[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.

Thad Coons(11:07:44) :I’m amazed to see people still floundering in this murk. The insistence of philosophers and logicians on classical two-valued logic has crippled their ability to deal clearly and consistency with negation and uncertainty. Using a 3-valued logic, (which has been available for some 80 years now), it’s easy to show that more than one kind of negation and more than one kind of uncertainty can be described. This makes a certain degree of confusion understandable if logicians are relying on rhetorical methods, but it’s ridiculous to waste verbiage when useful tools that contribute to clear and consistent reasoning about the subject have been lying unused this long.

Kenny(12:15:14) :Of course many things can be described with multi-valued logics. My comments on the paper this weekend showed that a few of the arguments used were invalid in a certain three-valued logic that seemed plausible enough. However, in the discussions by Dummett and Horwich, the focus is not on the technical matter of what can be done with various logical systems, but what logical system most accurately represents the reasoning processes that are actually justified for natural or scientific language.

I think it’s quite plausible that natural language will turn out to be too messy and pragmatic to have any particular logic, but that a sort of rigorized scientific language could well turn out to need intuitionist, classical, or possibly relevant logic.

Fuzzy logic and Lukasiewicz logics and various other systems all seem to have applications, but I don’t know of any convincing arguments that they accurately describe something in human language.

Thad Coons(11:58:14) :Using three valued logic, I can construct three different negations.

One of them is a strong negation (-), which could be interpreted as “impossible”. Negating this, in turn, gives a weak “not impossible” assertion, so that P implies -(-P), but not vice versa.

With this negation, the rule of bivalence(P v -P) (P is true or impossible) doesn’t necessarily hold, but the corresponding ~(P & ~P), (not both true and impossible) does.

Another is a weak negation, which could be regarded as “unconfirmed”. Negating this gives a strong assertion. Here we have -(-P) implies P, but not vice versa. This has (P v ~P) (P is true or unconfirmed) holding, but not necessarily ~(P v ~P) (not both true and unconfirmed)

Defining negation so that P is equivalent to ~(~P) gives a situation such that neither (P v ~P) nor ~(P & ~P) is necessarily true, but they are equivalent, just as classical two-valued logic would have it.

However, when the tertiam non datur is strongly asserted as a necessary truth, it cripples the ability to speak of the uncertain. Likewise, the ability to speak of the uncertain apparently forces a weakening of the tertiam non datur in some form, losing double negation, truth functionality, or some similar price.

I’m not sure that either natural language or intuition is required to be consistent, but having a formal language to speak about uncertainty seems better than not having one.

Kenny(14:39:16) :It’s true that having a formal language to speak about uncertainty would be quite helpful. But it’s not clear just what that formal language should be.

With your “strong negation” it’s not clear to me how double-negation gets interpreted as “not impossible” – it seems to me that it should say “it’s impossible that it be impossible” not just “it happens not to be impossible”. Also, I don’t understand how iteration of the “weak negation” gets a strong statement – it seems to me that it should say “it is not confirmed that it is unconfirmed”, which doesn’t seem to imply that it is in fact confirmed. Do you mean the outer negation in both cases to be a classical negation, rather than either weak or strong negation?

Of course, I’m not exactly sure what the details of your system are. The interpretation you’re speaking about with “impossible” and “necessary” truth suggest to me something basically similar to the S4 modal models of intuitionist logic, in which the primary negation is rendered as necessity of the subsidiary negation.

Thad Coons(15:40:37) :You are correct that the outer negation does act as a simple negation. With only three logical values, there is no way to distinguish “it is impossible to be impossible” from “It is not impossible”, or “it is possible”. Likewise, there is no way to distinguish “it is unconfirmed that it is unconfirmed” from simply “it is not unconfirmed”, or “it is confirmed”.

However, I welcome the advance in expressive power over two-valued logic, even if it doesn’t have the full flexibility of natural language.

There is some similarity to other modal logics, but it’s not identical to any of the Lewis systems.

Kenny(21:02:25) :So if the three truth values are taken to be something like 1, 1/2, 0, does the strong negation return the values 0,0,1 and the weak negation return the values 0,1,1?

Kenny(21:03:09) :What about a negation that returns the values 0,1/2,1? Is there a place for this sort of negation in the system?

Thad Coons(06:46:29) :Yes, to both statements. I use the classical negation which returns (0, 1/2, 1) by preference, and don’t use the strong or weak negations per se much at all. Instead, I use their corresponding assertive forms. Instead of the weak negation (0, 1, 1), I ordinarily use “Certainty” (1, 0, 0), and instead of the strong negation, (0, 0, 1), I use “possibility” (1, 1, 0).

These definitions result in a partial modal logic which includes the classical results “Not certainly = possibly not” and “not possible = certainly not”, and allows “If certain then true” and “if true then possible”, but not their converses.

When I began working with this scheme, I kept finding intuitively agreeable and familiar consequences falling out of it so easily and naturally that I was amazed to find little mention and less use of it in the standard sources in non-classical logic.

Chi(10:10:48) :How do you pronounce datur? I know how to properly say tertium in latin, but not datur.