We must then by the law of identity, as thus interpreted, mean to assert not any identity of properties, but the identity of the subject of properties amidst all the changes that may take place in the properties themselves. This may be regarded as a theory as to the nature of individuality and continuous identity in the midst of change, and is of great importance in its proper place. But it cannot properly stand as one of the traditional laws of thought which constitute the foundation of logical doctrine.
416. The Law of Contradiction.—The principle of contradiction is best regarded as expressing one aspect of the relation between contradictory judgments, namely, that they cannot both be true. The essential characteristic of a judgment is that it claims to be true. But we cannot declare anything to be true without implicitly declaring something else to be false. All affirmation implies denial; and we cannot clearly grasp the import of any given judgment unless we understand precisely what it denies.
The relation between a judgment and its denial is made explicit by the law of contradiction and the law of excluded middle, the 455 first of which declares that two contradictory judgments cannot both be true, and the second that they cannot both be false.
It is clear that the law of contradiction, as thus interpreted, does not carry us very far, and that it cannot fulfil the function, which Hamilton assigned to it, of serving as the principle of all logical negation. It serves, however, to express the significance of negation, and at the same time to set forth (from a different point of view from that taken by the law of identity) a fundamental postulate which must be granted if our processes of thought and reasoning are to be valid. For validity of thought and reasoning demand that false judgments shall be refuted; and only by the help of the law of contradiction is any such refutation possible. The refutation requires that another judgment contradictory of the first shall be established; but this would go for nothing, if two contradictories could be true together.
The law of contradiction thus takes its place by the side of the law of identity as a first principle of dialectic and reasoning: not indeed advancing us on our way, but serving as a postulate, without which it would not even be possible for us to make a start.
We may pass to a consideration of the formula A is not not-A, by which the law of contradiction is more usually expressed. Here, as Sigwart points out, we have no longer an expression of a relation between two judgments, but an affirmation that in a given judgment the predicate must not contradict the subject; and inasmuch as denial and contradiction have primarily no meaning except in relation to judgments, this interpretation of the principle of contradiction can at any rate not be regarded as equally fundamental with that which we have previously given. At the same time, it is clear that if any A were not-A, then, understanding not-A to denote whatever does not belong to the class A, we should have two contradictory judgments, for we should be able to assert of something both that it belonged to the class A and that it did not belong to the class A.
The formula A is not not-A need not, therefore, be rejected, if its secondary character is recognised.
Mill’s attitude towards the law of contradiction involves an apparent inconsistency. He begins by regarding it as a mode of defining negation.[455] It is, he says, a mere identical proposition that if the negative be true, the affirmative must be false; for the 456 negative asserts nothing but the falsity of the affirmative, and has no other sense or meaning whatever. He goes on, however, both in the Logic and in the Examination of Sir William Hamilton’s Philosophy, to speak of the law as a generalisation from experience. He finds its original foundation in the fact that belief and disbelief are two different mental states, excluding one another, this being a fact which we obtain by the simplest observation of our own minds. We observe, moreover, that light and darkness, sound and silence, equality and inequality, in short any positive phenomenon whatever and its negative, are distinct phenomena, pointedly contrasted, and the one always absent when the other is present. From all these facts the law of contradiction is, in Mill’s opinion, a generalisation.
[455] Logic, ii. 7 § 5.
Two distinct points appear to be involved in this argument. As regards the reference to belief and disbelief, we must agree that the foundation of the law of contradiction is to be found in the nature of judgment. The essential characteristic of a judgment is that it claims to be true, and the affirmation of a truth implies by its very nature a denial. It is, however, difficult to see where any generalisation comes in here.