Thursday, November 29, 2007

One Truth Or Two?

In the comments thread on the last post, Brandon made the following suggestion:

"What the dialetheist is essentially doing is suggesting that 'true' is equivocal: there is an unadulterated truth, so to speak, which is classical T; and there is another kind of truth, which is truth as found united to falsehood, which union some dialetheists represent by B."

I don't think this is entirely right, although after explaining why not, I'll circle back around to the question of why I think this might be getting at something important. I'll leave *that* as an open question for commenters, since I'm not entirely sure what to say about it.

This does seem to tie into a larger question about the relation of logical rules and the meaning of logical terms. Just because in classical logic it is taken as a given that something cannot be true or false, does it follow that "true" *means* "true, but not also false"? Quine, in his 1970 "Philosophy of Logic" book, seems to think so, suggesting (I think...I certainly don't have the book in front of me) that anyone who thinks that (P&~P) might be true is changing the *meaning* of the negation sign. Other philosophers and logicians, certainly, have seen these issues differently, as a matter of genuine disagreement between different theories about the same logical concepts, rather than a matter of the same terms being used to denote different concepts. Intuitively, I find the latter view a lot more compelling, but I'd be the first to admit that that's not an argument. In any case, for a moment at least, let's put that view to one side.

So, when a dialetheist says that a proposition is true *and* false, what do they mean by "true" and is it different from what they mean by "true" when they say that a proposition is just true?

Graham Priest, at least, in laying out his own "Logic of Paradox" (LP)--see, e.g. his brief exposition of this in his article "What is so bad about contradictions?"--is very clear on this point. Here's how he puts it. In classical logics, truth value is thought of as something like a function, where 1 (True) or 0 (False) is assigned to each proposition. In the LP, truth value is thought of as a relation, such that any given proposition can be related either to 1 (True), to 0 (False), to both 1 and 0 (True/False) or to neither.

So I think that, on this new understanding, when the dialetheist denies the Law of Non-Contradiction, claiming that some propositions are related to both 0 and 1 instead of only to one of them, they aren't really saying that there are two kinds of truth, one that excludes falsehood and one that is compatible with it, but that there's only one kind of truth, and that some propositions are true but not false and some propositions are true (in *exactly the same sense*) but also false., *that said,* everything I've said in the last couple of paragraphs has been internal to the dialetheist's description of what they are up to. I think that there *is* a legitimate question, from an outside perspective, about whether LP or any other logic featuring "truth value gluts" *really* models a situation in which some propositions are both really true and really false, or whether they are simply three-value logics where one truth value has been arbitrarily labeled "true and false," but could just as easily be thought of in some other way. This was, in fact, my initial reaction when glancing at "In Contradiction" for the first time, and, although I'm no longer convinced this is the case, at least a faint glimmering of suspicion remains in my mind on this point.

Put slightly less tendentiously, even if the third truth value really is related in some interesting way to the first two that captures some important element of paradoxical scenarios, is it, as Brandon suggests, a matter of a union between falsehood and a different thing that the word "true" can mean when altered to fit that context, or a union of what "true" always means and what "false" always means? Priest, at least, claims to be talking about the latter, but are there good reason to suspect that in practice he's really talking about the former?

I will note at least one unpleasant consequence that may hold for dialetheists if the "two truths" interpretation is correct.

"This sentence is not true" remains as puzzling as it was before the abandonment of the Law of Non-Contradiction.


Brandon said...

In the LP, truth value is thought of as a relation, such that any given proposition can be related either to 1 (True), to 0 (False), to both 1 and 0 (True/False) or to neither.

