their speech. This is illogical.
If
anything, it is the Buddhist proponents of
“paraconsistent” logic who are dualistic and divorced
from reality. They fail to take note of the actual order of
knowledge development from positive percepts to
negative concepts. Indeed, even at the level of
perception, one precedes two. Contrary to what many
philosophers imagine, we perceive a whole before we
mentally divide it into parts. Here, the confusion
involved is to conflate a given moment of perception and
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perception over time. In any given moment, what we
happen to perceive is a whole and this is quickly and
mostly automatically divided into parts by the conceptual
faculty
(note that the perception precedes the
subdivision, and only the latter involves negation, i.e.
saying ‘this part is not the same as that part’); but of
course, over time, many such moments of perception, or
more precisely their memories, are added together (again
by the conceptual faculty) to form a larger whole. These
two operations of the conceptual faculty – viz.
conceptual dissection of a present perceptual whole and
integration of many past percepts into a conceptual
whole – should not be confused.
I did not, unfortunately, note down every use of such
deviant logic that I came across over the years in
Buddhist literature. But I do still remember one relatively
early
instance in the Dhammapada, traditionally
attributed to the Buddha (India, ca. 563-483 BCE)96, “He
for whom there is neither this nor the further shore, nor
both….” The tetralemma plays a very important role in
the Madhyamika philosophy of Nagarjuna97, regarded as
96
The Dhammapada was probably compiled in the third
century BCE.
97
See my book Buddhist Illogic on this subject. It should
not be thought that Nagarjuna’s perverse thought has had no
equivalent in the West. For example, the Megaric school
(founded by Euclides of Megara in 4th century BCE Greece)
argued much like him that predication is either wrong (if the
predicate “differs” from the subject) or useless (if the predicate
is “the same as” the subject), ignoring the fact that such a
statement is itself an act of predication. I have over the years
spotted many such similarities between Eastern and Western
philosophies. This is a topic that still needs extensive study,
though there may already be good books on it that I am
unaware of.
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a forerunner of Zen. As for later, specifically Zen
writings, in China and then Japan, they are full of it.
Consider, for instance, the words of the third patriarch of
Zen, Seng Tsan (China, d. 606 CE):
“What is, is not; what is not, is. If this is not yet
clear to you, you’re still far from the inner truth.
One thing is all, all things are one; know this and
all’s whole and complete.” (Italics mine.)98
Eihei Dogen (1200-53 CE), who founded the Japanese
Soto Zen sect, often seems (to me, at least) maddeningly
obscure, if not insane, due to his frequent breaches of the
laws of thought. He indulges without hesitation in self
contradictory statements, such as “There is sitting letting
go of body-mind, which is not the same as sitting letting
go of body-mind.” Likewise, the law of the excluded
middle is no obstacle to his way of thinking. Consider,
for instance, this statement: “Active buddhas are neither
originally enlightened nor enlightened at some particular
time,
neither
naturally
enlightened, nor without
enlightenment” – what are they, then, I ask? Or again:
“practice-realization is neither existence nor beyond
existence” – what’s left, I ask? Surely, if all logical
possibilities are exhausted (as seems to be the intention,
here), then there are no other possibilities! Dogen pays
98
In his “Affirming Faith in Mind,” given in full in Roshi
Philip Kapleau’s Zen Merging of East and West (New York:
Doubleday, 1980), pp. 184-189. It is hard for me to believe
that illogic, the suppression of reason, is compatible with
enlightenment, let alone conducive or essential to it—just as it
is hard for me to believe that idolatry, the worship of inanimate
objects, is compatible with enlightenment, let alone conducive
or essential to it. Yet these are recurring theoretical teachings
within Zen Buddhism. Even so, paradoxically, I do believe that
Zen has much good to offer mankind on a practical level!
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no attention to such obvious restrictions, making his
discourse incomprehensible nonsense.99
Although Zen discourse is often antinomic, its favorite
form seems to be “neither this nor that.” That is to say,
although contradictions and exclusions of the middle are
both viewed as possible and do occur in practice, the
main emphasis is on denying any thesis whatever, and
‘logically’ enough also the contradictory of any thesis
whatever. For ultimate reality is considered by Zen
philosophers as essentially out of this world (even while
in it) – therefore, whether phenomena point to the
existence or to the non-existence of something, anything,
is irrelevant. No proposition is true, because none is
capable of describing reality as it really is. The
phenomenal world is inherently paradoxical; only
beyond it can all opposites be harmonized.100
This is the gist of the argument, however self
inconsistent and unconscionable it seems to us who are
not enlightened. Of course, some sense can be made of it
by thinking of ultimate reality as the ‘common ground’
of conflicting phenomena – and this sort of explanation is
99
Beyond Thinking, pp. 51, 79, 80. I should additionally
draw attention to the frequent use of tautology in some
Buddhist texts, as if this was informative. For example, Dogen
also enjoys tautologies like “sitting is sitting;” he also, I notice,
takes pleasure in reversing statements, as in “sitting is
buddha-dharma and buddha-dharma is sitting” (p. 51); and
reshuffling terms, as in “zazen is invariably the intention to
become buddha, and… zazen is invariably becoming buddha
with intention” (p. 39). Such discourse may of course be
informative, but I suspect the intention is more poetic.
100
The Vimalakirti Nirdesa Sutra, a Mahayana text some
consider as dating from about 100 CE (although there is no
mention of it till after Nagarjuna’s time, i.e. about a century
later), is a veritable litany of antinomies.
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often used (for example, see the above quoted statement
by Seng Tsan). So the ‘noumenon’ (i.e. that which is
beyond the phenomenon) may be thought of as both
transcendent and immanent. But a true Zen master would
disdainfully reject all such philosophizing as misleading
babble. Any resort to words as a means of rational
description or explanation is regarded as useless when it
comes to the “matter” of enlightenment. Consider for
instance the following remarks in The Blue Cliff
Record101:
“It’s wrong to say either that he had words or
didn’t have words; nor will it do to say that his
answer neither had nor didn’t have words. Chao
Chou left behind all the permutations of logic.
Why? If one discusses this matter, it is like sparks
struck from a stone, like flashing lightning. Only
if you set your eyes on it quickly can you see it. If
you hesitate and vacillate you won’t avoid losing
your body and life.”
All the above tends to the conclusion that Zen ‘logic’ is
illogical. However, that judgment can be considerably
mitigated, if we understand Zen ‘anti-dualistic’ discourse
not as theoretical but as pragmatic. Its purpose is not to
formulate a true philosophy, in the Western sense, but to
push people to a transforming mystical experience. Thus,
when a Zen advocate states: “This is neither true nor
false” or “This is neither good nor bad” or “This is
neither desirable nor repugnant” – his intent is really,
respectively: “Do not think or say that this is true and do
101
Pi Yen Lu, a Chinese Ch’an Buddhist classic. These
remarks were made by Yuan Wu K’e Ch’in (1063-1152),
relative to Case 59 (p. 339). Boston, MA: Shambala, 2005. Tr.
Thomas & J.C. Cleary.
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not think or say that it is false,” “Do not think or say that
this is good and do not think or say that it is bad,” and
“Do not think or say that this is desirable and do not
think or say that it is repugnant.” For example, Seng
Tsan says:
“When you assert that things are real, you miss
their true reality. But to assert that things are
void, also misses reality. The more you talk and
think on this, the further from the truth you’ll be.”
(Italics mine.)102
In other words, the Zen advocate is not really making
logical, prescriptive or descriptive judgments, but
advocating the suspension of all judgments, all discourse,
in order to arrive at the ultimate “truth.” There is no great
inconsistency in doing that. We may, of course, point out
that in claiming to be free of concepts he is using
concepts and that that is an inconsistency. However, he
would reply that he is doing that only in order to
communicate with us in our language, in an attempt to
allude to things beyond its scope. He is able to function
in both the phenomenal and noumenal worlds, whereas
we are not – so he has to find some way to reach out to
us. In that case, we can only criticize him for being rather
gauche in his discourse. He should make it more precise,
as just demonstrated. It would then be possible to speak
of Zen logic, without inverted commas.
Nevertheless, although a statement like “neither claim it
is nor claim it is not” is intended as a non-claim, it
objectively definitely does contain a factual claim – viz.
the claim that following this advice will facilitate or
result in enlightenment (“the truth”); and such a claim is,
102
The Blue Cliff Record. (I forgot to note the page
number.)
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THE LAWS OF THOUGHT
of course, subject to assessment as true or false, whether
the Zen advocate admits it or not.
I would say a very representative example of Zen logic is
the koan of Te-shan (China, 9th century CE), the Zen
master famous for presenting his students with the
following predicament: whether they ‘uttered a word’
(i.e. showed some evidence through word or deed of their
understanding of Zen) or not, they would get thirty
blows. Another master, Lin-chi (the founder of the
Rinzai sect), sent one of his own followers to him with
specific instructions. He told him to ask Te-shan why
someone who said a word would nevertheless get thirty
blows; then, when Te-shan struck him, the student was to
grab the stick and push Te-shan back with it. When the
student did as instructed, Te-shan responded by simply
walking away.103
What we have here is a logic of action, rather than
words104. There is, to start with, a seemingly inescapable
dilemma – whether you speak (rightly or wrongly) or
abstain from speech, the result will be the same: you will
be in error and punished by blows. There is, however, a
logical possibility of escape – grab the stick as it comes
down and push it back. This could be described as a
martial arts response to the attempted physical blow.
Logically, the dilemma has by this means been
effectively dissolved. There seemed to be no way out,
judging by Te-shan’s statement; but there was in fact a
103
See D. T. Suzuki, The Zen doctrine of no-mind
(Boston, MA: Weiser, 1972), p. 87.
104
Notice that the student did not try to dissuade Te
shan, saying “if you try to hit me, I will grab the stick and push
it.” Rather, he waited for Te-shan to actually strike and then
grabbed the stick and pushed it.
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way out, perceived by Lin-chi. The opponent is
neutralized, prevented from producing the threatened
consequences (blows) to either antecedent (speech or
silence).
Earlier in the present volume, in an attempt to more
accurately depict the logic of a fortiori reasoning, I
developed the notion of relative terms, say R1 and R2,
such that more R1 and less R2 (and vice versa) are
logically equivalent. This idea, I showed, can be
extended to the special case of complements, say R and
not-R. Although complements, taken as absolute terms,
are mutually exclusive – if we take them as relative
terms, they are compatible, indeed imply each other. That
is, we can define R and not-R so largely that each
includes the other, in the same continuum but in opposite
directions, i.e. in such a way that more R is less not-R
and less R is more not-R. This logical artifice of course
changes the meaning of R and not-R, but it is useful for
the development of a fortiori logic.
After I worked this idea out, it occurred to me that it
could help explain Zen logic. It could be that Oriental
philosophers who conceive of A and not-A as being
compatible are really thinking in relative terms. Perhaps
we in the West think of A and not-A in absolute terms,
while they in the East think in relative terms. This may
explain, at least in part, why the conjunction of A and
not-A does not repel the Oriental mind to the same
degree as it does the Western mind. Although, to be sure,
this theory is somewhat belied by the fact that Orientals
also accept the possibility of neither A nor not-A, which
this theory cannot explain.
Needless to say, the said insight does not change the fact
that A and not-A, taken in their absolute senses, are
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incompatible; the Aristotelian law of non-contradiction
remains true and unassailable. Relative terms are logical
artifacts that function consistently within that universal
framework – they do not erase it. The law simply
changes form, becoming a distinction between ‘more’
and ‘less’: What is more R is less (and not more) not-R,
and what is less R is more (and not less) not-R.
Moreover, it is interesting to note that when A and not-A
are intended as relative terms, everything falls under both
of them; there is no further possibility beyond them. That
is to say, the law of the excluded middle also remains
operative for relative terms, although it too is stated
slightly differently.
The Vanity of the Tetralemma
The most radical assault on reason consists in trying to
put in doubt the laws of thought, for these are indeed the
foundations of all rational discourse. First, the law of
identity is denied by saying that things are never quite
what they seem to be, or that what they are is closer to
grey than black and white. This is, of course, an absurd
remark, in that for itself it lays claim to utter certainty
and clarity. Then, the laws of non-contradiction are
denied by saying that things may both be and not-be, or
neither be nor not-be. This is the ‘tetralemma’, the
fourfold logic which is favored in Indian and Chinese
philosophies, in religious mysticism, and which is
increasingly referred to among some ‘scientists’. To
grasp the vanity of the tetralemma, it is necessary to
understand the nature of negation and the role of
negation as one of the foundations of human logic.
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The first thing to understand is that everything we
experience is positive phenomenon. Everything we
perceive through our senses, or remember or imagine in
our minds, or even cognize through ‘intuition’ – all that
has to have some sort of content to be at all perceived.
Each sense organ is a window to a distinct type of
positive phenomenon. We see the blue sky above, we
hear birds sing, we smell the fresh air, we taste a fruit, we
feel the earth’s texture and warmth, etc. Similarly, the
images and sounds in our heads, whether they come from
memory or are produced by imagination, are positive
phenomena; and even the objects of intuition must have
some content that we can cognize. Secondly, we must
realize that many positive phenomena may appear
together in space at a given moment. This is true for each
phenomenal type. Thus, the blue sky may fill only part of
our field of vision, being bounded by green trees and
grey buildings; we may at once hear the sounds of birds
and cars; and so on. Thirdly, many positive phenomena
may at any given time share the space perceived by us.
Thus, superimposed on visual phenomena like the sky
may be other types of phenomena: the sound of birds in
the trees, the smell of traffic in the streets, the feelings in
our own body, and so on. We may even hallucinate,
seeming to project objects of mental perception onto
physical space. For example, the image of one’s
eyeglasses may persist for a while after their removal.
Fourthly, each positive phenomenon, whatever its type,
varies in time, more or less quickly. Thus, the blue sky
may turn red or dark, the sounds of birds or traffic may
increase or decrease or even stop for a while, and so
forth.
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THE LAWS OF THOUGHT
In order to express all these perceptual possibilities –
differences in space and in time and in other respects, we
need a concept of negation, or more precisely an act of
negating. Without ‘negation’, we cannot make sense of
the world in a rational manner – it is the very beginning
of logical ordering of our experience. Thus, in a given
visual field, where (say) blue sky and trees appear, to be
able to say ‘the sky ends here, where the trees begin’ we
need the idea of ‘negation’ – i.e. that on one side of some
boundary sky is apparent and on the other side it is not,
whereas on the first side of it trees are not apparent and
on the other they are. Likewise, with regard to time, to be
able to describe change, e.g. from blue sky to pink sky,
we need the idea of ‘negation’ – i.e. that earlier on this
part of the sky was blue and not pink, and later on it was
pink and not blue. Again, we need the idea of ‘negation’
to express differences in other respects – e.g. to say that
‘the sounds of birds singing seem to emanate from the
trees, rather than from buildings’. Thus, negation is one
of the very first tools of logic, coming into play already
at the level of sorting of experiences.
Moreover, negation continues to have a central role when
we begin to deal with abstractions. Conceptual
knowledge, which consists of terms and propositions
based directly or indirectly on perceptual phenomena,
relies for a start on our ability to cognize similarities
between objects of perception: ‘this seems to resemble
that somewhat’ – so we mentally project the idea of this
and that ‘having something in common’, an abstract (i.e.
non-phenomenal, not perceived by any means) common
property, which we might choose to assign a name to.
However, to take this conceptual process further, we
must be able to negate – i.e. to say that ‘certain things
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other than this and that do not have the abstract common
property which this and that seem to have’, or to say that
‘this and that do not have everything in common’. That
is, we must be able to say not only that one thing
resembles another in some way, but also that these or
other things do not resemble each other in that way or in
another way. Thus, negation is essential for making sense
of
information also at the conceptual level of
consciousness.
Now, what is negation? To answer this question we first
need to realize that there are no negative phenomena in
the realm of experience. Everything we perceive is
positive phenomenon – because if it was not we
obviously would have nothing to perceive. We can only
‘perceive’ a negative state of affairs by first mentally
defining some positive state of affairs that we should
look for, and then look for it; if having looked for it
assiduously we fail to find it, we then conclude
inductively that it is ‘absent’, i.e. ‘not present’. Thus,
positive phenomena come before negative ones, and not
after.
Existence logically precedes non-existence.
Negative
phenomena
are
‘phenomena’
only
metaphorically, by analogy to positive phenomena – in
truth, negative phenomena are not: they do not exist.
‘Negation’ is not a concept in the sense of an abstraction
from many particular experiences having a certain
property in common. Negation is a tool of the thinking
observer, as above described. It is an act, an intention of
his.
To illustrate how confused some people – even some
scientists – are with regard to negation, I offer you the
following example drawn from Richard Dawkins’ The
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THE LAWS OF THOUGHT
Greatest Show on Earth: The Evidence for Evolution105.
He describes an experiment by Daniel J. Simons, in
which some people are asked to watch a brief video and
observe how many times a certain event takes place in it;
but at the end they are asked another question entirely,
viz. whether they noticed the presence of a man dressed
up as a gorilla in the course of the movie, and most of
them admit they did not106. According to Dawkins, we
may infer from this experiment how “eye witness
testimony, ‘actual observation’, ‘a datum of experience’ – all are, or at least can be, hopelessly unreliable.”
But this is a wrong inference from the data at hand,
because he confuses positive and negative experience.
