## Monday, August 4, 2008

### Can Dialetheists Make Sense of Monaletheias?

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.

Daniel Lindquist said...

I agree that there seems to be a problem here, and I don't see a way out for the dialetheist.

Allowing that sentences might be both true and false is one thing; allowing that they might be true and not true (and/or false and not false) is another. I've noticed that Priest almost never mentions the latter sort of contradiction, though it seems that any dialetheic logic is going to have to allow it.

Consider "This sentence is false and not true"; if, like Priest, you want to call Liars both true and false, and want to do the same here, then it must be (at least) true, false, and not true, and thus is at least true and not true.

There isn't even a way to express that in the notation he uses in "What's So Bad About Contradictions?"; you have to use two relations to express that a formula is both related to and not related to the truth-value True, but in Priest's notation an evaluation is supposed to be a relation between formulas and truth-values. (Presumably, Priest's description for how his notation works was not intended to itself be a dialetheia!)

I do not see a way to remedy this last problem: Even if evaluation-relations are allowed to relate formulas to four truth-values (True, False, Not True, and Not False), there are revenge problems ("This sentence is False and Not True and not True and not Not False" still can't be assigned a truth-value with a single evaluation-relation), and even if there weren't it would seem queer that a formula's not being true was not just a matter of it not having the truth-value "true"!

I think that you're right that the "classical recapture" is dialetheism's weakest point; it's hard for classical reasoning to be plausibly "recaptured" if being true and not false does not exclude being false and not true. (And even being "true and not false and neither both true and false nor both not true and not false" does not exclude being both true and false and not true and not false! Indeed, it seems that no truth-value excludes any others. That seems really bad. This last piece of reasoning seems like it has to be too hasty, since it would be devastating to dialetheism, but I can't see what a response would be like.)

Colin said...

We need to get clear on the sense of being true and not false and/or being false and not true that we are after here. Daniel says that it is a problem to allow sentences to be both true and not true, but on most renderings of falsity as truth of negation and on truth as a transparent device, being not true is simply the same as being false. So I don't see why this would be any deeper (in fact, even why it is any different) an issue than dialetheia were in the first place.

From the OP:

"If you want to even be able to express the claim that most statements are monaletheias... 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"

I agree that this the target. And I fail to see where the dialetheist runs into any problem on this front. Suppose I am a dialetheist and I assert P. You know that I might also believe not-P so you begin to wonder whether I am committed to just P or to both P and not-P. I tell you that I reject not-P. I have verified my belief in a monaletheia. Problem solved, no? Well you say it is not for the following reason:

"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... then there is no meaningful concept of a monaletheia."

I don't see this. I think there is a lot of hand-waving about meaningfulness going on here. My best stab at what you are saying is that a necessary condition on grasping the concept of monaletheia at all is to have a semantics in which no sentence both is and is not a monaletheia. Why? Do you have an argument for this?

Daniel Lindquist said...

"Suppose I am a dialetheist and I assert P. You know that I might also believe not-P so you begin to wonder whether I am committed to just P or to both P and not-P. I tell you that I reject not-P. I have verified my belief in a monaletheia. Problem solved, no?"

No. For I still don't know if you accept not-P, even if I know you reject it. You hold "not-P" false; but you might also hold it true.

Suppose I then ask you "Do you merely reject not-P, or do you both accept and reject it" and you respond that you merely reject it. This still does not tell me what I originally wanted to know (if I know you are a dialetheist), for you might reject, accept, and not accept it: in which case you both reject it and do not accept it, and so you merely reject it, though you still accept it.

(For anyone who rejects a claim and does not accept it merely rejects it.).

I don't see that there can ever be an end to such a dialectic. Whatever you say you are committed to, it remains an open question for me whether or not you're also committed to denying all of it, even if I take you at your word on everything you say.

Suppose I grant that you have affirmed your commitment to a monaletheia; if you take it to be possible for a claim to be both a monoletheia and not a monoletheia (in the sense of both a monoletheia and a dialetheia), then you've left open whether or not you're committed to what you take to be a monoletheia that is both true and false.

I suspect this is what Burgis had in mind when he said it seems to undermine the very idea of "monoletheia" as meaningful; having only one truth-value seems like it should exclude having two, if identifying something as a monoletheia is to do the work we wanted it to do (such as allowing you to make clear that you assigned only a single truth-value, True, to P).