I need to re-read LP; but when Priest discusses the matter elsewhere (e.g., Beyond the Limits of Thought), he often puts it in terms of multivalence, i.e., there are four truth values: true (true simpliciter, as he sometimes calls it), false (false simpliciter), neither true or false, and then both true and false. The problem with this is that, since connectives are usually understood truth-functionally, adding new truth values changes how the connectives are understood, and you can no longer talk about the inference rules as if they were the same (which Priest tends to do). New connectives, therefore new rules governing them, and therefore a different logical subject entirely. Of course, this could be treated as just a loose way of talking. This brings us to the option that there are only two truth values, 0 and 1, which may be given to propositions in the following ways: 0, 1, both 0 and 1, neither 0 and 1.

This gets us out of the problem of additional truth values; but it gets us right into the problem you note at the end of the post. Because classical logic really does mean by 1 or T, true to the exclusion of false, and it really does mean by 0 or F, false to the exclusion of true, so that you could just substitute those descriptions in without any change; and the dialetheist is introducing a new way of looking at truth and falsity, where true can be either classical logic's T or true-and-false, and ditto with false. And then the dialetheist still seems to be changing the subject. (And equivocating; although I'm not quite sure if equivocation is an accusation that has quite so much bite with dialetheists!)

Ben said...

Well, as I read him, Priest strongly favors the conception of philosophy of logic whereby we have pre-formal notions of truth, falsity, negation, conjunction, disjunction, etc., etc., etc., and different logics are just different theories of those pre-existing notions--Quine saying that anyone who thinks statements of the form (P&~P) might be true is changing the meaning of negation operator is like a Newtonian saying anyone who thinks space is relative is just changing the meaning of the word "space"--so there's an extent to which it's quite reasonable (if you think that's the right way to think about it) to talk about the forms of inference as if they were the same.

In any case, it's somewhat orthogonal to the main open question here--this might just make matters worse for the LP if you're right--but I will note that if we do read these as four truth values, rather than as four combinations of the two truth values, this is even more counter-intuitive when it comes to the "neither" than it is for the "both."

"Not having a truth value" really shouldn't be its own truth value.

Colin said...

Sorry to be responding to a post you wrote last month, but I can't help myself. I don't see how dialtheism faces any problems on the basis of the multi-valued model theory used to give, i.e. models for LP. Put the issue this way. Suppose you believe in the non-triviality of truth.

NT: not everything is true.

And suppose you believe that the semantic paradoxes are significant, as opposed to being meaningless sentences.

MS: the semantic paradoxes, such as the Liar, are meaningful sentences.

Now, suppose you believe the correct account of consequence is given by classical logic. This seems to get you into trouble. Take a sentence like L below and run the following argument.

(L) This sentence is not true.

1) L or ~L (LEM)
2) suppose L
3) then ~L
4) so from 2,3 we have L and ~L
5) suppose ~L
6) then L
7) so from 5,6 we have L and ~L
8) by either disjunct in (1), L and ~L
9) so, L and ~L
10) therefore, everything is true (EFQ)

Now you face a problem. It looks like classical logic is incompatible with the assumptions of NT and MS. You could deny one of these assumptions as a potential solution, probably MS, but that is no easy case to make either.

All of this is entirely independent of any details of model theory. It is simply a matter of the inferential rules condoned by one or another logic. Now, even if we accept the impetus to revise our logic, I've highlighted at least two places that revision might take place.

If, like the dialetheist, you think the implication to (9) is tolerable and something we want to understand better, then you need to deny EFQ. So you need a paraconsistent logic. So you need a model theory that involved either truth value relations in place of functions, or some kind of third truth value, etc. But at this stage, the model theory can be seen as instrumental to all the goals I have outlined.

On the other hand, you might take the route of Hartry Field and others who deny LEM. This would block the above inference at the first premise, but as is well known, it seems to raise further similar paradoxes. It remains to be seen who has the better solution here, but the motives are clear enough.

The main point is that the dialetheist needn't have a funny conception of truth and falsity, of there being more than one 'kind' of truth. The model theory which invokes a third truth value can just be an instrument to giving an account of consequence that is appropriate to their aims: to wit, taking NT and MS as fixed and figuring out how to resolve the problematic argument sketched above.