The people who watched the video were too busy
looking for what they had been asked to observe to
notice the gorilla. Later, when the video was shown them
a second time, they did indeed spot the gorilla. There is
no reason to expect us to actually experience everything
which is presented to our senses. Our sensory
experiences are always, necessarily, selective. The
validity of sense-perception as such is not put in doubt by
the limited scope of particular sense-perceptions. The
proof is that it is through further sense-perception that
we discover what we missed before. Non-perception of
something does not constitute misperception, but merely
incomplete perception. ‘I did not see X’ does not
deductively imply ‘I saw the absence of X’, even though
repetition of the former tends to inductively imply the
latter.
105
106
New York: Free Press, 2009. Pp. 13-14.
The video can be seen at:
www.theinvisiblegorilla.com/videos.html.
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A negative ‘phenomenon’ is not like a positive
phenomenon, something that can directly be perceived or
intuited. A negation is of necessity the product of indirect
cognition, i.e. of an inductive (specifically, adductive)
process. We mentally hypothesize that such and such a
positive phenomenon is absent, and then test and confirm
this hypothesis by repeatedly searching-for and not
finding the positive phenomenon107. If we were to at any
time indeed find the positive phenomenon, the
hypothesis of negation would immediately be rejected;
for the reliability of a negation is far below that of a
positive experience. We would not even formulate the
negation, if we already had in the past or present
perceived the positive phenomenon. And if we did
formulate the negation, we would naturally retract our
claim if we later came across the positive phenomenon.
Therefore, the content of negative phenomena is
necessarily always hypothetical, i.e. tentative to some
degree; it is never firm and sure as with (experienced)
positive phenomena.
Negative assertions, like positive assertions, can be right
or wrong. If one looked diligently for a positive
phenomenon and did not find it, then one can logically
claim its negation. Such claim is necessarily inductive –
it is valid only so long as the positive phenomenon is
actively sought and not found. The moment the positive
phenomenon is observed, the negation ceases to be
107
Not-finding is the non-occurrence of the positive act of
finding. Objectively, note well, not-finding is itself a negative
phenomenon, and not a positive one. But subjectively,
something positive may occur within us – perhaps a sense of
disappointment or continued relief. See more on this topic in
my Ruminations, chapter 9.
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THE LAWS OF THOUGHT
justified. If one did not look for the positive
phenomenon, or did not look with all due diligence,
perhaps because of some distraction (as in the example
cited above), then of course the claim of negation is open
to doubt; certainly, it is inductively weak, and one is very
likely to be proved wrong through some later
observation.
How, then, is negation to be defined? We could well say
that negation is defined by the laws of non-contradiction
and of the excluded middle. That is, with regard to any
term ‘X’ and its negation ‘not-X’, the relation between
them is by definition the disjunction “Either X or not-X” – which is here taken to mean that these terms (X and
not-X) cannot be both true and cannot be both false, i.e.
they are exclusive and exhaustive. What do I mean here
by ‘definition’? – is that an arbitrary act? No – it is
‘pointing to’ something evident; it is ‘intentional’. Here,
it points to the instrument of rational discourse which we
need, so as to order experience and produce consistent
conceptual derivatives from it. The needed instrument
has to be thus and thus constructed; another construct
than this one would not do the job we need it to do for us.
That is, the only conceivable way for us to logically
order our knowledge is by means of negation defined by
means of the laws of non-contradiction and of the
excluded middle. Without this tool, analysis of
experience is impossible.
Suppose now that someone comes along and nevertheless
objects to the preceding assertion. Well, he says, how do
you know that the dilemma “either X or not-X” is true?
You just arbitrarily defined things that way, but it does
not mean it is a fact! Could we not equally well claim the
tetralemma “Either X or not-X or both or neither” to be
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true? The reply to that objection is very simple. Suppose
I accept this criticism and agree to the tetralemma. Now,
let me divide this fourfold disjunction, putting on the one
side the single alternative ‘X’ and on the other side the
triple alternative ‘not-X or both or neither’. I now again
have a dilemma, viz. “either ‘X’ or ‘not-X or both or
neither’.” Let me next define a new concept of negation
on this basis, such that we get a disjunction of two
alternatives instead of four. Let us call the complex
second alternative ‘not-X or both or neither’ of this
disjunction ‘NOT-X’ and call it ‘the super-negation of
X’.
Thus, now, the objector and I agree that the disjunction
“either X or NOT-X” is exclusive and exhaustive. We
agree, presumably, that this new dilemma cannot in turn
be opposed by a tetralemma of the form “Either X or
NOT-X or both or neither” – for if such opposition was
tried again it could surely be countered by another
division and redefinition. We cannot reasonably repeat
that process ad infinitum; to do so would be tantamount
to blocking all rational thought forever. Having thus
blocked all avenues to thought, the objector could not
claim to have a better thought, or any thought at all.
There is thus no profit in further objection. Thus, the
tetralemma is merely a tease, for we were quite able to
parry the blow. Having come to an agreement that the
new disjunction “Either X or NOT-X” is logically
unassailable, we must admit that the original disjunction
“Either X or not-X” was logically sound from the first.
For I can tell you that what I meant by not-X, or the
‘negation of X’, was from the beginning what is now
intended by NOT-X, or the ‘super-negation of X’!
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I was never interested in a relative, weak negation, but
from the start sought an absolute, strong negation. For
such utter negation, and nothing less radical, is the tool
we all need to order experience and develop conceptual
knowledge in a consistent and effective manner. In other
words, whatever weaker version of negation someone
tries to invent108, we can still propose a strong version
such that both the laws of non-contradiction and of the
excluded middle are applicable without doubt to it. If
such negation did not exist, it would have to be invented.
No one can destroy it by denying it or diluting it. Those
who try to are merely sophists who do not understand the
source, nature and function of negation in human
discourse. They think it is a matter of symbolic
manipulation, and fail to realize that its role in human
discourse is far more fundamental and complex than that.
Negation is the indispensable instrument for any attempt
at knowledge beyond pure perception.
108
There are people who say that the law of non
contradiction is logically necessary, but the law of the
excluded middle is not. Clearly, this claim can be refuted in the
same way. If they claim the three alternatives “Either X or not
X or ‘neither X nor not-X’” – we can again split the disjunction
into two, with on one side “X” and on the other side “not-X or
‘neither X nor not-X’” – and then proceed as we did for the
tetralemma. The same can be done if anyone accepts the law
of the excluded middle but rejects the law of non-contradiction.
All such attempts are fallacious nonsense.
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CHAPTER TWENTY NINE
Drawn from A Fortiori Logic (2013),
Appendix 7.1-2.
MODERN LOGIC
Modern Symbolic Logic
Since the later decades the 19th century, and more and
more so throughout the 20th century, “modern symbolic
logic” has gradually discarded and displaced “classical
formal logic.” What is the essential difference between
them? Classical formal logic, which was discovered or
invented by Aristotle (4th century BCE) and further
developed and improved on over time by many
successors, is based on the idea of studying the logical
properties of propositions by replacing material
propositions with formal ones. A categorical proposition
is formal, if its terms are variables instead of constants –
e.g. “All X are Y” is formal, because the symbols X, Y
represent in theory any terms that might arise in practice.
A hypothetical proposition is formal, if its theses are
variables instead of constants – e.g. “If X, then Y” is
formal, because the symbols X, Y represent in theory any
theses that might arise in practice.
Now, whereas classical logic symbolized terms and
propositions, it did not similarly symbolize the other
components of propositions, such as their quantities, their
modalities or their relational operators. In “All X are Y,”
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THE LAWS OF THOUGHT
the words “all” and “are” remained in ordinary language
(in our case, plain English). Similarly, in “If X, then Y”
the words “if” and “then” were not symbolized. In
modern symbolic logic, on the other hand, the trend
developed to symbolize every aspect of every
proposition109. This was, to be sure, a new school of
logic, which considered that only in this way could utter
precision of language be achieved, and all ambiguity or
equivocation be removed from human discourse. Modern
symbolic logic, then, advocated the adoption of an
altogether artificial language comparable to the language
of mathematics.
Some of the pros and cons of this approach are
immediately obvious110. One advantage of symbolization,
already mentioned, is the sense of precision sometimes
lacking in natural languages. However, this impression is
surely illusory – for if one’s understanding of the matter
at hand is vague and uncertain to start with, how can
symbols improve on it? A patent disadvantage of
symbolization is the esoteric nature of artificial language.
Logic was originally intended as a teaching for the
masses, or at least the intellectuals, to improve their daily
thinking. Nowadays, logic has become the exclusive
domain of a few specialists, and has little to do with
human cognitive practice. Moreover, communication is
not always easy even among symbolic logicians, because
each of them quite naturally prefers a different set of
109
For examples: the word “all” became an upside down
capital A, the word “some” became a laterally inversed capital
E (for existence, as in ‘there are’), the words “if–then”
(implication) became an arrow pointing from antecedent to
consequent, and so forth.
110
For more on this topic, see my Future Logic, chapter
64: Critique of Modern Logic.
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symbols, so that there is in fact not one artificial
language, but many of them. Another disadvantage is the
slow adaptability of any artificial language to forms of
discourse newly discovered in everyday usage. An
example of this is the a fortiori argument, which is still
without convincing symbolic expression.
The main activity of ‘modern logicians’ nowadays seems
to be to translate ordinary language into their favorite
symbolic language. Most of the time, they seem content
to rewrite a perfectly comprehensible plain English
sentence into a purely symbolic one, as if this is some
great achievement that will earn them their place in
history, or at least in the profession. Just that act of
translation or rewriting in symbolic terms seems to
satisfy and thrill them tremendously, as if it confers
scientific status onto the sentence. Additionally, they
resort to pompous terminology for window-dressing and
intimidation purposes. One gets the impression that
symbols play for them the role of magic incantations in
ancient times – ‘abracadabra!’ they would chant in
pursuit of mystical insights and powers.
But, think about it a moment. In truth, when modern
logicians rewrite a sentence in symbolic terms they have
achieved exactly nothing other than to use shorter
‘words’ (i.e. the symbols they invent) in place of
ordinary words, and (ideally) drawn up a table telling us
what symbols correspond to what ordinary words. All
they have done is abbreviate the given sentence.
Apparently, they are too lazy to write long sentences and
prefer concise ones. They have produced no new
information or insight. They cannot credibly argue that
the ordinary language statement was essentially deficient,
since it must have been understandable enough for them
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THE LAWS OF THOUGHT
to
have translated it into symbols. If it was
understandable enough for them, why not for everyone
else? What absolute need have we of the artificial
language(s) they so insistently try to sell us?111
Moreover, note this well, whenever we (and they) read
the symbolic statement they have concocted, our minds
have to translate it back into ordinary language in order
to understand it. We have to mentally repeatedly refer
back to the ordinary language definitions of the symbols.
We have to remember: “Oh! This funny symbol means
so and so, and that weird doodle means this, and the
zigzag means that,” and so forth. This means that our
mental process of understanding is made more difficult
and slowed down considerably. We are further from the
object of study than we were to start with – more
removed from the reality we are trying to think about.
This increased distance and waste of time is not
accidental or incidental, however – it serves to cloud the
issues and prevent critical judgment. Errors are hidden
from sight, and if we spot them we hardly dare point to
them for fear of admitting we may have missed
something. In this way, foolishness is perpetuated and
spreads on.
But all that is not the worst of it. The worst of it is
twofold. First, modern logicians usually symbolize much
too soon, when their level of insight and analysis is still
in its early stages. They typically do not give the subject
111
Ideally, of course, symbols are useful to summarize
large amounts of information. I would dearly love to develop
terse symbolic formulas that summarize my findings in the
logic of causation. So I am not entirely rejecting symbolism.
What I am saying here is that it is not necessary (i.e. we can
well do without it) and indeed can be a serious hindrance to
logical thinking and logic theory.
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matter studied time to develop and mature in their own
minds, but impatiently rush into their more orderly
looking and comfortable world of symbols. The result is
that their symbols are usually representative of a very
naïve, elementary, immature level of understanding of
the object at hand. Secondly, once their symbolic
representation is done, it freezes all subsequent work at
that childish level. Since the symbolization is already
settled, all they can do is play around with it in different
ways. All they can do is manipulate and reorder and
recombine their symbols this way and that way, and this
is what they pass the rest of their time doing. They
cannot feed on new experiences and insights from the
world out there or actual human discourse, since they
have already confidently and reassuringly separated
themselves from all that. Their symbols thus blind them
and paralyze them. The paucity of their results testifies to
it.
Another aspect of modern symbolic logic important to
note is its pretensions of ‘axiomatization’. In classical
formal logic, the Laws of Thought (Identity, Non
contradiction and the Excluded Middle) were sovereign;
these were axioms in the original sense of irreducible
primaries of rational knowledge, together constituting the
very essence of logic. In syllogistic validation by
Aristotle, all syllogisms could be reduced directly or ad
absurdum to a minimum number of primary moods –
mainly the first figure positive singular syllogism: “This
S is M, and all M are P, therefore, this S is P.” The latter
argument was not perceived as an axiom in its own right
or even as an arbitrary convention, but as a logical
insight in accord with the laws of thought that what is
claimed applicable to a concept must be acknowledged to
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THE LAWS OF THOUGHT
apply to the things it subsumes. The relation of human
reasoning, and more deeply of formal logic, to the laws
of thought was progressively ‘systematized’, but it was
allowed to remain essentially open and flexible. Though
integrated, it was not rigidly fixed, so as to allow for its
constant evolution and adaptation as knowledge
developed.
Modern logicians, on the other hand, focusing on the
more geometrico, the method of proof used in Euclidean
geometry, sought a more predictable and definitive
arrangement of knowledge. Their simplistic minds
demanded rigid rules and perfect orderliness. A hierarchy
was established between thoughts – with those at the top
of the hierarchy (the laws of thought) being viewed as
‘axioms’ and those lower down (syllogisms, and
eventually similarly other arguments) as ‘theorems’. This
may work well for mathematics, which is a relatively
special science, but it caused havoc in general conceptual
logic, which is the science of science. The question
naturally arose as to where those apparent axioms came
from and whether they could be replaced by contrary
ones as was done in non-Euclidean geometry. It did not
take long for these simpleminded people to decide that
logic was a conventional mental game, with no apparent
connection to the empirical world. This philosophy
(known as Logical Positivism) was largely justified by
Immanuel Kant’s analytic-synthetic dichotomy112, so it
could hardly be doubted.
What is lacking in this model of knowledge is the
understanding that formal logic is not deduced from the
112
See my Logical and Spiritual Reflections, book 2,
chapter 2, posted online at:
www.thelogician.net/6_reflect/6_Book_2/6b_chapter_02.htm.
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laws of thought. The laws of thought are not premises of
formal logic; they are not contents from which other
contents are deduced. The laws of thought refer us to the
autonomous logical insights through which we naturally
judge what constitutes appropriate inference. They are
what justifies the processes of deduction (and more
broadly, of induction) from premises to conclusions. The
forms of syllogism and other arguments are not deduced
from the laws of thought. The forms are induced from
actual thought contents. The thought contents can be
judged correct or not without reference to the forms,
using ad hoc logical insights. What formal logic does is
simply collect under a number of headings recurring
types of thoughts, so that again using ad hoc logical
insights we can once and for all predict for each type of
thought (e.g. syllogism 1/AAA) whether it is correct or
not. There is in fact no appeal to general ‘laws of
thought’ in this validation (or invalidation) process;
honest ad hoc logical insights are sufficient. The ‘laws of
thought’ are merely ex post facto typologies of ad hoc
particular acts of logical insight. For that reason, they are
not top premises is a geometrical model of knowledge.
The question these modern pseudo-logicians did not ask
themselves, of course, is why their allegedly logical
insights in the course of ‘axiomatization’ (including their
skepticism towards the objectivity of the laws of thought)
should be preferred to the logical insights of the ‘non
axiomatization’ logicians. Is any discussion of logic
possible without use of logic? Can logicians ever rightly
claim to transcend logic? Can they logically deride and
nullify logic? For instance, some have argued that appeal
to the laws of thought is either circular argument or
infinite regression. They stopped their reflection there,
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THE LAWS OF THOUGHT
and never asked themselves why the rejection of circular
argument or infinite regression should be considered
primary logical acts not needing justification, while the
laws of thought are to be rejected precisely on the ground
that (according to them) there are no logical acts not
needing justification. Is that not a double standard
(another primary logical insight)?113
The radical blunder of the Kantian legacy is the belief
that there is such a thing a ‘purely analytic’ or ‘a priori’
knowledge. Logicians influenced by this inane idea
remain blind to the empirical aspects of all knowledge
development. Even apparently purely symbolic systems
of logic rely on perceptions. Some symbols refer to
concrete objects (e.g. individuals a, b) and some to
abstract ones (e.g. classes x, y); but every symbol is, as
well as a sign for something else, in itself a concrete
object (whether as a bit of ink on paper, or of light on a
computer screen, or as a shape conjured in our mind’s
eye). If follows that symbolic formulas, whether
inductively or deductively developed, always depend on
some empirical observation. The observation of symbols
is not a transcending of experience; it is an empirical
process just like the observation of cows; it requires
physical or mental perception. Thus, if I count symbolic
entities or I imagine them collected together, that is not
purely analytic work – it is quite synthetic work.
Moreover, such logicians tend to ignore the countless
memories, imaginations and rational insights that form
the wordless background of all discourse concerning
logic.
113
We should of course in this context mention Kurt
Gödel, who showed the incompleteness of axiomatic systems
like that of David Hilbert.
