Studies and Exercises in Formal Logic

by John Neville Keynes

APPENDIX B.

THE THREE FUNDAMENTAL LAWS OF THOUGHT.

414. The three Laws of Thought.—The so-called fundamental laws of thought (the law of identity, the law of contradiction, and the law of excluded middle) are to be regarded as the foundation of all reasoning in the sense that consecutive thought and coherent argument are impossible unless they are taken for granted. The function which they thus perform is, however, negative rather than positive. Whilst constituting necessary postulates, apart from which our thought would become chaotic, they do not by themselves advance us on our way. On the one hand, we cannot without their support proceed a step in reasoning; on the other hand, if we were to rely on their aid alone, thought would immediately come to a standstill.

This is at any rate the view taken of the three laws in the exposition that follows. It is true that many logicians have ascribed to them functions of a more positive character, and—starting from the position that they are the fundamental assumptions of logic—have gone on to regard them as the basis upon which alone all logical doctrine, at any rate in its more formal aspect, can be established. The attempt to justify this view has necessitated reading into the laws much more meaning than they can properly be made to contain, and their interpretation has in consequence become highly complex and even confused.

At the outset the question arises whether the laws are to be regarded as referring to terms (or concepts) or to judgments. My own view is that, in all three cases, the latter reference is the more fundamental, but that a reference of the former kind is involved secondarily. This I shall endeavour to bring out in dealing with 451 the laws individually. The distinction is one to which considerable importance is rightly attached by Sigwart.

The question of the mutual relations between the three laws may be briefly touched upon before we proceed to consider the laws separately and in detail; it is not, however, a question that can be disposed of until a later stage. The main point to which attention may conveniently be called at once is that it is only in relation to the other laws that the full force of each of them can be brought out. The laws of identity and contradiction may be regarded as positive and negative statements of the same principle, namely, the unambiguity of the act of judgment; and the laws of contradiction and excluded middle are supplementary to one another in so far as between them they express the nature of negation. At the same time, an endeavour will be made to establish the independence of the laws in the sense that they cannot be deduced one from another.

415. The Law of Identity.—Following Sigwart, I think it most convenient to interpret this law as expressing the unambiguity of the act of judgment. Truth is something fixed and invariable. In the words of Mr Bradley, “Once true always true, once false always false. Truth is not only independent of me, but it does not depend upon change and chance. No alteration in space or time, no possible difference of any event or context, can make truth falsehood. If that which I say is really true, then it stands for ever” (Logic, p. 133).[454] Hence, since a judgment is the expression of truth, the content of a judgment is fixed and invariable; and only when our judgments are so regarded can our thoughts and reasonings be valid. It is in this sense that the law of identity is a fundamental principle of logic (which is the science of valid thought and reasoning); for it is clear that if for a given judgment we were allowed—when it suited us—to substitute another, or if the content of a given judgment could be regarded as now this and now something else, all thought would become chaotic and reasoning would be a sham. Of the validity of no single step of reasoning 452 could we be sure, since as we took the step the content of the original judgment might change, and on this ground it would be open to anyone to admit the original judgment and at the same time deny the inference attempted to be drawn from it.

[⁴⁵⁴] Compare what has been already said in section [50] about the universality of judgments. In particular, the bearing of Mr Bosanquet’s distinction between the time of predication and the time in predication must be borne in mind. When we say that the truth affirmed in any judgment is independent of time, we mean the time of predication, and we assume that the judgment is fully expressed: in order that it may be fully expressed, the time in predication, if any, must be made explicit.

It may be said that, as thus interpreted, the law of identity merely states that we cannot both affirm a judgment and deny it, and that this is what is expressed in the law of contradiction. There is force in this, to the extent that the laws of identity and contradiction may be regarded as expressing the positive and negative aspects of the same principle. It is, as Sigwart has said, only through the rejection of simultaneous affirmation and negation that we become conscious of the unambiguity of the act of judgment. At the same time, the positive formulation of the principle in the form of the law of identity—apart from its negative formulation in the form of the law of contradiction—is justifiable and helpful.