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 self-contradictory. 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, 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 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’.
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.
POSSIBILITY OF: | P+Q | P+nonQ | nonP+Q | nonP+nonQ |
Implication | yes | no | ? | yes |
Incompatibility | no | yes | yes | ? |
Exhaustiveness | ? | yes | yes | no |
Unimplication | ? | yes | ? | ? |
Compatibility | yes | ? | ? | ? |
Inexhaustiveness | ? | ? | ? | yes |
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 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 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).
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.
[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.