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Clearly, axiomatization was a con-game on a grand scale,
through which shallow but cunning pseudo-logicians
wanted to take power in the domain of logic. And they
have indeed managed to do that almost completely. But
the fact remains that primary logical insights like the
laws of thought, or the rejection of circularity, infinite
regression and double standards, or again the acceptance
of subsumption (syllogistic reasoning) and the many,
many other foundations of human thought, are not open
to doubt. No amount of ‘axiomatization’ can either prove
or disprove them, because all proof and disproof depends
on such insights to at all convince us. The conclusion to
draw from that is certainly not relativism or
conventionalism, for that too would be a claim to the
validity of some logical insight – i.e. the insight that all
insights are arbitrary must itself be arbitrary. Logic
cannot be refuted by logic. Logic can however be
justified, by honest acknowledgment that some thoughts
are primary logical insights. And these insights, which
together constitute what we call ‘human reason’, cannot
all be listed in advance, but emerge over time as
knowledge develops.
Aristotle said all that long ago, but many have preferred
to ignore him or misrepresent him because they dearly
want to belittle him and supplant him, being envious of
his achievements. Consider for instance the following
statement about the law of non-contradiction drawn from
his Metaphysics (Book 4, part 3. Translated by W. D.
Ross.):
“For a principle which every one must have who
understands anything that is, is not a hypothesis;
and that which every one must know who knows
anything, he must already have when he comes to
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THE LAWS OF THOUGHT
a special study. Evidently then such a principle is
the most certain of all; which principle this is, let
us proceed to say. It is, that the same attribute
cannot at the same time belong and not belong to
the same subject and in the same respect.” (Italics
mine.)
As a result of symbolization and axiomatization, modern
logic is essentially a deductive logic enterprise. However
complicated or complex it may look, it is inevitably
superficial and simplistic. Even when modern logicians
pretend to discuss induction, they are stuck in deductive
activities. Their ‘logic’ is thus more and more divorced
from reality. For this reason their thinking on issues of
metalogic is thoroughly relativistic. Things have to be the
way they think they are, since symbols are somehow
omniscient and omnipotent. They see no idiocy or harm
in ‘paraconsistent logic’ (i.e. in breach of one or more of
the laws of thought), since to them it is all a game with
conventional symbols with no connection to any reality.
When things do not fit into their preconceived schemes,
they blithely force them in and use florid terminology to
keep critical judgment at bay. They do not look upon
practical deviations from their arbitrary theoretical
constructs as problems, as signals that they have made
mistakes somewhere on their way; they just add more
symbols and make their theories more abstruse. Please
don’t think I am exaggerating – that’s the way it is.
Why are so many people drawn to and impressed by
modern symbolic logic? Part of the problem is of course
that this is what the universities demand from their
teaching staff and teach their students; papers have to be
written in symbolic terms to be even considered. But
why this preference? Perhaps because pages filled with
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esoteric symbols seem more ‘scientific’, reminding
readers of mathematical formulae in the physical
sciences. It matters little that in logical science the
subjacent subject-matter becomes less transparent and
comprehensible when translated into symbols. Indeed,
part of the aim is to befuddle and intimidate the reader,
so as to conceal weaknesses and faults in the treatment.
The grandiloquent language is similarly useful as
eyewash. Modern symbolic logic boasts of superiority to
classical formal logic, to give itself authority; but the
truth is that most good ideas the former has it has stolen
from the latter, reworking them a little and renaming
them to seem original and independent. The whole
enterprise is a massive ongoing fraud; or, alternatively, a
collective delusion of epidemic proportions.
I am not, of course, saying (as, no doubt, some will rush
to accuse me of saying) that everything modern symbolic
logic tells us is false and irrelevant, or stolen. What I am
saying is that whatever is true and significant in it is
certainly not due to symbolization and axiomatization,
and can equally well be (could be and probably was)
developed by classical formal logic. Moreover, to repeat,
excessive symbolism tends to simplistically lump things
together and gloss over important nuances, and
condemns its users to rigid and abstract thinking
processes out of touch with the empirical domain.
We have seen, in the course of the present treatise on a
fortiori logic, how some budding or experienced
logicians strayed or failed due to their attempt to solve
problems by means of modern symbolic logic. In the
following pages, I present some more examples of the
relative ineffectiveness of modern symbolic logic
compared to classical formal logic. I show how the issue
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THE LAWS OF THOUGHT
of ‘existential import’ is far less significant that it is
touted to be; how attempts to bypass the laws of thought
are futile; how the liar paradox is not only due to self
reference; and how the Russell paradox is due to the
acceptance of self-membership.
The Existential Import Doctrine
A term is, nowadays, said to have ‘existential import’ if
it is considered to have existing referents; otherwise, it is
said to be ‘empty’ or a ‘null class’. For examples, ‘men’
has existential import, whereas ‘dragons’ does not. This
concept is considered original and important, if not
revolutionary, in modern symbolic logic; and it is often
touted as proof of the superiority of that school over that
of classical formal logic. We shall here examine and
assess this claim. As we shall see, although the concept
has some formal basis, it is in the last analysis logically
trivial and cognitively not innocuous.
The founder of formal logic, Aristotle, apparently did not
reflect on the issue of existential import and therefore
built a logical system which did not address it. The issue
began to be raised in the middle ages, but it was not till
the latter half of the nineteenth century that it acquired
the importance attached to it today by modern logicians.
a.
Based on Aristotle’s teaching, classical formal
logic recognizes six basic categorical forms of
proposition: the general affirmative, “All S are P” (A),
which means that each and every S is P; the general
negative, “No S is P” (E), which means that each and
every S is not-P; the particular affirmative, “Some S are
P” (I), which means that each of an indefinite number
(one or more) of S is P; the particular negative, “Some S
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are not P” (O), which means that each of an indefinite
number (one or more) of S is not-P; and the singular
affirmative, “This S is P” (R), and the singular negative,
“This S is not P” (G), which refer to a specifically
pointed-to or at least thought-of individual instance. Note
that general (also called universal) propositions and
particular
propositions
are
called
plural,
in
contradistinction to singular ones114. The labels A, E, I,
O, R and G come from the Latin words AffIRmo and
nEGO; the first four are traditional, the last two (R and
G) were introduced by me years ago115.
The symbols S and P stand for the subject and predicate.
The verb relating them is called the copula, and may
have positive (is or are) or negative (is not or are not)
polarity116. In the present context, the copula should be
understood very broadly, in a timeless sense117. When we
say ‘is’ (or ‘is not’) we do not mean merely “is (or is not)
now, at this precise time,” but more broadly “is (or is not)
at some time, in the past and/or present and/or future.”
The expressions ‘all’, ‘some’ and ‘this’ are called
quantities. Obviously, the general ‘all’ covers every
single instance, including necessarily ‘this’ specific
114
Singular propositions are often called particular, but
this usage is inaccurate, since they refer to an indicated
individual.
115
One can remember these six labels by means of the
phrase ARIEGO.
116
What I have called ‘polarity’ is traditionally called
‘quality’, but the latter term is inaccurate and confusing and
should be avoided.
117
This approach allows us to momentarily ignore the
issue of modality, and reflects common usage in many
contexts. A fuller treatment of categorical propositions must of
course deal with modality; I do that in my earlier work, Future
Logic.
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THE LAWS OF THOUGHT
instance; and ‘all’ and ‘this’ both imply the particular
‘some’, since it indefinitely includes ‘at least one’
instance. The ‘oppositions’ between the six forms, i.e.
their
logical
interrelationships,
are
traditionally
illustrated by means of the following ‘rectangle of
oppositions’:
Diagram 29.1 – Aristotelian oppositions
Although Aristotle did not, to our knowledge, represent
the oppositions by means of such a diagram, we can refer
to it as a summary his views. It is taken for granted that,
on the positive side A implies R and R implies I (so, A
implies I), and on the negative side E implies G and G
implies O (so, E implies O), although these implications
cannot be reversed, i.e. I does not imply R or A, and R
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295
does not imply A, and so forth. This is called
subalternation118. The core opposition in this diagram is
the
contradiction between R and G; from this
assumption, and the said subalternations, all else
logically follows119. A and O are contradictory, and so
are E and I; A and E, A and G, E and R, are pairs of
contraries; I and O, I and G, O and R, are pairs of
subcontraries. Two propositions are contradictory if they
cannot be both true and cannot be both false; they are
contrary if they cannot be both true but may be both
false; they are subcontrary if they may be both true but
cannot be both false.
b.
Shockingly, the above traditional interpretation of
the basic categorical forms (Diagram 29.1) has in modern
times been found to be problematic. The above listed
propositions are not as simple as they appear. The form
“Some S are P” (I) means “Something is both S and P,”
while the form “All S are P” (A) means “Something is
both S and P, and nothing is both S and not-P;” similarly,
the form “Some S are not P” (O) means “Something is
both S and not-P,” while the form “No S is P” (E) means
“Something is both S and not-P, and nothing is both S
118
The implying proposition being called the subalternant
and the implied one the subaltern, and the two being called
subalternatives.
119
If A is true, then R is true, then G is false, then E is
false; whence, the contrarieties shown on the diagram. If I is
false, then R is false, then G is true, then O is true; whence,
thus the subcontrarieties shown. Since R and G are
incompatible (cannot both be true) and exhaustive (cannot
both be false), it follows that A and O, and likewise E and I,
whose instances overlap somewhat, must be contradictory,
since, if they were both true or both false, R and G would in at
least one case be accordingly both true or both false (this is
proof by exposition).
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THE LAWS OF THOUGHT
and P.” Seeing the forms I, A, O, E, in this more detailed
manner, we can understand that A implies I since I is part
of A (and likewise for E and O), but then we realize that
A and O are not truly contradictories (and likewise for E
and I).
The exact contradictory of “Something is both S and not
P” (O) is “Nothing is both S and not-P” (i.e. only part of
A, with no mention of its I component) and the exact
contradictory of “Something is both S and P, and nothing
is both S and not-P” (A) is “Nothing is both S and P,
and/or something is both S and not-P” (i.e. a disjunction
including O, but also E). Note this well120.
It should be pointed out that “All S are P” (A) can be
defined more briefly as: “Something is S, and nothing is
both S and not-P;” for given this information, it follows
logically that the things that are S are P (for if this was
denied, it would follow that some things are both S and
not-P), Similarly, “No S is P” (E) can be defined more
briefly as: “Something is S, and nothing is both S and P,”
without need to specify explicitly that “Some things are
both S and not-P.” Thus, all four forms A, E, I, O, imply,
or presuppose (which is logically the same), that “some S
exist(s).” Also, the positive forms, A and I, imply that
“some P exist(s).” On the other hand, the negative forms,
E and O, do not imply that “some P exist(s),” since the
120
The Kneales propose a similar analysis of the problem
in The Development of Logic (Oxford, London: Clarendon,
1962), chapter II, section 5. Further on (on p. 211), they say
that Peter Abelard “should have the credit of being the first to
worry about the traditional square of opposition, though he did
not work out all the consequences of the change he
advocated.”
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negation of a term is not informative regarding its
affirmation121.
Thus, in the above diagram, the diagonal links between
the corners A and O, and between E and I, should not be
contradiction but contrariety. For, while to affirm one
proposition implies denial of its opposite, to deny one
proposition does not imply affirmation of the other. To
remedy this real problem of consistency, modern
logicians have proposed to redefine the general
propositions A and E as the exact contradictories of O
and I, respectively. That is to say, the new meaning of A
is only “Nothing is both S and not-P” and the new
meaning of E is only “Nothing is both S and P.” It
follows from this measure that A (in its new, slimmer
sense) no longer implies I, and likewise E (in its new,
slimmer sense) no longer implies O. This redefinition of
symbols A and E can, to my mind, lead to much
confusion. In my view, it would be better to re-label the
forms involved as follows:

Keep the traditional (old) labels A and E without
change of meaning; i.e. old A = A, old E = E.
Label the modern (new) senses of A and E as
respectively not-O and not-I.
That is, new ‘A’ = not-O ≠ old A. Whereas, old A
= new ‘A’ plus I = I and not-O.
Likewise, new ‘E’ = not-I ≠ old E. Whereas, old
E = new ‘E’ plus I = O and not-I.
Thus, when we say A or E in the present paper, we mean
exclusively the traditional A and E; and when we wish to
121
We could say that nothing in the world is conceivably
P, without affecting the truth of “Some S are not P” or “No S is
P.” Clearly, in the special case where “nothing is P,” the latter
propositions are true for any and every value of S.
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THE LAWS OF THOUGHT
speak of the modern ‘A’ and ‘E’ we simply say not-O
and not-I, respectively. Note this convention well122.
Actually, such propositional symbols are effectively
abandoned in modern logic and the propositions are
expressed by means of a symbolic notation, including the
existential and universal quantifiers, ∃ (there exists) and
∀ (for all), respectively; but we do not need to get into
the intricate details of that approach here, because we can
readily discuss the issues of interest to us in plain
English. Now, let us consider the formal consequences of
the above findings in pictorial terms.
One way for us to solve the stated problem is to merely
modify the traditional rectangle of oppositions, by
showing the diagonal relationships between A and O and
between E and I to be contrariety instead of
contradiction; this restores the traditional diagram’s
consistency, even if it somewhat dilutes its force
(Diagram 29.2). Another possibility, which is the usual
modern reaction, is to change the top two corners of the
rectangle to not-O and not-I, instead of A and E
respectively; this allows us to retain the contradiction
between diagonally opposed corners, although now the
lateral
relation
between
the
top
is
unconnectedness instead of contrariety, and the vertical
corners
122
Of course, we could introduce modified symbols for
the new A and E, such as A’ and E’, but I prefer to stress their
underlying meanings, viz. not-O and not-I. In my view, it is
dishonest and misleading to redefine the symbols A and E
themselves as meaning only not-O and not-I. This is like a
hostile takeover, permanently blocking further reflection and
debate.
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relations in the upper square are unconnectedness instead
of subalternation (Diagram 29.3).123
29.2 – modified traditional
123
A third possible approach is, of course, to draw a
rectangle with A and E in the top two corners, and not-E and
not-A (instead of I and O) in the bottom two corners. In that
case, it is the lower square that would suffer changes, with
not-E and not-A as unconnected to each other and to R and G
respectively. This possibility is however not very interesting,
as the forms not-E and not-A are disjunctive. That is, not-E =
not-(O and not-I) = not-O and/or I; and not-A = not-(I and not
O) = not-I and/or O. Note that this position is historically found
in Peter Abelard, who insisted on distinguishing between “Not
all S are P” (not-A) and “Some S are not P” (O), and who
apparently denied that “No S is P” (E) implies anything to be S
let alone P (even while regarding “All S are P” (A) as implying
that something is S); see Kneales, p. 210.
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THE LAWS OF THOUGHT
29.3 – modern version
Notice that the lower square of the modern version is
unchanged. This is due to the judgment that the forms I
and O, i.e. “Something is both S and P” and “Something
is both S and not-P,” both imply that “some S exist” (or
“some things are S” or “there are things which are S”)
meaning that if they are true, their subject ‘some S’ has
existential import. Moreover, in the case of I, the
predicate P is also implied to have existential import,
since it is affirmed; but in the case of O, the predicate P
is not implied to have existential import, since it is
merely denied. Until now, note well, we have not
mentioned the issue of existential import in our formal
treatment. Now, it comes into play, with this
interpretation of particular propositions.
The same applies to R and G – their subject ‘this S’ has
existential import, whereas the predicate P has it if
affirmed but lacks it if denied. On the other hand, since
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not-O (as distinct from A) is a negative statement, i.e.
means “Nothing is both S and not-P,” it has no
implication of existential import. Similarly, since not-I
(as distinct from E) is a negative statement, i.e. means
“Nothing is both S and P,” it has no implication of
existential import. Clearly, if not-O was thought to be
contrary to not-I, then if not-O were true, it would imply
the negation of not-I, i.e. it would imply I; but this being
erroneous, not-O and not-I cannot be contrary, i.e. they
must be unconnected. Similarly, if not-O was assumed to
imply R, it would then imply I, since R still implies I;
therefore, not-O must also be unconnected to R; and
similarly for not-I and G. On the other hand, not-O
remains contrary to G, since if not-O is true, then O is
false, in which case G must be false; similarly as regards
not-I and R.
It is now easier to see why the traditional rectangle of
oppositions (7.1) seemed right for centuries although it
was strictly-speaking wrong. It was tacitly assumed when
drawing it that the subjects of general propositions
always have existential import, i.e. imply that “some S
exist (s).” When this condition is granted, then in
combination with it not-O becomes A and not-I becomes
E, and A implies I and E implies O, and A exactly
contradicts O and E exactly contradicts I – in other words
we happily return to the original rectangle of oppositions
(7.1). The problem is that this condition is not always
satisfied in practice. That is, not-O or not-I can occur
without their subject S having existential import.
Effectively, the forms “Nothing is both S and not-P”
(not-O) and “nothing is both S and P” (not-I) signify
conditional propositions (“Whatever is S, is P” and
“Whatever is S, is not P”) which, without the minor
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THE LAWS OF THOUGHT
premise “this is S,” cannot be made to conclude “this is
P” or “this is not P” (respectively). In other words, they
record a ‘connection’ between an antecedent and a
consequent, but they have no ‘basis’, i.e. they contain no
information affirming the antecedent, and thence the
consequent. Obviously, if that information is provided,
the condition is fulfilled and the result follows. Once we
realize that the traditional rectangle remains true in the
framework of a certain simple condition (viz. that some S
exist), we see that its hidden ‘inconsistency’ is not such a
big problem for formal logic.
