[UPDATE 1/28/06: Since everyone seems to be finding this page (and my blog in general) by googling “Tertium non Datur”, I figure I’ll explain the principle right here. The Wikipedia page on the Law of Excluded Middle may be useful as well. “Tertium non Datur” is, I believe, Latin for “the third is not given”, meaning that there is no third option beyond true and false for sentences. Michael Dummett (in the introduction to Truth and Other Enigmas) distinguishes this from the Law of Bivalence, which says that not only does no sentence have a value other than true or false, but in fact every sentence does have one of those two values. They are very similar principles, but are importantly different if one doesn’t assume classical logic.]
I commented on a paper this weekend at the Berkeley-Stanford-Davis Graduate Philosophy Conference that was a defense of the law of excluded middle. However, it seems to me that it might be better to call it a defense of the law of tertium non datur, at least according to the confusing terminology Dummett adopts on page xviii-xix of the preface to Truth and Other Enigmas in an attempt to clarify the essay “Truth”. According to Dummett’s usage, the law of excluded middle is the logical law asserting that “A or not A” is always true. The closely related law of bivalence states that every statement is either true or false. He distinguishes these from a principle he calls tertium non datur, which states that no statement is neither true nor false.
It seems clear that the law of bivalence implies tertium non datur (if there were a statement that was neither true nor false, then bivalence would be false, so there must not be such a statement), but the reverse implication only works if one believes the law of double negation, which someone like Dummett rejects.
The reason Dummett felt obliged to include all this terminology in the preface to his collection is that in “Truth”, he takes himself to be defending tertium non datur while attacking the law of bivalence.
I won’t discuss the paper I was commenting on, but I will point out that Paul Horwich seems to have been victim of the same confusion in his book Truth. In Question 18, he seeks to show how his minimal conception of truth can preserve the fact that the product of ideal inquiry is true, while attacking the notion that all truth is the product of some ideal inquiry, which he takes to be a constructivist or anti-realist position of a sort. (Some of the same confusions arise in Questions 26-28, though perhaps Horwich is more clear about what’s going on there.)
He points out that there are facts beyond the reach of ideal investigation (he question-beggingly calls them “truths”), such as vague predicates, or perhaps Dummett’s example about the courage of a now-dead individual who was never exposed to danger in her life. In such a case, neither A nor not-A is verifiable, and thus A is neither true nor false. But (using “T” for the truth predicate and “~” for negation), this just means ~T(A)&~T(~A), which (using the T-schema (which Horwich supposes everyone should accept)) just comes to ~A&~~A, which is a clear contradiction.
But this just points up some of the dangers of supposing that a statement’s truth just consists in its ideal verification, not with a sort of anti-realism in general. I think means that an anti-realist who wants to preserve the T-schema should say that the truth of an atomic statement consists in its ideal verifiability, while the truth of a negation consists in the ideal verifiability that no actually possible investigation will establish the truth of the original proposition. Thus, rather than “X was not brave” meaning that it is verifiable that X would have acted cowardly had she been exposed to danger, this anti-realist should say that “X was not brave” means that no process of inquiry will reveal that X would have acted bravely had she been exposed to danger. Thus, it is conceivable to conclude that X was not brave and X was not cowardly, without concluding that neither was X brave nor was she not brave.
Thus, on this view, establishing that a particular atomic proposition is undecidable will establish its falsity.
Since the meaning of a disjunction is taken to consist in the idealized verification of one of the disjuncts, we see that the universalized law of excluded middle might fail, though tertium non datur holds. That is, we can’t guarantee that either there is a verification of a statement or a verification that it can’t be verified. This doesn’t mean that we leave it open that there is such a statement (because for such a statement to exist is to verify both that it can’t be verified, and that there is no verification that it can’t be verified, which is contradictory), but it also means that we can’t guarantee that the law holds in general. We can just show that in every particular instance, it won’t fail (though we don’t know in which one of the two ways it will not fail).
Of course, it might also be possible for an anti-realist to deny the T-schema, but I don’t think that’s necessary. At any rate, it would bring the discussion too far away from the positions that Horwich is contemplating here.