Colin said...

"This still does not tell me what I originally wanted to know (if I know you are a dialetheist), for you might reject, accept, and not accept it: in which case you both reject it and do not accept it, and so you merely reject it, though you still accept it."

But to be honest what does any of this have to do with dialetheism? I could be a classicist. I could tell you that I accept P, but nothing precludes that I might also accept not-P. My logical commitments do not render me a perfect agent of that logic.

Also, I take it that acceptance and rejection are mutually exclusive. If I am right about this, then in virtue of rejecting not-P it follows that I do not, in fact, accept not-P. There are subtle issues here concerning the relation between the logic we use, our actual commitments (which may always not be closed under the logic we use), the relation between believing true and accepting with that of believing false and rejecting. I don't think it is obvious how all of these things are related.

Also, this talk of truth values is sort of conflating things isn't it? We have truth values in our formal semantics, i.e. model theory. But our language does not *really* in any substantive sense stand in relation to things called "truth values". We just have a truth predicate and a (defined) falsity predicate. So I guess I don't see what you mean when you say "having only one truth-value seems like it should exclude having two..."

In one sense, this sort of thing is no problem for the dialetheist cum paraconsistent logician. Just use a many-valued semantics for your paraconsistent logic and you have a formal system in which every sentences gets a single truth value. In particular, we can "see" from the outside that many of the sentences have the single value "true". Does this answer your concern? If not, why not?

Daniel Lindquist said...

"But to be honest what does any of this have to do with dialetheism? I could be a classicist. I could tell you that I accept P, but nothing precludes that I might also accept not-P. My logical commitments do not render me a perfect agent of that logic."

For a non-dialetheist, both accepting and rejecting P would be irrational, since they hold that P cannot be both true and false. But both accepting and rejecting some P (such as "This sentence is false") is something the dialetheist can do without being irrational by their own lights. (And I follow Davidson in holding that massive irrationality is impossible without one's thoughts losing all content.)

"Also, I take it that acceptance and rejection are mutually exclusive."

If so, then either truth and falsity are mutually exclusive, or accepting is other than holding true or rejecting is other than holding false. If dialetheism is forced to endorse the latter, then I'm satisfied that the gig is up (though I recognize that others may differ in their judgement of the situation).

"Also, this talk of truth values is sort of conflating things isn't it? We have truth values in our formal semantics, i.e. model theory. But our language does not *really* in any substantive sense stand in relation to things called "truth values"."

I generally mean talk of "truth-values" as simply idiomatic: to stand in relation to the truth-value "True" is just to be true, where truth is just what satisfies the T-schema. So I don't think that my talk of truth-values floated free of ordinary language any more than all semantical discourse (such as this very sentence) does. Truth and falsity are properties of sentences. (Where I discussed Priest's notation from "What's So Bad About Contradictions?", I mean the phrase as employed in the explanation of that notation; Priest talks of relating and not relating to truth-values as being how evaluations are treated in his notation.)

"So I guess I don't see what you mean when you say "having only one truth-value seems like it should exclude having two...""

A dialetheist can affirm that she holds a given proposition to be a monoletheia without losing her right to be committed to it being a dialetheia, by her own logic. A dialetheist's "monolethia" might be a dialetheia, too. If the statistical infrequency of dialetheias is supposed to motivate the "classical recapture", then this is a problem: for anything we count as a monoletheia (from a dialetheist's point of view) might need counting as a dialetheia, too; this makes it hard to say whether or not dialetheias really are uncommon.

(Suppose a dialetheist says that he regards P as true, and not false. If not being false is simply being true, then it is an open question whether or not he regards P as false, even without allowing for irrrationality on his part.)

"In one sense, this sort of thing is no problem for the dialetheist cum paraconsistent logician. Just use a many-valued semantics for your paraconsistent logic and you have a formal system in which every sentences gets a single truth value. In particular, we can "see" from the outside that many of the sentences have the single value "true". Does this answer your concern? If not, why not?"

No. For this "view" from outside is not available to those involved in the actual game of affirming and denying things in conversation: I cannot simply "see" what your commitments are, but have to work them out from what you affirm and deny (or more properly, from what I take you to affirm and what I take you to deny). And if you were to make your commitments explicit in the course of conversation through the use of logical & semantic vocabulary, it falls to the same problem as all the rest of your vocabulary.