It is interesting to also consider the significance of the
above revisions in the field of eduction (i.e. immediate
inference). Whereas A, which implies I (“Some S are
P”), is convertible to “Some P are S” – not-O, which
does not imply I, is not so convertible. Also, whereas
not-I is convertible to “No P is S,” since “Nothing is S
and P” and “Nothing is P and S” are equivalent and have
no implication of existential import for S or P – E is not
likewise unconditionally convertible, since in its case
even if we are given that “some S exist” we cannot be
sure that “some P exist” (but only that “some not-P
exist”). Note well, just as O does not imply predicate P to
have existential import, since it merely negates it, so is it
true for E; therefore, the traditional conversion of E is
really only valid conditionally. We can also look into the
consequences of the above revisions in the field of
syllogistic reasoning; the main ones are pointed out
further on.
c.
Let us now go a step further in the possible
critique of Aristotelian oppositions, and suggest that all
terms may be denied to have existential import, whatever
the forms they occur in, and whatever their positions
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therein. That is to say, not only the subjects of general
propositions, but even the subjects of singular or
particular propositions might conceivably lack existential
import. Although R and G, and I and O, do formally
imply that some S exist(s), it is still possible to deny
them in pairs without self-contradiction. That is, R and G
cannot be claimed strictly-speaking contradictory,
because if “this S exists” is false then they are both false;
this means that their traditional relation of contradiction
is valid only conditionally (i.e. provided “this S exists” is
true) and their absolute relation is in truth only
contrariety. Similarly, I and O are only relatively
subcontrary and their unconditional relation is really
unconnectedness.
Indeed, it happens in practice that we reject a singular
subject altogether, when we find that some predicate can
be both affirmed and denied of it. This is dilemmatic
argument: finding both that ‘this S’ is P and that it is not
P, we must conclude that either one of these predications
is wrong, or both are wrong because ‘this S’ does not
exist. Particulars, of course, do not necessarily overlap;
but if we can show by other means that “no S exists,” we
can be sure that neither the set of S referred to by I nor
that referred to by O exist, and thus deny both
propositions at once. Granting all this, the above
diagrams (7.2 and 7.3) can be further modified as
follows:
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THE LAWS OF THOUGHT
29.4 re-modified traditional
29.5 – modified modern
version
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305
In both these diagrams (7.4 and 7.5), all relations are the
same as before, except the one between R and G
(contrariety), and those between R and O, G and I, and I
and O (which are now unconnected pairs). Notice that in
the second diagram (7.5), although R and G are no longer
contradictory, the pairs not-O and O, and not-I and I,
remain contradictory, since if we deny that “Something is
both S and P” (I) on the basis that “No S exists,” we can
all the more be sure that “Nothing is both S and P” (not
I), and likewise regarding O and not-O.
d.
We have thus proposed two successive dilutions
(weakening revisions) of the traditional rectangle of
oppositions. In the first, we followed modern logic in no
longer assuming with Aristotle that the subjects of
universal propositions have existential import. In the
second, we went further and additionally denied that
singular and particular propositions may well lack
existential import. Clearly, if our goal is to formulate an
absolute logic, one applicable equally to propositions
with existential import and those without, the successive
dilutions of the Aristotelian diagram are justified and
important. But are such logics of anything more than
academic interest – are they of practical interest? The
answer must clearly be no, as I will now explain.
is
A difficulty with the ideas of existential import and
emptiness
immediately
apparent:
these
are
characterizations that may be true or false. Different
people at the same time, or the same person at different
times, may have different opinions as to the existential
import or emptiness of a certain term. Some people used
to think that dragons exist, and maybe some people still
do, yet most people today think dragons never existed.
So, these characterizations are not obvious or fixed. Yet
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THE LAWS OF THOUGHT
modern logicians present the question of existence or
non-existence as one which has a ready answer, which
can be formally enshrined. They fail to see that the issue
is not formal but contentual, and thus in every given
material case subject to ordinary processes of testing and
eventual confirmation or disconfirmation.
It follows that the issue of existential import is not as
binary as it is made out to be. The issue is not simply
existence or non-existence, as modern logicians present
it. The issue is whether at a given time we know or not
that existence or non-existence is applicable to the case at
hand. A term with existential import may be said to be
‘realistic’, in that it refers (or is believed to refer) to some
existing thing(s). An empty term, i.e. one without
existential import, may be said to be ‘unrealistic’, in that
it refers (or is believed to refer) to a non-existent thing.
In between these two possibilities lies a third, namely
that of ‘hypothetical’ terms, for which we have not yet
settled the issue as to whether they are (in our opinion)
realistic or unrealistic. Moreover, this third possibility is
not monolithic like the other two, but comprises a host of
different degrees.
Our knowledge is mostly based on experience of
physical and mental phenomena, though also on logical
insights relating to such experience. Roughly put, we
would regard a term as realistic, if we have plentiful
empirical evidence as to the existence of what it refers to,
and little reason to doubt it. We would regard a term as
unrealistic, if we have little empirical evidence as to the
existence of what it refers to, and much reason to doubt
it. And we would regard a term as hypothetical if we are
thus far unable to decide whether it should be
characterized this way or that. In any case, the decision is
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usually and mostly inductive rather than purely deductive
as modern logicians effectively imagine it.
How are terms formed? Very often, a term is formed by
giving a name to a circumscribed phenomenon or set of
phenomena that we wish to think about. Here, the
definition is fixed. More often, a term is applied
tentatively to a phenomenon or set of phenomena, which
we are not yet able to precisely and definitively
circumscribe. In such case, we may tentatively define it
and affirm it, but such a term is still vague as well as
uncertain. Over time we may succeed in clarifying it and
making it more credible. Here, the definition is variable.
Thus, the formation of terms is usually not a simple
matter, but an inductive process that takes time and
whose success depends on the logical skills of the
thinker(s) concerned.
Of course, as individuals we mostly, since our childhood,
learn words from the people around us. This is
effectively
fixed-definition
terminology
for
the
individual, even if the term may have been developed
originally as a variable-definition one. In this context, if
we come across an obscure ready-made term, we cannot
understand it till we find some dictionary definition of it
or someone somehow points out for us the referent(s)
intended by it. But even then, inductive acts are needed
to understand the definition or the intent of the pointing.
When you point at something, I cannot immediately be
sure exactly what it is you are pointing at; I may have to
ask you: ‘do you mean including this, excluding that?’
and thus gradually zero in on your true intent.
Each of us, at all times, retains the responsibility to judge
the status of the terms he or she uses. The judgment as to
whether a term is realistic, or unrealistic is not always
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THE LAWS OF THOUGHT
easy. In practice, therefore, most terms are effectively
hypothetical, whether classed as more probably realistic
or more probably unrealistic. Even so, some terms are
certainly realistic or unrealistic. All terms that are truly
based exclusively on empirical evidence or whose denial
is
self-contradictory are certainly realistic, and all
manifestly counterfactual or self-contradictory terms are
certainly
unrealistic.
So,
all
three
characterizations are needed and effective.
of
these
Let us suppose the formation of realistic terms is obvious
enough, and ask how imaginary ones are formed.
Imaginary terms are not formed ex nihilo; they are
formed by combining old terms together in new ways. A
new term T is imagined by means of two or more
existing terms T1, T2…. We would call term T realistic,
if all the terms (T1, T2…) constituting it are realistic and
their combination is credible. But if all the terms on
which T is based are realistic, but their combination is
not credible (e.g. we know that no T1 is T2, so the
conjunction T1 + T2 is contrary to fact), we would call T
unrealistic; and of course, if one or more of the terms
constituting T is/are unrealistic, we would call T
unrealistic. If T is made up of hypothetical elements or if
its elements are realistic but their combination is of
uncertain status, we would call T hypothetical.
Now, our thinking in practice is aimed at knowledge of
reality. That is to say, when we come across a term
without existential import, i.e. when we decide that a
term is unrealistic, whether because it goes against our
empirical observations or because it is in some way
illogical—we normally lose interest in it and drop it. We
certainly do not waste our time wondering whether such
a subject has or lacks some predicate, since obviously if
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the subject is non-existent it has no predicates anyway. If
we regard a term as empty, the oppositions of its various
quantities and polarities in relation to whatever predicate
are henceforth totally irrelevant. An empty term, once
established as such, or at least considered to be such,
plays no further role in the pursuit of knowledge. This
attitude is plain common sense, except perhaps for
lunatics of various sorts. For this reason, the oppositions
between propositions involving empty terms are trivial.
That is, the above detailed non-Aristotelian models of
opposition are insignificant.
The net effect of the successive ‘dilutions’ is to make the
strong, Aristotelian rectangle of oppositions (concerning
propositions with existential import) seem like a special
case of little importance, and to give the weaker, non
Aristotelian rectangles (concerning variously empty
propositions)
a
disproportional
appearance
of
importance. The reason why this occurs is that the
weaker oppositions represent the lowest common
denominator between the Aristotelian and non
Aristotelian oppositions, which we need if we want to
simultaneously discuss propositions with and without
existential import. But the result is silly, for the
Aristotelian diagram (7.1) is the important one, teaching
us to think straight, whereas the non-Aristotelian ones are
really of very minimal and tangential academic interest.
Practical logic is focused on terms that are believed to be
realistic or at least hypothetical – it is not essentially
concerned with empty terms. Contrary to the accusations
made by modern logicians, Aristotelian logic is not only
concerned with realistic terms. It is in fact mainly used
with hypothetical terms, since (as already pointed out)
most of the terms which furnish our thoughts are
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THE LAWS OF THOUGHT
hypothetical – tentative constructs in an ongoing
inductive enterprise. We do not think hypothetical
thoughts by means of some special logic – we use the
same old Aristotelian logic for them. That is to say, in
accord with the principle of induction, we treat a
hypothetical term as a realistic term until and unless we
have reason to believe otherwise.
The reason we do so is that a hypothetical term, i.e. one
not yet proved to be realistic or unrealistic, is a candidate
for the status of realism. This being the case, we treat it
as we would any realistic term, subjecting it to the
strong, Aristotelian model of oppositions, rather than to
any watered-down model with wider aspirations, in the
way of an inductive test. If the hypothetical term is
indeed deserving of realistic status, it will survive the
trial; if, on the other hand, it does not deserve such status,
it will hopefully eventually be found to lead to
contradiction of some sort. In that event, we would
decide that the hypothetical term should rather be classed
as an unrealistic term, and we would naturally soon lose
interest in it. Thus, there is only one significant and
useful model for oppositions between propositions,
namely the Aristotelian one.
Indeed, we sometimes use Aristotelian logic even for
unrealistic terms. Very often, we remove the stigma of
unrealism by rephrasing our statement more precisely124.
124
For example, we might say (instead of “unicorns are
horses with a horn”) “the imaginary entities called unicorns
look like horses with a horn on their forehead” or (instead of
“some unicorns are white, some black”) “some of the unicorn
illustrations I have seen involve a white horse, but some
involve a black one”. Note that both the initial propositions
(given in brackets) have empty terms, even though one is
general and the other is particular. Clearly, after such
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Alternatively, we might just keep the imaginary intent in
mind: say a novelist wishes to write about fictional
people, or even science-fiction creatures, he would not
logically treat his subjects as empty terms – but rather
subject them to the logic applicable to realistic terms, so
as to enhance the illusion of realism in his novel. Thus,
the logic applicable to empty terms which we have above
investigated is in practice never used.
Whatever the alleged existential import of the terms
involved, our thoughts remain guided by the demanding
model of Aristotelian oppositions. The rational pursuit of
knowledge still indubitably requires the clear-cut logic of
Aristotle enshrined in the traditional rectangle of
oppositions (diagram 29.1). The reason why Aristotle
took the existential import of the subjects of categorical
propositions for granted is, I suggest, because naturally,
if there is nothing (i.e. no subject) to talk about (i.e. to
predicate something of) we will not talk about it; and if
we are talking, then that presumably means we do have
something to talk about, i.e. a subject as well as a
(positive or negative) predicate. This is manifest
common sense.
If Aristotle – as far as we know, or at least as far as
readers of his extant works have so far managed to
corrective rephrasing the two propositions do have existential
import, although they do so with reference to imaginary
(mental) entities rather than to real (physical) ones.
Consequently, while the initial propositions cannot be said to
be true, the more precise ones replacing them can be said to
be true, and we can apply Aristotelian logic to them without
qualms. Note also in passing that even a seemingly eternally
imaginary entity may one day become real – for example, we
might by artificial selection or by some genetic manipulation
one day produce real unicorns.
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THE LAWS OF THOUGHT
discern, or so we are told by historians of logic – did not
ask the question regarding the existence of the subject, it
is probably simply because he quite intelligently had no
interest in empty subjects. He was rightly focused on the
pursuit of knowledge of the world facing him, not some
non-existent domain. Modern logicians are rather, I
suggest, more intent on impressing the yokels with their
intellectual brilliance. With that overriding purpose in
mind, they fashion systems of no practical significance
whatever. They make mountains out of molehills,
presenting trivia as crucial discoveries, so as to draw
attention to their own persons.
e.
Modern logic is a complex web of static
relationships, most of them irrelevant. It ignores the
dynamics of human thinking, the fact that our knowledge
is constantly in flux. It is, we might say, a science of
space irrespective of time. In an effort, on the surface
praiseworthy, to formally acknowledge the issue of
existential import, it gives undue attention to empty
terms, elevating them from a very marginal problem to a
central consideration. Instead of dealing with existential
import parenthetically, as a side issue, it erects a logical
system that effectively shunts aside some of the most
important logical processes in the human cognitive
arsenal.
The traditional universal propositions are cognitively of
great importance. They cannot just be discarded, as
modern logic has tried doing under the pretext that
formal logic had to be expanded to include consideration
of counterfactual terms. There are logical processes
involving these propositional forms that are of great
practical importance, and which logic must focus on and
emphasize. It is absurd to henceforth effectively ignore
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these venerable and indispensable forms while making a
big thing of a theoretical consideration of no practical
significance whatever. The universals A and E cannot be
retired under any pretext; they are not mere conventional
conjunctions of more primitive forms.
For a start, universal propositions are essential to the
crucial logical processes of subsumption and non
subsumption, which are enshrined in Aristotle’s
syllogistic. First figure syllogisms serve to include an
instance in a class or a subclass in a wider class; they
teach us the notion that ‘all X’ includes every individual
‘this X’ and any possible set of ‘some X’. If, instead of
an argument such as “All X are P and this S is X,
therefore this S is P” (1/ARR) we propose the modern
major premise “Nothing is X and not-P,” with the same
minor premise, we obviously (even though the minor
premise implies the existential import of an X) can no
longer directly draw the desired conclusion! We are
forced to stop and think about it, and infer that “this S is
not not-P” before concluding that “this S is P.” Similarly,
second figure syllogisms serve to exclude an instance
from a class or a subclass from a wider class, and third
figure syllogisms to identify overlaps between classes;
and the moods of these figures become inhibited or
greatly
distorted
if
universal
propositions
reinterpreted as modern logicians suggest.
are
Again, universal propositions are essential to the crucial
logical processes of generalization and particularization.
If ‘this X’ and ‘some X’ are not implied by ‘all X’, then
we cannot generalize from the former to the latter. Of
course, given ‘this X’ or ‘some X’, we do have
existential import, and thus can anyway generalize to ‘all
X’. But the fact remains that if, in accord with modern
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THE LAWS OF THOUGHT
logic, we conceive our generalization as a movement of
thought from “This/Some X is/are Y” to “Nothing is X
and not-Y,” we miss the point entirely, even if admittedly
the existential import of X is implied by the premise. For
in such case, the formal continuity between premise and
conclusion is lost, there being two inexplicable changes
of polarity (from something to nothing and from Y to
not-Y)! Similarly, particularization requires formal
continuity. To move freely from I to A, and then possibly
to IO, we need the traditional opposition (contradiction)
between A and O.
Another issue that is ignored by modern logicians is
modality. Although modern logic has developed modal
logic to some extent, it has done so by means of
symbolic notations based on very simplistic analyses of
modality. Although it has conventionally identified the
different categories of modality (necessity, impossibility,
actuality, inactuality, possibility, unnecessity), it has not
thoroughly understood them. It has not clearly identified
and assimilated the different types of modality (the
logical, extensional, natural, temporal, and spatial
modes), even if human discourse has included them all
since time immemorial. Notably lacking in its treatment
is
the awareness that modality is an expression of
conditioning and that the different types of modality give
rise to different types of conditioning.
Consideration of modality is manifestly absent in the
doctrine of existential import. The latter (as we saw) is
built around the timeless (or ‘omnitemporal’) forms of
categorical proposition, which are non-modal. It does not
apply to modal categorical propositions, for these do not
formally imply (or presuppose) the actuality of their
subject but only its possibility. Thus, a universal
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proposition with natural-modality, “All S can (or must)
be P,” does not formally imply that “Some things are S”
but only that “Some things can be S;” likewise, one with
temporal modality, “All S are sometimes (or always) P”
does not imply that “Some things are S” but only that
“Some things are sometimes S;” and so forth.
This may be called ‘existential import’ in a broadened
sense, acknowledging that being has degrees; but it is
certainly not the actual sense intended by modern
logicians: they apparently imagine that use of such modal
propositions implies belief that “Some things are S.”
And of course, the modality of subsumption, as I have
called this phenomenon in my book Future Logic
(chapter 41), is very relevant to the processes of
opposition, eduction (immediate inferences), syllogistic
deduction (mediate inference) and induction. Regarding
the latter, see my detailed theory of factorial induction in
the said work. Thus, we may well say that the proponents
of the doctrine of existential import constructed an
expanded system of logic based on a rather narrow vision
of the scope of logic. Even if their expansion (for all it is
worth—not much, I’d say) is applicable to non-modal
propositions, it is not appropriate for modal ones.
f.
