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.
...now, *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.