The Law of Excluded Middle
In general, in this book, I am avoiding logical questions, but in this chapter, as in the last, I shall be concerned with a logical topic, namely the law of excluded middle. As every one knows, Brouwer has challenged the law, and has done so on epistemological grounds. He, in common with many others, holds that "truth" can only be defined in terms of "verifiability", which is obviously a concept belonging to theory of knowledge. If he is right, it follows that the law of excluded middle, and the law of contradiction also, belong to epistemology, and must be reconsidered in the light of whatever definition of truth and falsehood epistemology permits. We considered truth and falsehood in a preliminary manner in Chapter XVI, and discussed the attempt to define them epistemologically. It is fairly obvious that, if an epistemological definition is adhered to, the law of excluded middle, in its usual form, cannot be true, though the law of contradiction may be. We have to consider, in this chapter and the next, whether to sacrifice the law of excluded middle or to attempt a definition of truth which is independent of knowledge.*
The difficulties of either view are appalling. If we define truth in relation to knowledge, logic collapses, and much hitherto accepted reasoning, including large parts of mathematics, must be rejected as invalid. But if we adhere to the law of excluded middle, we shall find ourselves committed to a realist metaphysic which may seem, in the spirit if not in the letter, incompatible with empiricism. The question is fundamental, and of the greatest importance.