The critique of the Aristotelian rectangle of
oppositions began apparently in the middle ages, with
Peter Abelard (France, 1079-1142). According to the
Kneales, further input on this issue was made over time
by William of Shyreswood, by Peter of Spain and St.
Vincent Ferrer, and by Leibniz. They also mention
Boole’s interest in it, and many people attribute the
modern view of the issue to this 19th century logician.
However, E. D. Buckner suggests that the modern idea
stems rather from Franz Brentano (Austria, 1838-1917),
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THE LAWS OF THOUGHT
in a paper published in 1874125. And of course, many big
name logicians such as Frege and Russell have weighed
in since then.
Even though the new logic that ensued, based on the
concept of existential import, is today strongly
entrenched in academia, the switchover to it was
epistemologically clearly not only unnecessary but ill
advised. The doctrine of existential import has been
woefully misnamed: it is in fact not about existential
import, but rather about non-existential import. It gives
to empty terms undue importance, and thus greatly
diminishes the real importance of non-empty terms. To
be sure, this innovation fitted the anti-rational ‘spirit of
the times’, and it kept many people happily busy for over
a century, and thus feeling they existed and were
important – but it was in truth emptiness and vanity.
Apparently, none of these people reflected on the
obvious fact that once a term is identified as empty, it is
simply dumped – it does not continue affecting our
reasoning in any significant manner. This being so, there
is no need to abandon the universal forms A and E
because they imply (presuppose) the existential import of
their subject. Even if the Aristotelian framework, which
is built around non-empty terms, occasionally ‘fails’ due
to the appearance of an empty term in discourse, such
event is taken in stride and dealt with by summarily
eliminating the discredited term thenceforth, and
125
For Buckner’s account of the history, see:
www.logicmuseum.com/cantor/Eximport.htm. Notice his
pretentious characterization of “the traditional ‘syllogistic’” as
“a historical curiosity.” Brentano’s position is to be found in his
Psychologie vom empirischen Standpunkt, II, ch. 7. The
Kneales do mention the latter reference in passing, in a
footnote on p. 411.
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317
certainly not by switching to a non-Aristotelian
framework as modern logicians recommend to do. In any
case, the issue of existential import does not apply to
modal logic, and so lacks generality.
Moreover, these people failed to realize that Aristotelian
logical processing relates not only to realistic terms, but
more significantly to hypothetical terms, i.e. terms in
process. They viewed logic as a deductive activity; they
did not realize its essentially inductive character. If, due
to an immoderate interest in empty terms, the science of
logic abandons the universal forms A and E, it deprives
people of a language with which to accurately express
the movements of thought inherent in the processes of
syllogistic
inference and of generalization and
particularization. The science of logic must acknowledge
the forms of actual human thinking, and not seek to
impose artificial contraptions of no practical value.
Otherwise, natural processes essential to human
cognition cannot be credibly expressed and logic will
seem obscure and arbitrary.
Modern logic has sown confusion in many people’s
minds, turning the West from a culture of confident
reason to one of neurotic unreason. The purpose of logic
studies ought to be to cognitively empower people, not
incapacitate them. If logicians err in the forms of thought
they describe and prescribe, they betray their mission,
which is to intelligently and benevolently guide and
improve human thinking. If they err, whether out of
stupidity or malice, they turn logic from a responsible
science and a fine art to a vain and dangerous game.
They do not merely cease benefitting mankind; they
positively harm people’s minds.
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CHAPTER THIRTY
Drawn from A Fortiori Logic (2013),
Appendix 7.4-5.
THE LIAR AND RUSSELL PARADOXES
(REDUX)
The Liar Paradox (Redux)
I dealt with the Liar paradox previously, in my Future
Logic126, but now realize that more needs to be said about
it.
This paradox is especially difficult to deal with
because it resorts to several different discursive ‘tricks’
simultaneously.
a.
The statement “This proposition is false” looks
conceivable offhand, until we realize that if we assume it
to be true, then we must admit it to be indeed false, while
if we assume it to be indeed false, then we must admit it
to be true – all of which seems unconscionable.
Obviously, there is a contradiction in such discourse,
since nothing can be both true and false. But the question
is: just what is causing it and how can it be resolved? We
are not ‘deducing’ the fact of contradiction from a ‘law
of thought’ – we are ‘observing’ the fact through our
rational faculty. We cannot, either, ‘deduce’ the
resolution of the contradiction from a ‘law of thought’ –
we have to analyze the problem at hand very closely and
126
See there chapter 32.2. (See also Ruminations 5.1.)
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creatively propose a satisfying solution to it, i.e. one
which indeed puts our intellectual anxiety to rest. As we
shall see, this is by no means a simple and
straightforward matter.
The proposition “This proposition is false” is a double
paradox, because: if it is true, then it is false; and if it is
false, then it is true. Notice the circularity from true to
false and from false to true. The implications we draw
from the given proposition seems unavoidable at first
sight. But we must to begin with wonder how we know
these implications (the two if–then statements) to be true.
How do we know that “it is true” implies “it is false,”
and that “it is false” implies “it is true”? Apparently, we
are not ‘deducing’ these implications from some unstated
proposition. We are, rather, using ad hoc rational insight
of some sort – i.e. in a sense directly ‘perceiving’
(intellectually cognizing) the implications of the given
proposition. But such rational insight, though in principle
reliable, is clearly inductive, rather than deductive, in
epistemological status. That is to say, it is trustworthy
until and unless it is found for some reason to be
incorrect. This means, there may be one or more errors in
our thinking, here; it is not cast in stone. And indeed
there must be some error(s), since it has led to double
paradox. Therefore, we must look for it.
Perhaps use of the pronoun “it” is a problem, for it is a
rather vague term. Let us therefore ask the question:
more precisely what does the pronoun “it” refer us to,
here?
At first sight, the “it” in “if it is true, then it is false; and
if it is false, then it is true” refers to the whole given
statement, “This proposition is false.” In that event, we
must reword the double paradox as follows: if ‘this
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proposition is false’ is true, then ‘this proposition is
false’ is false; and if ‘this proposition is false’ is false,
then ‘this proposition is false’ is true. Here, the subject
of the two if–then statements is more clearly marked out
as “this proposition is false,” and so remains constant
throughout. But this clarification reveals an abnormal
changes of predicate, from “true” to “false” and from
“false” to “true,” which cannot be readily be explained.
Normally, we would say: if ‘this proposition is false’ is
true, then ‘this proposition is false’ is true; and if ‘this
proposition is false’ is false, then ‘this proposition is
false’ is false. The reason we here reverse the predicates
is
that we consider the original proposition, “this
proposition is false,” as instructing such reversal.
However, whereas a proposition of the form “‘this
proposition is false’ is true” is readily interpretable in the
simpler form “this proposition is false,” a proposition of
the form “‘this proposition is false’ is false” cannot
likewise be simplified. How would we express the
double negation involved? As “this proposition is true”?
Clearly, the meaning of the latter is not identical to that
of the former, since the subject “this proposition” refers
to different propositions in each case. So the formulation
of the liar paradox in full form, i.e. as “if ‘this
proposition is false’ is true, then ‘this proposition is
false’ is false; and if ‘this proposition is false’ is false,
then ‘this proposition is false’ is true,” does not make
possible the reproduction of the initial formula expressed
in terms of the pronoun “it.”
b.
Let us therefore try something else. If the
pronoun “it” refers to the term “this proposition”, then
the double paradox should be reformulated as follows: if
‘this proposition’ is true, then ‘this proposition’ is false;
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and if ‘this proposition’ is false, then ‘this proposition’ is
true. But doing that, we see that in each of these two if
then statements, though the subject (“this proposition”)
remains constant throughout, the predicate (“true” or
“false,” as the case may be) is not the same in the
consequent as it was in the antecedent. There is no
logical explanation for these inversions of the predicate.
Normally, the truth of a proposition P does not imply its
falsehood or vice versa.
We might be tempted to use the given “This proposition
is false” as a premise to justify the inference from the
said antecedents to the said consequents. We might try to
formulate two apodoses, as follows:
If
this proposition is
true, then it is false
(hypothesis),
and this proposition is
false (given);
therefore,
proposition
this
is
true
(putative conclusion).
If
this proposition is
false, then it is true
(hypothesis),
and this proposition is
false (given);
therefore,
proposition
this
is
true
(putative conclusion).
Obviously, in the first case we have invalid inference, in
that we try to deny the antecedent to deny the
consequent, or to affirm the consequent to affirm the
antecedent. In the second case, the putative conclusion
does follow from the premises; but we can still wonder
where the major premise (the hypothetical proposition)
came from, so we are none the wiser. So, this approach
too is useless – i.e. it proves nothing.
Alternatively, we might try formulating the following
two syllogisms:
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THE LAWS OF THOUGHT
This proposition is false
(given),
and this proposition is
true (supposition);
therefore,
this
proposition is false
(putative conclusion).
This proposition is false
(given).
and this proposition is
false (supposition);
therefore,
proposition
this
is
true
(putative conclusion).
Clearly, these arguments are not quite syllogistic in form;
but they can be reworded a bit to produce syllogisms.
The first two premises would then yield the conclusion
“there is a proposition that is true and false” (3/RRI),
which is self-contradictory (whence, one of the premises
must be false); the second two premises, however, being
one and the same proposition, would yield no syllogistic
conclusion other than “there is a proposition that is false
and false” (3/RRI), which is self-evident (and trivial).
But these are not the conclusions we seek, which must
concern “this proposition” and not merely “some
proposition.”
A better approach is to look upon the latter two
arguments as follows. In the first case, the premises “this
proposition is false” (given) and “this proposition is true”
(supposition) seem to together imply “this proposition is
both true and false;” and the latter paradoxical
conclusion in turn indeed suggests that “this proposition
is false,” since contradiction is impossible. And in the
second case, the premises “this proposition is false”
(given) and “this proposition is false” (supposition) agree
with each other that “this proposition is false,” and so
this is their logical conclusion. Since both arguments
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conclude with “this proposition is false,” the latter must
be the overall conclusion.
However, the latter result is not as conclusive as it seems,
because upon closer scrutiny it is obvious that “this
proposition is false” and “this proposition is true” do not
refer to the same subject, since the predicate changes.
The first “this proposition” refers to the proposition “this
proposition is false” and the second “this proposition”
refers to the proposition “this proposition is true.” So,
these two propositions in fact have different subjects as
well as different predicates (viz. false and true,
respectively). The subjects superficially look the same,
because they are verbally expressed in identical words;
but their underlying intent is not the same, since they
refer to significantly different propositions (propositions
with
manifestly
different,
indeed
contradictory,
predicates). This means that when the predicate changes,
the subject effectively changes too. When the predicate is
“true,” the subject means one thing; and when the
predicate is “false,” the subject means something else.
Although the words “this proposition” are constant, their
underlying intent varies. That is to say, the term “this
proposition” does not have a uniform meaning
throughout, and therefore cannot be used as a basis for
the inferences above proposed.
c.
Let us now try another angle. If we examine our
initial reasoning in terms of the pronoun “it” more
carefully, we can see what is really happening in it.
Given that ‘this proposition is false’ is true, we can more
briefly say: ‘this proposition is false.’ Also, given ‘this
proposition is false’ is false, we can by negation educe
that ‘this proposition is not false’ is true, which means
that ‘this proposition is true’ is true, or more briefly put:
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‘this proposition is true’127. In this way, we seem to
argue, regarding the subject “this proposition is false,”
from ‘it is true’ to ‘it is false’, and from ‘it is false’ to ‘it
is true’. But in fact the use of the pronoun “it” or the term
“this proposition” as abbreviated subject is a sleight of
hand, for the underlying subject changes in the course of
the second transition (that ending in “this proposition is
true”). When abbreviation is used throughout, we seem
to be talking about one and the same proposition
throughout as being both true and false. But seeing that
this is based on hidden equivocation, the paradoxes
disappear.
It is interesting to note that when the reasoning is viewed
more explicitly like that, the proposition “this proposition
is true” also becomes paradoxical! We can argue: if ‘this
proposition is true’ is true, then obviously ‘this
proposition is true’. And: if ‘this proposition is true’ is
false, then its contradictory ‘this proposition is not true’
must be true, which means that ‘this proposition is false’
is true, i.e. more succinctly: ‘this proposition is false’.
Here, superficially, there seems to be no paradox,
because we seem to argue, regarding the subject “this
proposition is true,” from ‘it is true’ to ‘it is true’, and
from ‘it is false’ to ‘it is false’. But if we look at the final
conclusion, viz. “this proposition is false,” we see that it
corresponds to the liar paradox!128 And here again, the
127
Some logicians have tried to deal with the liar paradox
by denying that true and false are contradictory terms, i.e. that
not-true = false and not-false = true. Such a claim is utter
nonsense; the attempt to shunt aside the laws of non
contradiction of the excluded middle so as to resolve a
paradox is self-contradiction in action.
128
That ‘this proposition is true’ is implicitly (if only
potentially) as paradoxical as ‘this proposition is false’ is, so
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explanation of the double paradox is that the apparent
subject “it” or “this proposition” changes significance in
the course of drawing the implications.
Notice that, in both these lines of reasoning, the first leg
is ordinary self-implication, mere tautology, while the
second leg is the operative self-contradiction, the
paradox. If the given proposition (whether “this
proposition is false” or “this proposition is true”) is true,
we merely repeat the proposition as is (without need to
add the predication “is true”). But if the given
proposition is false, we cannot drop the additional
predication (i.e. “is false”) without changing the original
proposition. Thus, we could say that the two
propositions, “this proposition is false” or “this
proposition is true,” present no problem when taken as
true; and it is only when they are hypothetically taken as
false that the problem is created. So we could say that the
way out of the liar paradox (and its positive analogue) is
simply to accept the two claims as true, and not imagine
them to be false!
We could furthermore, if we really want to, argue that
“this proposition is false” and “this proposition is true”
differ in that the former explicitly appears to put itself in
doubt whereas the latter does not do so. On this basis, we
could immediately reject the former and somewhat
accept the latter, even while admitting that the latter is
equally devoid of any useful information. That is to say,
since the former appears ‘more paradoxical’ than the
far as I know, a new discovery. Note well how both paradoxes
occur through quite ordinary eductions: viz. if ‘P is Q’ is
affirmed, then P is Q; and if ‘P is Q’ is denied, then ‘P is not Q’
is affirmed, then P is not Q (where P stands for ‘this
proposition’, and Q for ‘false’ or ‘true’ as the case may be).
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latter, the latter is to be preferred in extremis. But this,
note well, ignores the equally insurmountable difficulties
in it. It is better to resolutely reject both forms as vicious
constructs.
d.
To grasp the illusoriness of the liar paradox, it is
important to realize that the two forms, “this proposition
is false” and “this proposition is true,” are not each
other’s contradictory; and that, in fact, neither of them
has a contradictory! This is a logical anomaly, a fatal
flaw in the discourse of the liar paradox; for in principle,
every well-formed and meaningful proposition is
logically required to have a contradictory. If a
propositional form lacks a contradictory form, it cannot
be judged true or false, for such judgment depends on
there being a choice. We do not even have to limit our
propositions to the predicates “true” or “false” – any
predicate X and its negation not-X would display the
same property given the same said subject. That is, “this
proposition is X” and “this proposition is not-X” are not
each other’s contradictory, and are therefore both equally
deprived of contradictory.
We could, of course, remark that “this proposition is X”
can be denied by “that proposition (i.e. the preceding
one) is not X,” or even introduce a symbol for the
original proposition in the new proposition. In such case,
although the subjects would be verbally different, their
intents would surely be the same. But the form “that
proposition is not X” is more akin to the form “‘this
proposition is X’ is not X,” in which the whole original
proposition is given the role of subject and its predicate
is given the role of predicate. However, though these two
forms somewhat equivalent in meaning to each other and
to the original proposition, their logical behavior patterns
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are not identical with that of the original proposition, as
we have already seen. The fact remains that “this
proposition is not X” is not the contradictory of “this
proposition is X.”
Clearly, any proposition involving the special subject
“this proposition” exhibits a very unusual property, and
may be dismissed on that basis alone. The reason why
such a proposition lacks a contradictory is that its subject
refers to the proposition it happens to be in, and that
proposition is evidently different when the predicate in it
is the term “false” and when it is the term “true” (or more
generally, any pair of predicates ‘X’ and ‘not-X’). When
the predicate changes, so does the subject; so the subject
cannot be pinned-down, it is variable, it is not constant as
it should be. The term “this proposition” has a different
reference in each case, which depends on the predicate;
consequently, each subject can only be associated with
one predicate and never with the other (i.e. its negation).
From this we see that when at the beginning we thought,
looking upon the statement “This proposition is false,”
that if we take it at its word, then it is must be regarded
as false, and so we have to prefer to it “This proposition
is not false,” i.e. “This proposition is true,” and so forth,
we did not realize that we were in fact, due to the
ambiguity inherent in the term “This proposition” or “it,”
changing its meaning at every turn. This change of
meaning passes by unnoticed, because the term used is
by its very nature not fixed. The pronouns “this” and “it”
can be applied to anything and its opposite without such
change of meaning being verbally signaled in them. They
are not permanently attached to any object, but are
merely contextual designations. In the technical
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terminology of linguistics, they are characterized as
‘deictic’ or ‘indexical’.
