A few weeks ago I was in Australia, at the Fourth World Congress of Paraconsistency, and I saw Graham Priest give a talk entitled "Inclosures, Vagueness and Self-Reference." The point of the talk was, as far as I remember, was that he now thinks the paradoxes of vagueness are in the same 'family' of paradoxes as the so-called 'paradoxes of self-reference' (i.e. the semantic and set-theoretic paradoxes), since (he now thinks) they all fit within the Inclosure Schema. I won't be talking about that in this post, but for anyone who's interested in finding out what the IS is or how it fits into Priest's views, I'd recommend that they read his book "Beyond the Limits of Thought."
Before getting to the point of this post, let's define some terms. Since true contradictions are called 'dialetheias,' and the theory that true contradictions exist is called dialetheism, my friend Ryan decided that a good word for the opposite view, that there are no true contradictions, would be 'monaletheism.' Dialetheists believe that a single statement can have as many as two truth-values, whereas a monaletheist is someone who believes that, whether or not there are gappy statements with zero truth-values, the maximum number of truth-values that any one statement can have is one. I prefer to talk about 'monaletheism,' rather than the Law of Non-Contradiction, since if the LNC is just the formula that (∀α)¬(α ∧ ¬α), or even (∀α)¬◊(α ∧ ¬α), then a dialetheist can happily accept that the LNC is true. It just means that every time they assert a contradiction (α ∧ ¬α), they also have to accept the truth of the further contradiction [(α ∧ ¬α)] ∧ ¬(α ∧ ¬α)]. Moreover, in many paraconsistent logics--including Priest's favored logic LP--(∀α)¬(α ∧ ¬α) is a basic rule, derivable from other important basic rules like the law of identity. In fact, in Australia, in a talk entitled "Making Sense of Paraconsistent Logic - Classical Logic, Paraconsistent Logic and the Nature of Logic," Koji Tanaka said that this was a feature of all the paraconsistent logics that he approved of. When I presented my paper later that afternoon, I saw that Koji was in the audience, so when I got to this point I described this as "a feature of all logics that Koji likes," and he nodded enthusiastically.
Anyway, that's monaletheism. So by analogy to true contradictions as "dialetheias," I'll be calling statements with only one truth value 'monaletheias.' A monaletheist doesn't believe in the existence of dialetheias, but a (non-trivialist) dialetheist certainly believes in the existence of monaletheias. (So does the trivialist, technically, but of course the trivialist also disbelieves in monaletheias, which the regular dialetheist does not.) In fact, dialetheists typically believe that the vast majority of statements are monaletheias. For example, 'the Axis powers won World War II' seems to be just false and not true, and 'the Allied powers won World War II' seems to be just true and not false, so those are both monaletheias. I won't be talking much about them in this post, but we could analogously call statements without truth-values (if, unlike Priest, you think that a statement can be gappy without being glutty) nonaletheias.
OK, back to the Priest talk on vagueness and inclosure. One of his examples kind of bugged me, although at the time I couldn't quite articulate why. I'm going to take this out of context, because I'm not interested in the point he was ultimately building toward in that paper, but in whether this this example is actually a serious problem for the dialetheist. Moreover, I'm doing this from memory, so I may be misrepresenting Priest's comments. Still, whether Priest said it or not, it's an interesting point.
One of his power point slides contained a Liar-type sentence that said of itself that it was a false monaletheia. "This sentence is false and not true." If I'm remembering correctly, Priest did a few quick logical derivations to reduce this to a standard Liar, and concluded that it was both true and false. Although of course he wasn't using this terminology, the implicit problem he was dealing with was that this result seems to show that the sentence in question is both a monaletheia and not a monaletheia. (To get the same effect even more starkly, one could try standard Liar paradox reasoning on "this sentence is not true and not a dialetheia." If you think that the usual reasoning from Liars to true contradictions is sound and if, like Priest, you believe that gaps entail gluts, so the only way to not be a dialetheia is to be a monaletheia, "this sentence is true and not a dailetheia" both is and is not a monaletheia.) Priest's response in the Australia talk was to say that he didn't think this was a problem for dialetheism. "After all, the point of dialetheism was never to be consistent, it was just to contain inconsistencies..."
I think this response may be too quick, and miss the real potential objection here, which is roughly this:
If you want to even be able to express the claim that most statements are monaletheias (i.e. that 'the rate of true contradictions is very low, so we are entitled to assign a very low epistemic probability to any particular contradiction,' which is the foundation stone of the 'classical recapture,' the prohibition against multiplying contradictions beyond necessity and whatever plausibility moderate dialetheism has) you need to be able to make sense of the idea of a monaletheia, i.e. that we can assert the truth of some statements in a way that rules out our also accepting their falsity, or vice versa. If, on the other hand, you believe, based on the usual reasoning, that we can derive contradictions from Liar-like sentences, then applying uniform standards means that you have to allow the possibility that something can both be a monaletheia and not be a monaletheia, as in the two cases discussed above. If that's a possibility, then there is no meaningful concept of a monaletheia.
Anyone else see a problem here? If you do, do you see any way that the dialetheist could (non-arbitrarily) get around it without sacrificing a hugely important part of the traditional case for dialetheism? It looks to me like they can't.