(I'm also not sure that dialetheias can plausibly be expressed with a single truth-value; if being "true and false" is not a matter of being true and being false (but is having a third value) then it's not clear to me that anything taking the value "true and false" should count as a contradiction, since it takes neither the value "true" nor the value "false", and certainly doesn't take the both of them.)

Colin said...

Well it strikes me that we disagree about too many things here to reach common ground. I think that it is plausible that rejection is not believing to be false. You say that means the gig is up by your lights, so be it.

One issue that was on the table is whether or not you could suss out the dialetheists commitments with respect to a monaletheia. You argued that we cannot because even if the dialetheist accepts P and rejects not-P she may still accept not-P. My point was that this is perfectly open to the classicist as well. Nothing about their commitment to classical logic can prevent them from having inconsistent beliefs. But you then claim that this would be irrational for the classicist and not for the dialetheist. I don't know what that has to do with the issue of what her commitments are in the first place. Not to mention that it seems rationality has got to be independent of personal logical commitments. If it is irrational for the classicist to hold incosnsitent beliefs it should be irrational for everyone, no? And of course there are lots of ways to be irrational which have nothing to do with logic. It would be irrational for the dialetheist to believe that it is both Monday and Tuesday, i.e. not Monday. This is not because such a thing is logically impossible by her lights, but simply because she has absolutely no reason to accept that something so bizarre is true and plenty of reasons to reject it. But as I said, I don't see what questions of rationality have to do with anything. We were talking about sussing people's commitments, right?

Ben said...

Colin,

(1) Since the discussion was about dialetheism vs. monaletheism (i.e. the "Law of Non-Contradiction" when it's meant in a more than the purely formal sense), I'm not sure why you broad in 'the classsicist.' I'd say that if part of the reason that the classicist in question endorses classical logic is that they are committed to monaletheism (so, e.g. they believe that it's fine if contradictions imply everything, since they are never true, so all inferences from them are vacuously truth-preserving), then they can't believe something inconsistent without revising this belief.

Saying that the 'classicist' (assuming that we don't mean a trivialist who is committed to classical logic, committed to some inconsistent belief, and actually accepts all entailments of that combination) can believe something inconsistent is like saying that the act utilitarian can believe that some action is morally right even though it would not result in the greatest possible amount of happiness. There's a really uninteresting sense in which this is true in both cases--the act utilitarian could say that an action of that type is right because they don't realize that it wouldn't lead to the greatest happiness, just as the (monaletheist) classicist could believe something inconsistent without realizing that it is inconsistent. Saying that if it is irrational for the monaletheist to believe an inconsistency, it should be irrational for everyone to believe in an inconsistency is the exact equivalent of saying that if it is immoral for the act utilitarian to act in a way that won't bring about the greatest happiness, it is immoral for everyone to. In other words, it's technically true, but misses the point in a really basic way. Yes, if act utilitarianism is true, it is wrong even for the Kantian to act in a way that won't bring about the greatest happiness, but it won't be wrong according to the Kantain. Similarly, if monaletheism is true, then it is irrational for anyone to believe any inconsistency, even the dialetheist (i.e. dialetheism is irrational), but it won't be irrational according to the dialetheist.

(2) Similarly, I totally agree that being a dialetheist doesn't commit you to believing any particular contradiction, and that there are lots of ways to be irrational that have nothing to do with consistency. So far, so good. The problem is that I think this misses the target. I don't think Daniel was arguing--and I know I wasn't--that any particular dialetheist is likely to believe the negation of any particular claim they've expressed belief in, but merely that it's a problem for the dialetheist that, on their lights, they don't seem to have a way to express the information that they are ruling out the truth of the negation of that claim. This is what I take Daniel to have been getting at with the bit about the view from outside the game not being available.

(3) The problem is that, although you are absolutely right, falsehood is often defined (even by Priest) as truth of negation, phrases like 'true and not false' are precisely the ones Priest and other dialetheists often use to try to indicate that they think some statement only has one truth value. It's fine if this is just a kind of shorthand for a more precise way of putting things, but I'd like to know what that more precise way is. I do think that if it's possible for something to be a monaletheia and not a monaletheia, then the term monaletheia is rendered meaningless, or at least that it doesn't have the intended meaning. For any term X that we come up with, if it's rationally possible to affirm that a sentence is X but not affirm that it's not-X, or only to affirm that it's X but not affirm that it's a not-X, then being-an-X is neutral between being a monaletheia and being a dialetheia. The problem, as I see it, is that to even express the idea that most statements are just true or false but not both, which is crucial to the classical recapture and the rest, you have to come up with some term X that this is not true about. This doesn't look possible, if dialetheism is true and Liar-like sentences are dialetheias, since we can construct things like:

"This sentence is just false."