Thus, it appears that the liar paradox arises, however we
understand its terms, as a result of some sort of
equivocation in the subject. Although we seem
superficially to refer to one and the same subject in the
antecedent and consequent of our if–then reasoning,
there is in fact a covert change of meaning which once
we become aware of it belies the initial appearance of
contradiction. The suggested impossible implications are
thus put in doubt, made incredible. The contradictions
apparently produced are thus defused or dissolved, by
virtue of our inability to make them stick.
e.
Another, and complementary, way to deal with
the liar paradox is to point out the logical difficulty of
self-reference. This is a tack many logicians have
adopted, including me in my first foray into this topic in
Future Logic. The argument proposed here is that the
term “this proposition” refers to an object (viz. “This
proposition is false” or “This proposition is true”) which
includes the term itself. A finger cannot point at itself,
and “this” is the conceptual equivalent of a finger.
Effectively, the expression “this” has no content when it
is directed at itself or at a sentence including it. It is
empty, without substance. It is as if nothing is said when
we indulge in such self-reference.
Thus, “This proposition is X” (where X stands for false,
or true, or indeed anything) is in fact meaningless; and a
meaningless sentence cannot be true or false. Such a
sentence can reasonably be described as neither true nor
false, without breach of the law of the excluded middle,
because neither of these logical evaluations is applicable
to meaningless sentences. “This proposition is false”
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looks meaningful because its four constituents (i.e.
“this,” “proposition,” “is” and “false”) are separately
normally meaningful. But in this particular combination,
where one of the elements (viz. “this”) does not refer to
anything already existent, the sentence is found to be
meaningless.
The apparent contradictions that self-reference produces
help us to realize its meaninglessness. And it is through
the intellectual realization of the meaninglessness of self
reference that we explain away and annul the apparent
contradictions. On this basis, we can say that even
though the sentence “This proposition is true” does not at
first sight give rise to any paradox (as people think: “if it
is
true, it is true; and if it is false, it is false”),
nevertheless, since it involves self-reference as much as
“This proposition is false,” it is equally meaningless and
cannot be characterized as true or false. In fact, as I have
shown above, “This proposition is true” does also give
rise to double paradox.
Someone might object: What about the propositions:
“this statement is self-referential” and “this statement is
not self-referential”? Surely, we can say that these are
meaningful and that the former is true while the latter is
false! The retort to that objection is that the two
propositions “this proposition refers to itself” and “this
proposition does not refer to itself” are not mutual
contradictories, because (just like in the liar paradox)
their subjects differ radically, each referring to the
proposition it is in and not to the other. Thus, while the
positive version may seem more self-consistent than the
negative one, and therefore to be preferred in extremis,
they are in fact both fundamentally flawed, because (just
like in the liar paradox) neither of them has a
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contradictory, and without the logical possibility of
negating a discourse it is impossible to judge whether it
is right or wrong.129
f.
Not long after the preceding reflections, I
happened to come across another interesting example of
paradoxical self-reference, namely “Disobey me!”130 This
involves the ‘double bind’ – if I obey it, I disobey it and
if I disobey it, I obey it. To resolve this paradox, we need
to first put the statement in more precise form, say: “you
must disobey this command!” We can then disentangle
129
Another objection (which was actually put to me by a
reader) would be propositions like “this statement has five
words” and “this statement has six words” – even though they
contain the demonstrative “this,” the former looks true and the
latter false! Here, we might in reply point out that though the
propositions “this statement has five words” and “this
statement does not have five words,” seem to mean opposite
things, they cannot be contradictories, since both appear true.
Also compare: “this statement has five words” and “this
statement does have five words” – the former is true while the
latter is false, though both mean essentially the same. Clearly,
the behavior of these propositions is far from normal, due to
their unusual dependence on the wording used in them. On
one level, we get the message of the proposition and count
the number of words in it, and then check whether this number
corresponds to the given number: if yes, the proposition is
judged ‘true’, and if no, it is judged ‘false’. But at the same
time, we have to be keep track of the changing reference of
the demonstrative “this,” which complicates matters as already
explained, and additionally in this particular context we must
beware of the impact of wording. The Kneales give “What I am
now saying is a sentence in English” as an example of
“harmless self-reference” (p. 228).
130
I found this example in Robert Maggiori’s La
philosophie au jour le jour (Paris: Flammarion, 1994); the
author does not say whether it is his own invention or
someone else’s (p. 438).
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the knot by realizing that the order being given has
outwardly imperative form but inwardly lacks content. It
does not define a specific, concrete action that is to be
done or not-done. If we wished to obey it, or to disobey
it, we would not know just what we are supposed to do or
not-do! It is therefore an order that can neither be obeyed
nor be disobeyed. Ruminating on this case led me to
what I now believe is the trump card, which convincingly
finalizes the resolution of the liar paradox, even as the
preceding reflections all continue to be relevant.
It occurred to me then that this is precisely the problem
with the liar paradox. It says “this proposition is false” –
but it does not tell us anything about the world that can
be judged as true or false. A ‘proposition’ is a statement
that makes some claim about the world. If the statement
makes no such claim, if it ‘proposes’ nothing, it cannot
be logically assessed as true or false. If it refers to
nothing – whether physical, mental or spiritual,
perceptual, intuitive or conceptual – it has no meaning. A
meaningless statement does not qualify as a
‘proposition’. The attributes of ‘true’ or ‘false’ are not
ordinary predicates, like ‘white’ or ‘black’, which can be
attached to any subject and then judged to be truly or
falsely attached. The attributes of ‘true’ or ‘false’ require
a precise claim to be made before they can at all be used.
The truth of this explication can be seen with reference to
the ‘propositional forms’ used in logic theory. Take, for
example, “All X are Y.” Such a propositional form
cannot be judged true or false because it manifestly has
no content. Only when such an abstraction is given some
specific content, such as “All men are mortal,” can we
begin to ask whether it is true or false. A propositional
form is too vague to count as a proposition. It does not
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THE LAWS OF THOUGHT
tell us anything about the world, other than implying that
there are (or even just that there may be) concrete
propositions which have this form. Just as we cannot
disobey or even obey an imperative without content, so
we cannot judge a purely formal expression true or false.
The same applies to the liar paradox: like a formal
proposition, it has no concrete content, and therefore
cannot be judged true or false. The liar paradox has no
content partly due to its having a self-referential subject
(“this proposition”). But the truth is, even if its subject
was not self-referential, it would still have insufficient
content. This is so, because its predicate “false” (and
likewise its opposite, “true”) is not an ordinary predicate;
it is more like a formal predicate. It can only be used if
another, more concrete predicate has already been
proposed for the subject at hand. For example, “this
proposition is interesting” could be judged true or false
(if it was not self-referential) because it already has a
predicate (viz. “interesting”). Thus, the problem with the
liar paradox is not only the self-reference it involves but
also its lack of a predicate more concrete than the logical
predicate “false” (or “true”).
All this illustrates how the ‘laws of thought’ are not
axioms in the sense of top premises in the knowledge
enterprise from which we mechanically derive other
premises. Rather the expression ‘laws of thought’ refers
to recurring insights which provide us with some
intellectual guidance but cannot by themselves determine
the outcome. The individual in pursuit of knowledge, and
in particular the logician, is driven by the obviousness or
by the absurdity of a situation to look for creative
solutions to problems. He or she must still think of
possible solutions and test them.
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The Russell Paradox (Redux)
Logic is what helps us transmute scattered concrete
perceptions into well-ordered abstract concepts. Human
knowledge, or opinion, is based on experience,
imagination and rational insight. The latter is a kind of
‘experience’ in the larger sense, a non-phenomenal sort
of experience, call it logical ‘intuition’. Reason was for
this reason called by the ancients, in both West and East,
the ‘sixth sense’ or ‘common sense’, i.e. the sense-organ
which ties together the other five senses, those that bring
us in empirical contact with phenomenal experience:
colors, shapes, sounds, smells, tastes, touch-sensations,
etc., whether they are physically perceived or mentally
imagined. The five senses without the sixth yield chaotic
nonsense (they are non-sense, one cannot ‘make sense’
of them); and conversely, the sixth sense is useless
without the other five, because it has nothing about
which to have rational insights. Imagination reshuffles
past experiential data and reasoning, making possible the
formation of new ideas and theories which are later
tested with reference to further experience and reasoning.
Elements of class logic. Logic initially developed as a
science primarily with reference to natural discourse,
resulting in what we today refer to as predicate logic. In
natural human discourse, we (you and me, and everyone
else) routinely think of and discuss things we have
perceived, or eventually conceived, by means of
categorical propositions involving a subject (say, S) and
a predicate (say, P) which are related to each other by
means of the copula ‘is’. Such propositions have the
form “S is P,” which may be singular or plural, and in the
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THE LAWS OF THOUGHT
latter case general (or universal) or particular, and
positive or negative, and moreover may involve various
modes and categories of modality131.
A proposition of the form ‘S is P’ is really a double
predication – it tells us that a thing which is S is also P;
thus, S and P are really both predicates, though one (the
subject S) is given precedence in thought so as to
‘predicate’ the other (the predicate P) of it132. Primarily, S
refers to some concrete phenomenon or phenomena (be
it/they physical, mental or spiritual), i.e. an individual
entity or a set of entities, and P to a property of it or of
theirs. For examples, “John is a man” and “All men are
human beings” are respectively a singular predication
(about one man, John) and a plural one (about all men).
Additionally, still in natural discourse, the subject of our
thoughts may be predicates as such, i.e. predicates in
their capacity as predicates; an example is: “‘men’ may
be the subject or predicate of a proposition.” The latter
occurs in specifically philosophical (or logical or
linguistic) discourse; for example, in the present essay.
Now, logicians through the ages, and especially in
modern times, have effectively found natural discourse
somewhat inadequate for their needs and gradually
developed a more artificial language, that of ‘classes’133.
131
We need not go into the details of these distinctions
here, for they are well known. There are also many fine
distinctions between different sorts of terms that may appear
in propositions as subjects or predicates; but let us keep the
matter simple.
132
‘Predication’ refers to the copula and the predicate
together as if they were an action of the speaker (or the
statement made) on the subject.
133
The following account of class logic is based on my
presentation in Future Logic, chapters 43-45. The word ‘class’
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335
This type of discourse exactly parallels natural discourse,
but is a bit more abstract and descriptive so as to
facilitate philosophical (or logical or linguistic) discourse
and make it more precise. In this language, instead of
saying “this S is P,” we say “this S is a member (or
instance) of P” (note well the lengthening of the copula
from ‘is’ to ‘is a member (or instance) of’. If ‘this S’
symbolizes a concrete individual, then ‘P’ here is called a
‘class’; but if ‘this S’ symbolizes an abstract class, then
‘P’ here is called a ‘class of classes’.
A class, then, is an abstraction, a mental constructs in
which we figuratively group some concrete things (be
they physical, mental or spiritual). Although we can and
do temporarily mentally classify things without naming
the class for them, we normally name classes (i.e. assign
them a distinctive word or phrase) because this facilitates
memory and communication. Naming is not the essence
of classification, but it is a great facilitator of large-scale
classification. The name of a class of things does not
‘stand for them’ in the way of a token, but rather ‘points
the mind to them’ or ‘draws our attention to them’; that is
to say, it is an instrument of intention.
A class in the primary sense is a class of things in
general; a class in the secondary sense is more
specifically a class of classes. Membership is thus of two
kinds: membership of non-classes in a class, or
membership of classes in a class of classes.
comes from the Latin classis, which refers to a “group called to
military service” (Merriam-Webster). I do not know whether the
Ancients used that word in its logical sense, or some such
word, in their discourse, but they certainly thought in class
logic mode. Examples of class thinking are Aristotle’s
distinction between species and genera and Porphyry’s tree.
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Alternatively, we may speak of first-order classes and
second-order classes to distinguish these two types.
There are no other orders of classes. When we think
about or discuss more concrete things, we are talking in
first-order class-logic; when we think about or discuss
first-order classes, we are talking in second-order class
logic, and the latter also applies to second-order classes
since after all they are classes too. The two orders of
classes should not be confused with the hierarchy of
classes within each order.
The relation between classes of classes and classes is
analogous to the relation between classes and concretes;
it is a relation of subsumption. When a lower (i.e. first
order) class is a member of a higher (i.e. second-order)
class, it does not follow that the members of the lower
class are also members of the higher class; in fact, if they
are members of the one they are certainly not members
of the other. Thus, for example, you and me, although we
are members of the class ‘men’ because we are men, we
are not members of the class ‘classes of men’ because we
are not ‘men’. Also, the class ‘men’ is not a man, but is a
member of the class ‘classes of men’. The members of
the class ‘classes of men’ (or more briefly put, ‘men
classes’), which is a class of classes, are, in addition to
the broad class ‘men’, the narrower classes ‘gardeners’,
‘engineers’, ‘sages’, ‘neurotics’, and so on.134
134
Note that saying or writing the word men without
inverted commas refers to a predicate. When we wish to refer
to the corresponding class, we say the class of men, or the
class men; if we are writing, we may write the same with or
without inverted commas, or simply ‘men’ in inverted commas.
When dealing with classes of classes, we say the class of
classes of men, or the class of men-classes, or the class men
classes, and we may write the same with or without inverted
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Hierarchization, on the other hand, refers to classes
within a given order that share instances, not merely by
partly overlapping, but in such a way that all the
members of one class are members of the other (and in
some but not all cases, vice versa). For example, since all
men are animals, though not all animals are men, the
class ‘men’ is a subclass (or species) of the class
‘animals’, and the class ‘animals’ is an overclass (or
genus) of the class ‘men’. If two classes have the same
instances, no more and no less, they may be said to be
co-extensive classes (a class that serves as both species
and genus in some context is said to be sui generis). If
two classes merely share some instances, they may be
said to be intersecting (or overlapping) classes, but they
are not hierarchically arranged (e.g. ‘gardeners’ and
‘engineers’). If two classes of the same order have no
instances in common, they may be said to be mutually
exclusive classes.
It is important to grasp and keep in mind the distinction
between hierarchy and order. Since you and I are men,
each of us is a member of the class ‘men’; this is
subsumption by a first-order class of its concrete
instances. Since all men are animals, the class ‘men’ is a
subclass of the class ‘animals’; this is hierarchy between
two classes of the first order. Since ‘men’ is a class of
animals, it is a member of the class ‘classes of animals’
(or ‘animal-classes’); this is subsumption by a second
order class (i.e. a class of classes) of its first-order-class
instances (i.e. mere classes). Since all ‘classes of men’
are ‘classes of animals’, the class ‘men-classes’ is a
subclass of the class ‘animals-classes’; this is hierarchy
commas, or simply ‘classes of men’ or ‘men-classes’ in
inverted commas.
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THE LAWS OF THOUGHT
between two classes of the second order, i.e. between
two classes of classes. The relation between classes of
the first order and classes of the second order is never
one of hierarchy, but always one of subsumption; i.e. the
former are always members (instances) of the latter,
never subclasses. Hierarchies only occur between classes
of the same order.
Thus, in class logic, we have two planes of existence to
consider. At the ground level is the relatively objective
plane of empirical phenomena (whether these are
physical, mental or spiritual in substance); above that,
residing in our minds, is the relatively subjective plane of
ideas (which are conceived as insubstantial, but do have
phenomenal aspects – namely mental or physical images,
spoken or written words, and the intentions of such
signs), comprising ideas about empirical phenomena and
ideas about such ideas. Classes are developed to facilitate
our study of empirical phenomena and classes of classes
are developed in turn to facilitate our study of classes –
for classes (including classes of classes) are of course
themselves empirical phenomena of sorts. Classification
is a human invention helpful for cognitive ordering of the
things observed through our senses or our imaginations
or our introspective intuitions. Although classes are
products of mind, this does not mean that they are
arbitrary – they are formed, organized and controlled by
means of our rational faculty, i.e. with the aid of logic.
Clearly, to qualify as a class, a class must have at least
one member (in which case the sole member is “one of a
kind”). Usually, a class has two or more members,
indeed innumerable members. A class is finite if it
includes a specified number of instances; if the number
of instances it includes is difficult to enumerate, the class
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is said to be open-ended (meaning infinite or at least
indefinite). What brings the instances of a class together
in it is their possession of some distinctive property in
common; the class is defined by this property (which
may of course be a complicated conjunction of many
properties). A class without instances is called a null (or
empty) class; this signifies that its defining property is
known to be fanciful, so that it is strictly speaking a non
class.
Thus, note well, the term ‘class’ is a bit ambiguous, as it
may refer to a first-order class (a class of non-classes, i.e.
of things other than classes) or a second-order class (a
class of classes, i.e. a mental construct grouping two or
more such mental constructs). A class (of the first order)
is not, indeed cannot be, a class of classes (i.e. a class of
the second order). There is, of course, a class called ‘non
classes’; its instances are principally all concrete things,
which are not themselves classes; for example, you and I
are non-classes. ‘Non-classes’ is merely a class, not a
class of classes, since it does not include any classes.
Thus, ‘non-classes’ may be said to be a first-order class,
but does not qualify as a second-order class.135
135
Note that, whereas positive terms are easy enough to
translate into class logic language, negative terms present a
real difficulty. For example, whereas the term men refers only
to non-classes, its strict antithesis, the term non-men in its
broadest sense, includes both non-classes (i.e. concrete
things other than men) and classes (i.e. more abstract things).
Again, whereas the term finite classes refers only to classes,
its strict antithesis, the term non-finite-classes in its broadest
sense, includes both open-ended classes (abstracts) and non
classes (concretes). Thus, we must, for purposes of
consistency, admit that some terms do cover both non-classes
and classes (including classes of classes). Practically, this
means we have to make use of disjunctives which reveal the
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The realm of classes of classes is very limited as an
object of study in comparison to the realm of mere
classes. For what distinctions can we draw between
classes? Not many. We can distinguish between classes
and classes of classes, between finite and open-ended
classes, between positive and negative classes136, and
maybe a few more things, but not much more.
