The other point that Mill raises, namely, the fact that all our knowledge is of contrasts is a generalisation which is ordinarily known as the psychological law of relativity. The fact, however, that we cannot apprehend light except as distinguished from darkness, sound except as distinguished from silence, etc., cannot be regarded as equivalent to the law of contradiction. What that law asserts is, as Mill himself puts it, that “the same proposition cannot be both false and true.”
Boole maintains that “the axiom of metaphysicians which is termed the principle of contradiction, and which affirms that it is impossible for anything to possess a quality and at the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is x2 = x.” The law of contradiction is expressed in Boole’s system in the form x(1 − x) = 0, where x may stand either for the truth of a judgment or for a term; and it is of course clear that x(1 − x) = 0 follows from x2 = x. It will, however, be observed that the converse also holds good, so that the question as to which of the two laws is really the more fundamental remains open to discussion. Apart from this, any attempt to deduce the law of contradiction from any other principle whatsoever is open to the fundamental objection that unless the law of contradiction is 457 accepted as a postulate no single step in reasoning is possible: for as soon as it is open to us to affirm a judgment and at the same time to deny it, it is à fortiori open to us to affirm a judgment and to deny any inference that may be drawn from it. To the question of the interdependence of the laws of thought we shall return.
It has been denied that the law of contradiction is a necessary law of thought, on the ground that not only do we often meet with self-contradiction, but that sometimes people have even boasted of holding contradictory opinions.[456] If, however, the law of contradiction is to be rejected, it must be shewn not merely that we sometimes contradict ourselves, but that we do so with perfect clearness of thought, and that we do not thereby stultify ourselves.
[456] Compare Bain, Logic, Deduction, p. 223.
The mere fact of our holding contradictory opinions goes for nothing so long as the self-contradiction is not realised by us. In such cases it may be assumed that one or other of the contradictory doctrines will be given up as soon as the contradiction between them is made manifest. If the truth of both is still maintained, it will probably be found that there is some reservation—as, for example, by means of a distinction between different kinds of truth, one doctrine being held to be true literally and the other in some poetical or allegorical sense—whereby consistency is restored at the expense of ambiguity and want of clearness. Apart from some explanation of this kind, the problem of accounting for the way in which some of us appear to hold inconsistent beliefs is one for the psychologist rather than the logician. The ultimate explanation must be sought in confusion of thought, or lack of intellectual sincerity, or in these two causes combined. From a logical point of view to rest in an unresolved contradiction is to stultify ourselves and to confess failure.
417. The Sophism of “The Liar.”—The sophism known as Ψευδόμενος or The Liar has been thought by some writers to present an exception to the universal applicability of the law of contradiction.[457]
[457] Compare Ueberweg, Logic, p. 245.
“Epimenides, the Cretan, says that all Cretans are liars. He is, therefore, himself a liar. Hence what he says is not true, and the Cretans are not liars. But if so, his statement may be accepted, and they are liars. And so on, ad infinitum.”
The solution is simple if we interpret the statement of Epimenides 458 to mean merely that Cretans usually speak falsehood. Let his assertion then be understood in a stricter sense than this, and as meaning that Cretans are always in all things liars, that no assertion made by a Cretan is ever by any chance true.
Again the solution is simple if we merely suppose the assertion false. Epimenides here speaks falsely, but Cretans frequently or sometimes speak the truth. We are obviously confusing the contradictory with the contrary if we pass from the position that it is not true that Cretans are always in all things liars to the position that what a Cretan says must therefore be true.