(4) Since you seem to agree that it is impossible to express, just in terms of assertion and negation, the right kind of belief in P such that you are also showing that you fail to believe ~P, you fall back on the distinction between asserting/denying and accepting/rejecting. The problem is--as Greg Restall likes to point out in a few of his papers on this--it's horribly unclear what the relationship is between denial and rejection once you allow in truth-value gluts, or even truth-value gaps. Now, if you don't allow either, it's all terribly clear, since what-should-be-rejected and what-we-should-assert-the-negation-of is necessarily co-extensive, as of ourse is what-should-be-accepted and what-should-be-affirmed. You seem to think that the dialetheist ought to keep assertion and acceptance co-extensive, but break the link for negation and rejection, and only reject things that they don't also accept. Daniel seems to think that the dialetheist ought to keep both co-extensive, such that they should sometimes simultaneously accept and reject. I have no idea who's right about this, but in any case, I'm not sure how much breaking the link between negation and rejection would help the dialetheist here. Let's say that we reserved rejection for monaletheically false sentences, so that we should accept all other sentences, i.e. sentences that are dialetheically false (i.e. also true) and sentences that are monaletheically true. (That's you're proposal, right) Then we encounter the following Liar-like sentence:

"This sentence deserves rejection."

Given that we accept the account of acceptance rejection just given, and the usual moves employed to derive contradictions from Liars, this seems like a pretty straightforward case. If it is true, it follows that it is false, since this would only be true of (some) false sentences. If it is false, then it is in one of the two categories that we've decided don't deserve rejection, (a) monaletheic truths or (b) dialetheias. The problem with (a) is that we've already seen that if it is true, it's false, so one way or the other, it's a dialetheia. It both deserves rejection and does not deserve rejection.

...so I'm not sure falling back from assertion/negation talk to acceptance/denial talk helps.

Ben said...

"For any term X that we come up with, if it's rationally possible to affirm that a sentence is X but not affirm that it's not-X, or only to affirm that it's X but not affirm that it's a not-X, then being-an-X is neutral between being a monaletheia and being a dialetheia."

....was a typo, obviously. I meant to write something like:

"For any term X that we come up with, if it's rationally possibly to affirm that a sentence is both X and not-X, or only to affirm that it's X and fail to affirm that it's not-X, then being-an-X is neutral between being a monaletheia and being a dialetheia."

Colin said...

"I'm not sure why you broad in 'the classsicist.'"

Sorry should have stuck to the present terminology, but I think it is sort of doing the same work. After all, the classical logician and the monalethetist have in common that they think it is impossible for anything contradictory to be true. In particular I would assume that the monaletheist believes this in part because she adopts a non-paraconsistent (although to be fair, it needen't specifically be classical) truth theory.

"... if monaletheism is true, then it is irrational for anyone to believe any inconsistency, even the dialetheist (i.e. dialetheism is irrational), but it won't be irrational according to the dialetheist."

I realized that I was sidestepping the intended target when I said all that about inconsistency, but part of the reason I was intentionally sidestepping is because I don't see why this above issue is the more interesting issue. I thought that the discussion had turned to the question whether or not a dialetheist could believe in monaletheia and communicate this in a way that we can tell they believe of this particular sentence that it is just true and not false. With respect to that topic I don't see why the dialetheist's commitments to this-or-that logical or semantic theory has any bearing on what we can suss of her commitments.

"...they don't seem to have a way to express the information that they are ruling out the truth of the negation of that claim."

I assume, though, that you know enough of the literature to know that this is not true. There are, for instance, stronger-than-material conditionals which can be used to express the simple falsity of P by expressing something very roughly along the lines of "if P is true, then everything is true". Assuming that we can take it for granted that most dialethetists are not trivialists, we can infer from such an expression that they mean to say that P is false and not true.