An apparent double paradox. Bertrand Russell
(Britain, 1872-1970) proposed a distinction between ‘a
class that is a member of itself’ and ‘a class that is not a
member of itself’. Although every class is necessarily co
extensive with itself (and in this sense is included in
itself), it does not follow that every class is a member of
itself (evidently, some are and some are not). Such a
distinction can be shown to be legitimate by citing
convincing examples. Thus, the class ‘positive classes’ is
a member of itself, since it is defined by a positive
property; whereas the class ‘negative classes’ is not a
member of itself, since it is also positively defined (albeit
with general reference to negation). Again, the class
‘finite classes’ is not a member of itself, since it has
innumerable members; while the class ‘open-ended
classes’ is a member of itself, since it too has
innumerable members.
What about the class ‘classes’ – is it a member of itself
or not? Since ‘classes’ is a class, it must be a member of
implicit alternatives. This of course complicates class logic
considerably.
136
Positive classes are defined by some positive property
and negative classes are defined by a negative one. For
examples, ‘men’ is defined with reference to rational animals
(positive), whereas ‘bachelors’ is defined with reference to not
yet married men (negative).
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‘classes’ – i.e. of itself. This is said without paying
attention to the distinction between classes of the first
and second orders. If we ask the question more
specifically, the answer has to be nuanced. The class
‘first-order classes’ being a class of classes and not a
mere class, cannot be a member of itself, but only a
member of ‘second-order classes’; the members of the
‘first-order classes’ are all mere classes. On the other
hand, since the class ‘second-order classes’ is a class of
classes, it is a member of itself, i.e. a member of ‘second
order classes’. Thus, the class ‘second-order classes’
includes both itself and the class ‘first-order classes’, so
that when we say that the wider class ‘classes’ is a
member of itself, we mean that it is more specifically a
member of the narrower class ‘classes of classes’. As
regards, the class ‘non-classes’, since it is a class and not
a non-class, it is not a member of itself. Note however
that Russell’s paradox does not make a distinction
between classes of the first and second orders, but
focuses on ‘classes’ indiscriminately.
Russell asked whether “the class of all classes which
are not members of themselves” is or is not a member
of itself. It seemed logically impossible to answer the
question, because either way a contradiction ensued. For
if the class ‘classes not members of themselves’ is not a
member of the class ‘classes not members of
themselves,’ then it is indeed a member of ‘classes not
members of themselves’ (i.e. of itself); and if the class
‘classes not members of themselves’ is a member of
‘classes not members of themselves,’ then it is also a
member of ‘classes which are members of themselves’
(i.e. of its contradictory). This looked like a mind
blowing double paradox.
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The solution of the problem. The pursuit of knowledge
is a human enterprise, and therefore one which proceeds
by trial and error. Knowledge is inductive much more
than deductive; deduction is just one of the tools of
induction. There are absolutes in human knowledge, but
they are few and far between. When we formulate a
theory, it is always essentially a hypothesis, which might
later need to be revised or ruled out. So long as it looks
useful and sound, and does so more than any competing
theory, we adopt it; but if it ever turns out to be belied by
some facts or productive of antinomy, we are obliged to
either reformulate it or drop it. This is the principle of
induction. When we come upon a contradiction, we have
to ‘check our premises’ and modify them as necessary. In
the case at hand, since our conception of class logic is
shown by the Russell paradox to be faulty somehow, we
must go back and find out just where we went wrong. So,
let us carefully retrace our steps. We defined a class and
membership in a class by turning predication into
classification, saying effectively:
If something is X, then it is a member of the class
‘X’, and not a member of the class ‘nonX’.
If something is not X, then it is not a member of
the class ‘X’, but a member of the class ‘nonX’.
Where did we get this definition? It is not an absolute
that was somehow cognitively imposed on us. We
invented it – it was a convention by means of which we
devised the idea of classes and membership in them.
Knowledge can very well proceed without recourse to
this idea, and has done so for millennia and continues to
do so in many people’s mind. It is an idea with a history,
which was added to the arsenal of reasoning techniques
by logicians of relatively recent times. These logicians
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noticed themselves and others reasoning by means of
classification, and they realized that this is a useful
artifice, distinct from predication and yet based on it
somehow. They therefore formally proposed the above
definition, and proceeded to study the matter in more
detail so as to maximize its utility. The ‘logic of classes’,
or ‘class logic’, was born.
However, at some stage, one logician, Bertrand Russell,
realized that there was an inherent inconsistency in our
conception of classification, which put the whole edifice
of class logic in serious doubt. That was the discovery of
the paradox bearing his name. That was a great finding,
for there is nothing more important to knowledge
development, and especially to development of the
branch of knowledge called formal logic, than the
maintenance of consistency. Every discovery of
inconsistency is a stimulation to refine and perfect our
knowledge. Russell deserves much credit for this finding,
even if he had a lot of difficulty resolving the paradox in
a fully convincing manner. Let us here try to do better,
by digging deeper into the thought processes involved in
classification than he did. What is classification, more
precisely?
If we look more closely at our above definition of a class
‘X’ and membership of things in it by virtue of being X,
we must ask the question: what does this definition
achieve, concretely? Are we merely substituting the
phrase ‘is a member of’ for the copula ‘is’, and the class
‘X’ for the predicate X? If this is what we are doing,
there is no point in it – for it is obvious that changing the
name of a relation or a term in no way affects it. Words
are incidental to knowledge; what matters is their
underlying intent, their meaning. If the words change, but
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not the meaning, nothing of great significance has
changed. No, we are not here merely changing the words
used – we are proposing a mental image.
Our idea of classification is that of mental entities called
classes in which things other than classes (or lesser
classes, in the case of classes of classes) are figuratively
collected and contained. When we say of things that they
are members of class ‘X’, we mean that class ‘X’ is a sort
of box into which these things are, by means of
imagination, stored (at a given time, whether temporarily
or permanently). That is to say, our ‘definition’ of
classification is really a formal convention used to
institute this image. What it really means is the
following:
If something is X, then it is in the class ‘X’, and
out of (i.e. not in) the class ‘nonX’.
If something is not X, then it is out of (i.e. not in)
the class ‘X’, but in the class ‘nonX’.
Clearly, to ‘be’ something and to ‘be in’ (within, inside)
something else are not the same thing. Our definition
conventionally (i.e. by common agreement) decrees that
if X is predicated of something, then we may think of
that thing as being as if contained by the mental entity
called class ‘X’. But this decree is not an absolute; it is
not a proposition that being subject to predication of X
naturally and necessarily implies being a member of
class ‘X’. For the whole idea of classification, and
therefore this definition of what constitutes a class and
membership therein, is a human invention. This
invention may well be, and indeed is, very useful – but it
remains bound by the laws of nature. If we find that the
way we have conceived it, i.e. our definition of it,
inevitably leads to contradiction, we must adjust our
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definition of it in such a way that such contradiction can
no longer arise. This is our way of reasoning and acting
in all similar situations.
As we shall presently show, since the contradiction is a
consequence of the just mentioned defining implication,
we must modify that implication. That is to say, we must
decree it to have limits. Of course, we cannot just
vaguely say that it has limits; we must precisely define
these limits so that the practical value of our concept of
classification is restored. We can do that by realizing that
our definition of classification with reference to
something ‘being in’ something else means that class
logic is conceived of as related to geometrical logic. This
is obvious, when we reflect on the fact that we often
‘represent’ classes as geometrical figures (notably,
circles) and their members as points within those figures.
This practice is not accidental, but of the very essence of
our idea of classification. Classification is imagining that
we put certain items, identified by their possession of
some common and distinctive property, in a labeled
container137.
Let us now examine the concept of self-membership in
the light of these reflections. What is the idea of self
membership? It is the presupposition that a class may be
a member of itself. But is that notion truly conceivable?
If we for a moment put aside the class logic issue, and
reformulate the question in terms of geometrical logic,
we see that it is absurd. Can a container contain itself? Of
course not. There is no known example of a container
containing itself in the physical world; and indeed we
cannot even visually imagine a container containing
137
This is a pictorial ‘representation’, an analogical image
not to be taken literally.
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THE LAWS OF THOUGHT
itself. So the idea of self-containment has no empirical
basis, not even in the mental sphere. It is only a fanciful
conjunction of two words, without experiential basis. For
this reason, the idea strikes us as illogical and we can
safely posit as a universal and eternal ‘axiom’ that self
containment is impossible. A nonsensical term like ‘the
collection of all collections’ is of necessity an empty
term; we are not forced to accept it, indeed we are
logically not allowed to do so; we can only consistently
speak of ‘the collection of all other collections’138.
A container is of course always co-extensive with itself,
i.e. it occupies exactly the space it occupies. But such
‘co-extension’ is not containment, let alone self
containment, for it does not really (other than verbally)
concern two things but only one; there is no ‘co-’ about
it, it is just extended, just once. We refer to containment
when a smaller object fits inside a larger object (or in the
limit when another object of equal size neatly fits inside
a certain object). The concept of containment refers to
two objects, not one. There has to be two distinct objects;
it does not suffice to label the same object in two ways.
To imagine ‘self-containment’ is to imagine that a whole
object can somehow fit into itself as a smaller object (or
that it can somehow become two, with one of the two
inside the other). This is unconscionable. A whole thing
cannot be a part (whether a full or partial part) of itself;
nothing can be both whole and part at once. A single
thing cannot be two things (whether of the same or
138
To give a concrete image: a bag of marbles (whether
alone or, even worse, with the marbles in it) cannot be put
inside itself, even if the bag as a whole, together with all its
contents, can be rolled around like a marble and so be called
a marble.
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different size) at once; nothing can simultaneously exist
as two things.
You cannot decide by convention that something is both
whole and part or that one thing is two. You cannot
convene something naturally impossible. You can only
convene something naturally possible, even though it is
unnecessary. Thus, the concept of self-containment is
meaningless; it is an inevitably empty concept, because it
assumes something impossible to be possible. There is no
such thing as self-containment; a container can never
contain itself. If this is true, then it is of course equally
true that no class includes itself, for (as we have seen)
classification is essentially a geometrical idea. Given that
a container cannot contain itself, it follows that the
answer to the question as to whether a class can be a
member of itself is indubitably and definitely: No.
Because to say of any class that it is a member of itself is
to imply that a container can be a content of itself. Just as
no container which is a content of itself exists, so no
class which is a member of itself exists!
Now, this is a revolutionary idea for class logic. It
applies to any and every class, not just to the class
‘classes not members of themselves’ which gave rise to
the Russell paradox. Moreover, note well that we are
here denying the possibility of membership of a class in
itself, but not the possibility of non-membership of a
class in itself. When we say that no container contains
itself, we imply that it is true of each and every container
that it does not contain itself. Similarly, when we say that
no class is a member of itself, we imply that it is true of
each and every class that it is not a member of itself.
What this means is that while we acknowledge the
subject of the Russell paradox, namely the class ‘classes
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THE LAWS OF THOUGHT
that are not members of themselves’, we reject the notion
that such a class might ever, even hypothetically for a
moment, be a member of itself (and therefore
paradoxical) – for, we claim, no class whatever is ever a
member of itself.
How can this be, you may well ask? Have we not already
shown by example that some classes are members of
themselves? Have we not agreed, for example, that the
class ‘classes’ being a class has to be a member of the
class ‘classes’, i.e. of itself? How can we deny something
so obvious? Surely, you may well object further, if the
class ‘classes that are not members of themselves’ is not
a member of itself, then it is undeniably a member of
itself; and if it is a member of itself, then it is undeniably
not a member of itself? To answer these legitimate
questions, let us go back to our definition of
classification, and the things we said about that
definition. As I pointed earlier, our definition of classes
and membership in them has the form of a conventional
implication. It says:
If and only if something is X, then it is a member
of the class ‘X’.
Now, since this conventional implication leads us
inexorably to paradox, we must revise it, i.e. make it
more limited in scope, i.e. specify the exact conditions
when it ‘works’ and when it ceases to ‘work’. What is
essentially wrong with it, as we have seen, is that it
suggests that a class can be a member of itself. For
example, since the class ‘classes’ is a class, then it is a
member of ‘classes’; in this example, the variable X has
value class and the variable ‘X’ has value ‘classes’. But,
as we have shown, the claim that a class can be a
member of itself logically implies something
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geometrically impossible; namely, that a container can be
a content of itself. So, to prevent the Russell paradox
from arising, we need to prevent the unwanted
consequences of our definition from occurring. Given
that our concept of classification is problematic as it
stands, what are the conditions we have to specify to
delimit it so that the problem is dissolved?
The answer to this question is that when the subject and
predicate of the antecedent clause are one and the same,
then the consequent clause should cease to be implied.
That is to say, if the antecedent clause has the form “if
the class ‘X’ is X” then the consequent clause “then the
class ‘X’ is a member of ‘X’ (and thus of itself)” does
not follow. This ‘does not follow’ is a convention, just as
the general ‘it follows’ was a convention. What we have
done here is merely to draw a line, saying that the
consequent generally follows the antecedent, except in
the special case where the subject and predicate in the
antecedent are ‘the same’ (in the sense that predicate X is
applicable to class ‘X’ which is itself based on predicate
X). This is logically a quite acceptable measure, clearly.
If an induced general proposition is found to have
exceptions, then it is quite legitimate and indeed
obligatory to make it less general, retreating only just
enough to allow for these exceptions.
Since the initial definition of classification was a general
convention, it is quite permissible, upon discovering that
this convention leads us into contradiction, to agree on a
slightly narrower convention. Thus, whereas, in the large
majority of cases, it remains true that if something is X,
then it is a member of the class ‘X’, and more
specifically, if a class (say, ‘Y’) is X, then it (i.e. ‘Y) is a
member of the class of classes ‘X’ – nevertheless,
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exceptionally, in the special case where the class that is
X is the class ‘X’ (i.e. where ‘Y’ = ‘X’), we cannot go on
to say of it that it is a member of ‘X’, for this would be
to claim it to be a member of itself, which is impossible
since this implies that a container can be a content of
itself. Note well that we are not denying that, for
example, the class ‘classes’ is a class; we are only
denying the implication this is normally taken to have
that the class ‘classes’ is a member of the class ‘classes’.
We can cheerfully continue saying ‘is’ (for that is mere
predication), but we are not here allowed to turn that ‘is’
into ‘is a member of’ (for that would constitute illicit
classification).
In this way, the Russell paradox is inhibited from arising.
That is to say, with reference to the class ‘classes not
members of themselves’: firstly, it is quite legitimate to
suppose that the class ‘classes not members of
themselves’ is not a member of itself, since we know for
sure (from geometrical logic) that no class is a member
of itself; but it is not legitimate to say that this fact (i.e.
that it is not a member of itself) implies that it is a
member of itself, since such implication has been
conventionally excluded. Secondly, it is not legitimate to
suppose, even for the sake of argument, that the class
‘classes not members of themselves’ is a member of
itself, since we already know (from geometrical logic)
that no class is a member of itself, and therefore we
cannot establish through such supposition that it is not a
member of itself, even though it is anyway true that it is
not a member of itself.
As can be seen, our correction of the definition of
classification, making it less general than it originally
was, by specifying the specific situation in which the
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implication involved is not to be applied, succeeds in
eliminating the Russell paradox. We can say that the
class ‘classes not members of themselves’ is not a
member of itself, but we cannot say that it is a member of
itself; therefore, both legs of the double paradox are
blocked. In the first leg, we have blocked the inference
from not-being ‘a member of itself’ to being one; in the
second leg, we have interdicted the supposition of being
‘a member of itself’ even though inference from it of not
being one would be harmless. Accordingly, the answer to
the question posed by Russell – viz. “Is the class of all
classes which are not members of themselves a member
of itself or not?” – is that this class is not a member of
itself, and that this class not-being a member of itself
does not, contrary to appearances, make it a member of
itself, because no class is a member of itself anyway.
Thus, to be sure, though it is true that the class ‘classes’
is a class, it does not follow that it is a member of itself;
though it is true that the class ‘classes of classes’ is a
class of classes, it does not follow that it is a member of
itself; though it is true that the class ‘positive classes’ is a
positive class, it does not follow that it is a member of
itself; though it is true that the class ‘open-ended classes’
is an open-ended class, it does not follow that it is a
member of itself; though it is true that the class ‘classes
that are not members of themselves’ is a class that is not
a member of itself, it does not follow that it is a member
of
itself.
As for the class ‘classes members of
themselves’, it has no members at all. It should be
emphasized that the restriction on classification that we
have here introduced is of very limited scope; it hardly
affects class logic at all, concerning as it does a few very
borderline cases.
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THE LAWS OF THOUGHT
The above is, I believe, the correct and definitive
resolution of the Russell paradox. We acknowledged the
existence of a problem, the Russell paradox. We
diagnosed the cause of the problem, the assumption that
self-membership is possible. We showed that self
membership is unconscionable, since it implies that a
container can contain itself; this was not arbitrary
tinkering, note well, but appealed to reason. We proposed
a solution to the problem, one that precisely targets it and
surgically removes it. Our remedy consisted in
uncoupling predication from classification in all cases
where self-membership is assumed, and only in such
cases. This solution of the problem is plain common
sense and not a flight of speculation; it is simple and
elegant; it is convincing and uncomplicated; it does not
essentially modify the concept of class membership, but
only limits its application a little; it introduces a
restriction, but one that is clearly circumscribed and quite
small; it does not result in collateral damage on areas of
class logic, or logic in general, that are not problematic,
and therefore does not call for further adaptations of
logic doctrine. Note moreover that our solution does not
resort to any obscure ‘system’ of modern symbolic logic,
but is entirely developed using ordinary language and
widely known and accepted concepts and processes.
