Studies and Exercises in Formal Logic

by John Neville Keynes

The sophism is a little more puzzling if we begin by assuming it to be true that Cretans never speak the truth. Such an assumption contains no self-contradiction, and there is therefore nothing to prevent our taking it as our starting-point. This being so, let Epimenides make his assertion. Because it is true, here is a Cretan who has spoken the truth, and therefore it is false. Its own truth proves its own falsity. But, again, because it is true, Epimenides cannot be speaking the truth, and therefore it is false. Once more its own truth proves its own falsity.

The argument may also be put as follows. Assume it to be true that Cretans are always in all things liars, and then let Epimenides, the Cretan, make this assertion. Either he speaks truly or he speaks falsely. But if he speaks truly, it thereby follows that he speaks falsely; whilst, on the other hand, if he speaks falsely, he merely affords additional evidence of the truth of what he says.

The problem offering itself for solution is how an apparently valid argument can thus yield as its result nothing but a bare contradiction. The explanation is that we have commenced with premisses that are implicitly contradictory, and that our subsequent reasoning has fulfilled its proper function in making the contradiction explicit. There is nothing self-contradictory in assuming that Cretans never speak the truth; but having commenced with this assumption, we cannot without implicit contradiction suppose a Cretan to make the assertion. In other words, the two premisses—Cretans are always in all things liars; and Epimenides, the Cretan, said so—cannot be true together.

418. The Law of Excluded Middle.—The law of excluded middle supplements the law of contradiction in explaining the nature of the relation between two contradictory judgments. The law of contradiction tells us that, of two contradictory judgments one or other 459 must be false, the truth of either implying the falsity of the other; the law of excluded middle tells us that of two contradictory judgments one or other must be true, the falsity of either implying the truth of the other. It is only by the aid of the two laws combined that the meaning of negation can be fully expressed.

Sigwart regards the law of excluded middle as a derivative principle dependent upon the principle of contradiction and another principle which he designates the principle of twofold (or double) negation. He observes that to interpret the nature of negation completely we must add to the principle of contradiction the further principle that the negation of the negation is affirmative, that to deny a negation is equivalent to affirming the same predicate of the same subject. To this further principle he gives the name of double negation; and it is, he says, only because the denial of the negation is the affirmation itself that there is no medium between affirmation and negation.

The deduction is as follows. Let X = A is B, and X = A is not B. The principle of contradiction tells us that of the two judgments X and X, one is necessarily false. It follows that one is necessarily true. For if I deny X then by so doing I maintain X, while if I deny X then (by the principle of double negation) I maintain X. Therefore, the denial of both is equivalent to the affirmation of both, that is, it involves a contradiction. Hence there is no middle statement between affirmation and negation.

In criticism of the above it may be questioned whether the bare law of contradiction justifies us in passing explicitly from the denial of X to the affirmation of X. Sigwart’s own statement of the principle of contradiction is that X and X cannot be true together. This enables us to pass from the affirmation of X to the denial of X, or from the affirmation of X to the denial of X ; but nothing more. There appears, moreover, to be a want of symmetry in Sigwart’s treatment of the matter. He makes the law of contradiction yield (1) affirmation of X is denial of X, (2) affirmation of X is denial of X, (3) denial of X is affirmation of X ; while the principle of double negation yields only (4) denial of X is affirmation of X.

All four of these relations are required in order that the nature of contradiction may be fully expressed; but unless we sum up all four in a single statement, it seems better to express (1) and (2) by means of the principle of contradiction, and (3) and (4) by means of a second principle, whether we call the latter by the name of the 460 principle of excluded middle or by any other name. It will be observed that we can express (1) and (2) together in the form Not both X and X, and (3) and (4) together in the form Either X or X.

Sigwart’s principle of double negation thus appears to express one-half of what is ordinarily expressed by means of the law of excluded middle; and its separate recognition may be regarded as unnecessary. I agree with Sigwart, however, in holding that the law of excluded middle does no more than help to unfold the meaning of negation.

It is not necessary to occupy space in discussing the relation of the formula Every A is B or not-B to the principle of excluded middle as above described. This formula expresses a secondary relation between so-called contradictory terms which follows from the corresponding, but more fundamental, relation between contradictory judgments.