Another idea which is really hard to work out would be to define a proper sub-contrary of truth that we can call "untruth". Then the monaletheia are the sentences which are true and not untrue. But this is just a suggestion which no one to my knowledge has worked out.

On the last long bit about assertion/acceptance and the rest, I agree this all a bit hazy, but it deserves consideration. I don't think I have much to say that will clear it up so I'll leave it for now.

Ben said...

Colin,

"There are, for instance, stronger-than-material conditionals which can be used to express the simple falsity of P by expressing something very roughly along the lines of "if P is true, then everything is true". Assuming that we can take it for granted that most dialethetists are not trivialists, we can infer from such an expression that they mean to say that P is false and not true."

Good point, but surely for the moderate dialetheist, there are lots of statements (e.g. "the Axis powers won World War II") for which (if the if-then is read relevantly and not just materially) it would not be true that "if P, then everything is true." It would, actually, only be with a classical conditional that one could use "if the the Axis powers won World War II, then everything is true" to express (given that they weren't a trivialist) that they didn't think "the Axis powers won World War II" was dialetheic.

It seems to me that almost no normal false statements P are such that it would be (relevantly) true that "if P, then everything is true."

...and of course, this is all, 'given that the dialetheist isn't a trivialist,' and whereas there's a sense in which I agree that there's no reason not to assume this, one might object that it just puts the question back. How does the non-trivialist dialetheist express the falsity of trivialism in a way that rules out trivialism also being true?

"Another idea which is really hard to work out would be to define a proper sub-contrary of truth that we can call 'untruth'. Then the monaletheia are the sentences which are true and not untrue. But this is just a suggestion which no one to my knowledge has worked out."

My problem is that it seems to me that the reason it would be so difficult to work out, given the assumption that we can derive true contradictions from Liar-like sentences, is that it looks very much like you could always construct a Liar that attributed untruth or whatever to itself, and there seems to be no non-arbitrary way of saying that the moves that allegedly allow to get true contradictions from normal Liars wouldn't work for the new one.

Colin said...

Ben, good point on the conditional, this sort of thing worries me a little bit too... but on the upside I think we can agree that at least there are some P for which this strategy works. And if all it takes to express that non-triviality of your theory is to be able to express that some sentences are monaletheia, then this might do the trick. But you might be looking for a stronger result.

Also on this issue: "My problem is that ... it looks very much like you could always construct a Liar that attributed untruth or whatever to itself..."

I agree this will happen, so then you maybe have a stronger 'really untruth' and so on. The idea here is to sort of mimic Hartry Field's paracomplete approach to the "revenge problem" only in a dialetheic setting. But it remains to be seen how to do it at all and whether it is really a satisfactory resolution of the initial worry.

Unknown said...

First, I'd like to note that a comment (I think it was one of Colin's) seems to have disappeared.

As for Colin's proposal to use non-material conditionals to express that some proposition p is a monaletheia, I think Ben's criticism hits the mark. But why not just say something like this: Call the proposition expressed by the sentence "Every proposition is true" 'T'. Stipulate that a proposition p and a proposition q "have the same truth status" if and only if every truth value had by p is had by q and every truth value had by q is had by p.* Then a dialetheist can express "p is a monaletheia" by saying either that that p has the same truth status as T (if they wish to say that p is false only), or that p does not have the same truth status as T (if they wish to say p is true only). Of course, as Ben points out, the dialetheist is then faced with the problem of how to express that T itself is a monaletheia, which may be difficult but perhaps not insurmountable.

*[Why not just say that p and q "have the same truth value"? Because I think statements of the form "p has the same truth value as q", if p is a monaletheia and q is a dialetheia, should also count as a dialetheia by the dialetheist, whereas by my above definition, if p is a monaletheia and q a dialetheia the statement that p and q have the same truth status is false only.]

Unknown said...
This comment has been removed by the author.
Ben said...

Hey, Jason....that's actually the best suggestion I've heard yet, but you did anticipate exactly what my objection would be, so I won't spell it out.

Joseph Lurie said...

I'm missing something here. Why can't we just define being a monaletheia as not being a dialetheia? We already know that the concept of dialetheia is immune to the liar construction, as "This sentence is not a diaelethia" can be taken as simply true, and its negation can be taken as simply false. So if we claim that "This sentence is not a monalethia" simply means "This sentence is a dialetheia," then the apparent liar paradox should be averted.

Ben said...

Joseph,