A bit of the history. Let us now look briefly at some of
the history of the Russell paradox, and see how he and
some other modern symbolic logicians dealt with it139.
Georg Cantor had already in 1895 found an antinomy in
his own theory of sets. In 1902, when Gottlob Frege
139
I am here referring principally to the account by
William and Martha Kneale in The Development of Logic
(Oxford, London: Clarendon, 1962), ch. XI.1-2.
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353
(Germany, 1848-1925) was about to publish the second
volume of his Grundgesetze, he was advised by Russell
of the said paradox. Frege was totally taken in and could
not see how to get out of the self-contradictions inherent
in “the class of classes that do not belong to themselves.”
He perceived this as very serious, saying: “What is in
question is … whether arithmetic can possibly be given a
logical foundation at all.” Frege first tried to fix things by
suggesting that there might be “concepts with no
corresponding classes,” or alternatively by adjusting one
of his “axioms” in such a way that:
“Two concepts should be said to have the same
extension if, and only if, every object which fell
under the first but was not itself the extension of
the first fell likewise under the second and vice
versa”140.
Clearly, Frege’s initial suggestion that there might be
“concepts with no corresponding classes” can be viewed
as an anticipation of my uncoupling of predication and
classification in specific cases. However, Frege did not
identify precisely in what cases such uncoupling has to
occur. This is evident in his next suggestion, which,
though it points tantalizingly to the difficulty in the
notion of self-membership, does not reject this notion
outright but instead attempts to mitigate it. He speaks of
two concepts instead of one, and tries to conventionally
exclude the extension as a whole of each from the other,
while of course continuing to include the objects falling
under the extension; this shows he has not realized that
self-inclusion by an extension is not even thinkable.
It should be stressed that Russell’s paradox pertains to a
certain class (viz. that of all classes not members of
140
Kneale and Kneale, p. 654. Italics theirs.
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THE LAWS OF THOUGHT
themselves) being or not-being a member of itself – not
of some other class. Frege tries to resolve this paradox
with reference, not to a single class, but to a pair of equal
classes, even though (to my knowledge) he has not
demonstrated that co-extensive classes result in a
paradox comparable to the Russell paradox. It follows
that his attempted solution to the problem is not germane
to it. Moreover, Frege seems to have thought that if all
items that fall under one class (say, ‘Y’) fall under
another class (say, ‘X’), then the class ‘Y’ may be
assumed to fall under the class ‘X’; and vice versa in the
event of co-extension. This is suggested by his attempt to
prevent such assumption, so as to avoid (in his estimate)
the resulting Russell paradox. But in truth, it does not
follow from the given that all Y are X that the class ‘Y’
is a member of the class ‘X’ – it only follows that the
class ‘Y’ is a subclass of the class ‘X’, or an equal class
if the relation is reversible. Thus, it appears that Frege
confused the relations of class-membership and
hierarchization of classes, using a vague term like
‘falling’ to characterize them both.
We may well ask the question whether an equal class, or
a subclass, or even an overclass, might be a member of
its hierarchically related class. Offhand, it would seem to
be possible. For example, all positive classes are classes
and therefore members of the class ‘classes’, and the
class ‘positive classes’ is a subclass of the class ‘classes’;
however, although not all classes are positive classes
(some are negative classes), nevertheless the class
‘classes’ is a positive class (being defined by a positive
statement), and so is a member of the class ‘positive
classes’. But although this example suggests that an
overclass might be a member of its subclass (and
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355
therefore, all the more, an equal class or a subclass might
be a member of its hierarchical relative), we might still
express a doubt by means of analogy, as Frege perhaps
intended to do. We could argue inductively, by
generalization, that if a class cannot be a member of
itself, then maybe it cannot be a member of any
coextensive class (as Frege suggests), and perhaps even
of a subclass or an overclass. For the issue here is
whether the instances referred to by the first class can be
thought to occur twice in the second class (as members
of it in their own right, and as constituents of a member).
So, Frege may have raised a valid issue, which could
lead to further restrictions in class logic. However, this
need not concern us further in the present context, since
(as already explained) it is not directly relevant to
resolution of the Russell paradox.
Russell described his paradox in his book Principles of
Mathematics, published soon after. Although at first
inclined to Frege’s second approach, he later preferred
Frege’s first one, proposing that there might be “some
propositional functions which did not determine genuine
classes.” Note here again the failure to pinpoint the
precise source and remedy of the problem. Subsequently,
Russell thought that “the problem could never be solved
completely until all classes were eliminated from logical
theory.” This, in my view, would be throwing out the
baby with the bath water – an overreaction. But then he
found out (or rather, he thought he did, or he convinced
himself that he did) that the same paradox could be
generated without “talk of classes,” i.e. with reference to
mere predicates – that is, in terms of predicate logic
instead of in terms of class logic. As Kneale and Kneale
put it (p. 655):
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THE LAWS OF THOUGHT
“Instead of the class which is supposed to contain
all classes that are not members of themselves let
us consider the property of being a property
which does not exemplify itself. If this property
exemplifies itself, then it cannot exemplify itself;
and if it does not exemplify itself, then it must
exemplify itself. Clearly, the nature of the trouble
is the same here as in the original paradox, and
yet there is no talk of classes.”
But even if classes are not explicitly mentioned here, it is
clear that they are tacitly intended. How would a
property ‘exemplify’ itself? Presumably, property X
would be ‘a property which exemplifies itself’ if property
X happens to be one of the things that have property X.
That is to say, X exemplifies X if X is a member of the
class of things that are X. We cannot talk about
properties without resorting to predication; and once we
predicate we can (given the initial definition of
classification) surely classify. So, this attempt is just
verbal chicanery; the same thought is intended, but it is
dressed up in other words. It is dishonest. Moreover, the
way the paradox is allegedly evoked here does not in fact
result in paradox.
We cannot say, even hypothetically, “if this property [i.e.
the property of being a property which does not
exemplify itself] exemplifies itself” for that is already
self-contradictory. To reconstruct a Russell paradox in
‘property’ terms, we would have to speak of ‘the
property of all properties which do not exemplify
themselves’; for then we would have a new term to chew
on, as we did in class logic. But clearly, this new term is
quite contrived and meaningless. Here again, we must
mean ‘the class of all properties which do not exemplify
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themselves’ – and in that event, we are back in class
logic. Thus, note well, while Russell was right in looking
to see whether his paradox was a problem specific to
class logic, or one also occurring in predicate logic, and
he claimed to have established that it occurred in both
fields, in truth (as we have just demonstrated) he did not
succeed in doing that. In truth, the paradox was specific
to class logic; and he would have been better off
admitting the fact than trying to ignore it.
In response to certain criticisms by his peers, Russell
eventually “agreed that the paradoxes were all due to
vicious circles, and laid it down as a principle for the
avoidance of such circles that ‘whatever involves all of a
collection must not be one of the collection’.” Thus,
Russell may be said to have conceded the principle I
have also used, namely that a collection cannot include
itself as one of the items collected, although in truth the
way he put it suggests he conceived it as a convention
designed to block incomprehensible vicious circles rather
than a logical absolute (notice that he says ‘must not’
rather than ‘cannot’). He viewed the paradoxes of set
theory as “essentially of the same kind as the old paradox
of Epimenides (or the Liar).” This suggests that, at this
stage, he saw his own paradox as due to self-reference,
somehow. It does look at first sight as if there is some
sort of self-reference in the proposition ‘the class of all
classes that are not members of themselves is (or is not) a
member of itself’, because the clause ‘member of itself’
is repeated (positively or negatively, in the singular or
plural) in subject and predicate141. But it cannot be said
that self-reference is exactly the problem.
141
Note that if self-reference were the crux of the
problem, then the proposition ‘the class of all classes that are
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THE LAWS OF THOUGHT
A few years later, in a paper published in 1908, Russell
came up with a more elaborate explanation of the Russell
paradox based on his ‘theory of types’. Russell now
argued that “no function can have among its values
anything which presupposes the function, for if it had, we
could not regard the objects ambiguously denoted by the
function as definite until the function was definite, while
conversely … the function cannot be definite until its
values are definite”142. In other words, the question “the
class of all classes that are not members of themselves, is
it or is it not a member of itself” is inherently flawed,
because the subject remains forever out of reach. We
cannot take hold of it till we resolve whether or not it is a
member of itself, and we cannot do the latter till we do
the former; so, the conundrum is unresolvable, i.e. the
question is unanswerable. Effectively, the subject is a
term cognitively impossible to formulate, due to the
double bind the issue of its definition involves for any
thinker.
Here, we should note that the purpose of Russell’s said
explanation was effectively to invalidate the negative
class ‘classes not members of themselves’, since this is
the class giving rise to the double paradox he was trying
to
cure. The positive class ‘classes members of
themselves’ clearly does not result in a double paradox:
if
we suppose it is not a member of itself, self
contradiction does ensue, but we can still say without
self-contradiction that it is a member of itself. In fact, if
Russell’s explanation were correct, the positive class
members of themselves is (or is not) a member of itself’ would
be equally problematic, even though it apparently does not
result in a similar paradox.
142
Quoted by the Kneales, p. 658.
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359
ought to be as illicit as the negative one. For if we claim
the impossibility of a class referring to something that is
not yet settled, as Russell did with reference to the
negative class, then we must admit this characteristic is
also found in the positive class, and we must reject it too.
Russell does not seem to have realized that, i.e. that his
remedy did not technically differentiate the two classes
and so could be applied to both. For this reason, his
attempt to solve the Russell paradox with reference to
circularity or infinity must be judged as a failure. In my
own theory, on the other hand, it is the positive class
(that of self-membership) which is invalid (and empty),
since it is geometrically unthinkable, while the negative
class (that of non-self-membership) remains quite
legitimate (and instantiated), as indeed we would expect
on the principle that all claims (including that of self
membership) ought to be deniable.
Anyway, Russell concluded, briefly put, that a function
could not be a value of itself; and proposed that function
and value be differentiated as two ‘types’ that could not
be mixed together indiscriminately. But this theory is, I
would say, too general, and it complicates matters
considerably. As we have seen, we cannot refuse to
admit that, for instance, ‘classes’ is a class; the most we
can do is to deny that this implies that ‘classes’ is a
member of itself. This is a denial of self-membership, not
of self-predication or of self-reference. As regards the
notion of ‘types’ and later that of ‘orders within types’,
these should not be confused with the more traditional
ideas of hierarchies and orders of classes, which we laid
out earlier in the present essay. In truth, the resemblance
between Russell’s concepts and the latter concepts gives
Russell’s theory a semblance of credibility; but this
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THE LAWS OF THOUGHT
appearance is quite illusory – these are very different sets
of concepts. Russell’s notion of ‘types’ is highly
speculative and far from commonsense; while it might
appear to solve the Russell paradox, it has ramifications
that range far beyond it and incidentally invalidate
traditional ideas that do not seem problematic143. In short,
it is a rough-and-ready, makeshift measure, and not a
very convincing one.
Every paradox we come across is, of course, a signal to
us that we are going astray somehow. Accordingly, the
Russell paradox may be said to have been a signal to
Frege, Russell, and other modern logicians, that
something was wrong in their outlook. They struggled
hard to find the source of the problem, but apparently
could not exactly pinpoint its location. All the intricacy
and complexity of their symbolic and axiomatic approach
to logic could not help them, but rather obscured the
solution of the problem for them. This shows that before
any attempt at symbolization and axiomatization it is
essential for logicians to fully understand the subject at
hand in ordinary language terms and by means of
commonsense. To my knowledge, the solution of the
problem proposed in the present essay is original, i.e. not
to be found elsewhere. If that is true, then the theory of
class logic developed by modern symbolic logicians,
which is still the core of what is being taught in
universities today, needs to be thoroughly reviewed and
revised.
A bit of self-criticism. As regards the resolution of the
Russell paradox that I proposed over two decades ago in
my Future Logic, the following needs to be said here.
143
See for a start the Kneales’ critique of the ‘theory of
types’ in ch. XI.2.
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361
While I stand, in the main, by my theory of the logic of
classes there (in chapters 43-44), I must now distance
myself somewhat from my attempted resolution of the
Russell paradox there (in chapter 45).
I did, to my credit, in that past work express great
skepticism with regard to the notion of self-membership;
but I did not manage to totally rule it out. I did declare:
“Intuitively, to me at least, the suggestion that something
can be both container and contained is hard to swallow,”
and I even postulated, in the way of a generalization from
a number of cases examined, that “no class of anything,
or class of classes of anything, is ever a member of
itself,” with the possible exception of “things” or “things
classes” (although it might be said of these classes that
they are not members of any classes, let alone
themselves144); but still, I did not reject self-membership
on principle, and use that rejection to explain and resolve
the Russell paradox, as I do in the present essay.
This is evident, for instance, in my accepting the idea
that “‘self-member classes’ is a member of itself.” The
reason I did so was the thought that “whether self
membership is possible or not, is not the issue.”
Superficially, this is of course true – the Russell paradox
concerns the ‘class of all classes that are not members of
themselves’, and not ‘the class of all classes that are
members of themselves’. But in fact, as I have shown
today, this is not true; acceptance of self-membership is
the true cause of the Russell paradox, and non-self
membership is not in itself problematic.
144
Note that in this context I come up with the idea that
the definition of membership might occasionally fail. But I did
not at the time pursue that idea further, because I did not then
analyze what such failure would formally imply.
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THE LAWS OF THOUGHT
Anyway, not having duly ruled out self-membership, I
resorted to the only solution of the problem that looked
promising to me at the time – namely, rejection of
‘permutation’ from “is (or is not) a member of itself” to
“is (or is not) {a member of itself}” (notice the addition
of curly brackets). That is to say, I proposed the logical
interdiction of changing the relation of self-membership
or non-self-membership into a predicable term. Now I
see that this was wrong – it was an action taken too late
in the process of thought leading up to the Russell
paradox. It was a superficial attempt, treating a symptom
instead of the disease. I did that, of course, because I
thought this was “of all the processes used in developing
these arguments, [the] only one of uncertain
(unestablished) validity.” But in truth, it was not the only
possible cause of the effect – there was a process before
that, one of deeper significance, namely the transition
from ‘is’ to ‘is a member of’. I did not at the time notice
this earlier process, let alone realize its vulnerability; and
for that reason, I did not attack it.
Clearly, I was on the right track, in that I sought for a
place along the thought process at which to block
development of the Russell paradox. But my error was to
pick a place too late along that process. In fact, the right
place is earlier on, as advocated in the present essay. The
Russell paradox does not arise due to an illicit
permutation, but due to the illicit transformation of a
predicate into a class in cases where a claim of self
membership would ensue. And while the remedy
proposed is even now in a sense ‘conventional’, the flaw
it is designed to fix is quite real – it is that self
membership is in fact impossible and therefore can never
be assumed true. My previous proposed solution to the
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363
problem only prevented the Russell paradox; it did not
prevent self-membership, which is the real cause of the
paradox. Thus, the solution I propose in the present essay
is more profound and more accurate.
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THE LAWS OF THOUGHT
MAIN REFERENCES
A Dictionary of Philosophy. Anthony Flew, ed. London:
Pan, 1979.
Blofeld, John. Trans. The Zen Teaching of Huang Po:
On the Transmission of Mind. New York: Grove, 1958.
Bodhidharma. The Zen Teaching of Bodhidharma. Red
Pine, trans. New York: North Point, 1989.
Cheng, Hsueh-li. Empty Logic: Madhyamika Buddhism
from Chinese Sources. Delhi: Motilal Banarsidass, 1991.
Dhammapada,
The.
Juan
Harmondsworth, Mx.: Penguin, 1973.
Mascaro,
trans.
Dogen. Enlightenment Unfolds: The Essential Teachings
of Zen Master Dogen. Kazuaki Tanahashi, ed. Boston:
Shambala, 2000.
Eracle, Jean. Enseignements du Bouddha. Paris: Librio,
Gould, Stephen Jay. The Lying Stones of Marrakech:
Penultimate Reflections in Natural History. London:
Vintage, 2001.
Hume, David. A Treatise of Human Nature. Project
Gutenberg E-text (Internet).
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Kamalashila (Anthony Matthews). Meditation: The
Buddhist Way of Tranquility and Insight. Birmingham:
Windhorse, 1992.
Kneale, William and Martha. The Development of Logic.
Oxford, London: Clarendon, 1962.
Suzuki, Shunryu. Zen Mind, Beginner’s Mind. New
York: Weatherhill, 1970.
Trungpa, Chögyam.
Shambhala, 1994.
Illusion’s Game.
Other references are mentioned in footnotes.
Boston:
366
THE LAWS OF THOUGHT
REFERENCES
367
Works by Avi Sion
Avi Sion is the author of several works on logic
and philosophy: Future Logic (1990); Judaic
Logic (1995); Buddhist Illogic (2002) ;
Phenomenology (2003); The Logic of Causation
(1999, 2003, 2010); Volition and Allied Causal
Concepts
(2004);
Ruminations
(2005);
Meditations (2006); Logical and Spiritual
Reflections (2008-9); A Fortiori Logic (2013).
These works can be freely studied at:
www.TheLogician.net.
368
THE LAWS OF THOUGHT
ISBN: 978-1495973147

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