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The Logician © Avi Sion All rights reserved

FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 3.
LOGICAL RELATIONS.
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 '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.
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.[1]
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
selfcontradictory. The only way to resolve this conflict and paradox is to
admit the opposite thesis, viz. that some 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.
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,
notthen Q", or "If nonQ, notthen 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, notthen nonQ"., or "If Q,
notthen nonP".
We can also say of more than two propositions that they are incompatible;
meaning, if any one of them is true, all 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,
notthen Q"., or "If nonQ, notthen 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'.
The following table summarizes the above through analysis of the
possibilities of combination of the affirmations and denials of two
propositions, P and Q, which are given as having a certain logical relation,
specified in the left column. 'No' indicates logically impossible combinations, 'yes'
combinations specified as possible, and '?'
signifies that the status of the combination as it stands, without further
specification, is undetermined by the logical relation concerned. Table
3.1
Definitions of Logical Relations.
We shall have occasion to review these relations in more detail later,
and also define what we mean by logical possibility or impossibility. Their
study is a big part of logic. For now, it is enough to just point them out, for
practical purposes.
We can now restate 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 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
selfimplication 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 selfcontradiction, 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 selfevidence, and proves the assertion; not all
true assertions have this property, however).
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
selfcontradictory or selfevident.
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, excludedmiddle 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
excludedmiddle, 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. [1]
I would like to mention here, in passing, the topic of the Logic
of Questions, which some logicians have analyzed in considerable detail.
Some of the features of interrogations are: they are signaled by a written
question mark, or a certain intonation of speech. One question may conceal
several subsidiary questions, whose answers together lead to the whole
answer. Questions cannot as such be said to be true or false, though they
often intend or logically imply some tacit assertion. 'Every question
delimits a range of possible answers'a yes or no, a case in point or
example, an instruction on how to do something (the
New Encyclopaedia Britannica, 23:283). But some rhetorical questions are
so constructed that only false answers to them are possible. A compound
question which it is difficult to answer tersely correctly is a case in
point (called the 'fallacy of the many questions'). In such case, the
question posed should of course be challenged. A teacher may well ask a
leading question of a pupil, hinting at the true answer; but in some cases,
this technique is abused, and we see for instance a journalist generating a
false answer with propaganda value from an unaware respondent.
