Studies and Exercises in Formal Logic

by John Neville Keynes

The unambiguity of the act of judgment may be expressed somewhat differently (and its positive aspect, as distinct from what is expressed by the law of contradiction, may thereby be made more clear) by saying that the repetition of a judgment neither adds to nor alters its force. On this basis we may perhaps justify the passage of thought which consists in the repetition, not of a complete judgment, but of part of its content only. In other words, we may thus justify formal reasoning, so far as it involves mere elimination; and in the majority of formal reasonings elimination is involved, though it may be questioned whether mere elimination from a single proposition (as in passing from All S is MP to All S is P) is by itself entitled to the name of reasoning at all.

Mill gives an enunciation of the law of identity which must be distinguished from the above: “Whatever is true in one form of words is true in every other form of words which conveys the same meaning” (Examination of Sir William Hamilton’s Philosophy, p. 466). This is a postulate which it is necessary to make in connexion with the use of language as an instrument of thought. So long as the judgment expressed is the same, the form of expression which we give to it is immaterial; and, since in logical doctrine we cannot explicitly recognise more than a limited number of distinct propositional forms, we have to claim to be allowed to substitute for any non-recognised form that recognised form which 453 expresses the same judgment. Mill’s postulate, however, goes beyond the law of identity regarded as expressing the unambiguity of the act of judgment, and it cannot be regarded as equally fundamental. It is sometimes given as the justification of immediate inferences: to this point we shall return.

We may now turn to the law of identity in the form in which it is more ordinarily stated, namely, A is A, Everything is what it is. This form is open to criticism if regarded as professing to give information with regard to objects. In another sense, however, it may be taken to express an unambiguity of terms or concepts which is involved in the unambiguity of the act of judgment. For it is clear that unless in any given process of thought or reasoning our terms or concepts have a fixed signification and reference, the unambiguity of the act of judgment cannot be realised. We have here the secondary reference to terms or concepts which is contained in all the laws of thought in addition to their primary reference to judgments.

As the repetition of a judgment neither adds to nor alters its force, so we may say the same of terms (or concepts), meaning thereby that to refer to anything as both A and A is the same thing as to refer to it simply as A. This yields Boole’s fundamental equation x² = x (which itself admits of a twofold interpretation according as x stands for a term or a proposition).

The reasons why we should not interpret the formula A is A as expressing a judgment respecting the object A have to be considered. The fundamental difficulty is that this so-called judgment is, if interpreted literally, not thinkable at all. For all actual thought implies difference of some kind. Whenever we think of anything, it is as distinguished from something else, or as having properties in common with other things, or at any rate as itself existing at different times. Hence in no case can we think pure identity.

There are two ways of avoiding this difficulty.

(a) We may say that what is intended by identity is not pure identity, but exact likeness in some assigned respect or respects, the likeness sometimes amounting, so far at any rate as our apprehension is concerned, to indistinguishableness except in the property of occupying different portions of space (as, for example, in the case of a number of pins or bullets of the same make and size). On this interpretation, the law of identity may be regarded 454 as equivalent to Jevons’s principle of the Substitution of Similars—“Whatever is true of a thing is true of its like”—or to the axiom that “Things that are equal to the same thing are equal to one another.” Mansel indeed explicitly gives this axiom as equivalent to the law of identity.

It seems clear, however, on reflection that it is a misnomer to speak of these principles as laws of identity, and that at any rate they cannot be adequately expressed by the bare formula A is A. Nor can any analogous interpretation be given to the laws of contradiction and excluded middle. We must, therefore, reject this interpretation of the law of identity regarded as one of the three traditional laws of thought.

(b) We may attempt to evade the difficulty by explaining that by identity we mean continuous identity, as when I say “This pen is the same as the one with which I was writing yesterday.” Here there is no longer pure identity, since there is a difference of time.

If, adopting this interpretation, we mean by the law of identity that what is true of anything at a given time is true of it at other times also, we have no self-evident law, but a fallacy. For the properties of objects are not constant. In other words, the possession by an object of any given property is not, like the truth of a judgment (fully expressed), independent of time.