Monday, February 22, 2010

A Basic Question: Lying Curries?

What follows takes us a bit off-course from the recent discussion about J.C. Beall's maneuvers about "just true," but we'll return to that on Wednesday, with a discussion about a problem for his claim that "just true" can be made sense of through acceptance/rejection talk....which is also, I think, a problem for the general use of acceptance/rejection talk by many contemporary paradox-solvers (particularly parcomplete theorists). In any case, stay tuned for that on Wednesday. For today, let's talk about Curry...


Dialetheists take sentences like:

"This sentence is false."
"This sentence is not true."
"This sentence is false or gappy."

...etc., etc., to be both true and false, as a result of the following familiar argumentative steps (which, given bivalence, and the background assumption that such sentences are meaningful, truth-evaluable, etc., are pretty hard to find fault with):

(1) If the sentence in question is true, it's false, so it's both.
(2) If the sentence in question is false, it's true, so it's both.
(3) It must be one or the other, so it really is both.

Now, sentences like:

"If this sentence is true, the moon is made of green cheese."
"If this sentence is true, Hitler won World War II."

...etc., etc., are in quite a different category. Generally speaking, dialetheists try to get around the Curry Paradox by fiddling with the rules for their conditionals, and I've extensively criticized that strategy here in the past, but right now the question I want to focus on is a more basic one.

Besides weakening their conditionals, a necessary part of any dialetheist solution to Curry is that sentences like the ones above are just false. After all, such sentences can be constructed with anything you like in their consequents, so if the antecedents are true (whether they're just true or both true and false), then everything is true and all possible reasoning goes up in flames. Given the assumption that self-referential truth talk is meaningful, truth-evaluable, etc., the claim that Curry sentences are just false is the only way to avoid triviality here.

Hence, for example, in his book Spandrels Of Truth, JC Beall starts by classifying all "ttruth-ineliminable sentences"--i.e. sentences where the talk of truth and falsity doesn't ground out in a subject other than truth--as being both true and false. Then, when he gets to Curry, he qualifies this by saying that it's only his position on the the "conditional-free" fragment of his language.

OK. But wait.

What about the following combined Liar/Curry sentence?

Sentence LC: "If LC is true, then it's false."

Apply exactly the steps discussed earlier.

(1) Obviously, if LC is false, it's true, so it's both.

(2) Less obviously, if LC is false, it's true, so it's both.

(3) It must be one or the other, so it's both.

The second step is less obvious because one might think that it's possible to say that LC is false, but to deny that if it's true, it's false. (After all, in this case, to say that the sentence is false just is to deny that, right?) Sure. Of course, if the "if...then" is understood as the simple truth-functional conditional of classical logic, then the fact that the antecedent is false would suffice to make the conditional true, so step (2) would be obvious.

The problem, of course, is that in any of the sorts of logics suitable for adoption by a dialetheist, the conditionals won't be truth-functional. (One reason is that, in classical logic, "if P, then Q" has the same truth-table as "either not-P or Q."* Given double negation, that makes Modus Ponens logically equivalent to Disjunctive Syllogism. If dialetheism is right, Disjunctive Syllogism isn't universally truth-preseving unless everything is true, but if "if P, then Q" and "P" are going to be true in all the same circumstances that "either not-P or Q" and "not-not-P" are true, then Disjunctive Syllogism can't fail to be universally truth-preserving without Modus Ponens failing to be truth-preserving in all the same contexts.) So (2) is less obvious than all that.**

That said, it looks to me like (2) still goes through. To see why, assume for the sake of argument that the LC does somehow manage to be(just) false. Given that assumption, what do we want to say about it's status if it's true? In other words, do we want to say...

(I) The LC's (just) false, and if it's true, it's false.


(II) The LC's (just) false, and if it's true, it's true.

It's pretty obvious that we don't want to say (I), since the second conjunct is just what the LC hence, hence, if (I) is right, the LC's both (just) false and true, and hence not really "just" anything.

So let's try (II). If you say that if the LC's true, it's true, then if it's true, then what it says is right. What it says is that if it's true, it's false. Hence, from the assumption that if it's true, it's true, it follows that if it's true, it's false, which is all LC says!

Hence, if it's false, it's true, whether (I) or (II) is correct.

Maybe we could try to argue that neither (I) nor (II) is right. The dialetheist can hardly deny Excluded Middle, given its central role in delivering dialetheias from standard liar sentences, but once we stop accepting that conditionals are entirely truth-functional, we can pry apart "either Q or ~Q" from "either 'if P, then Q' or 'if P then ~Q'". For example, if we put relevance constraints on conditional formation, then it could be the case that P and Q simply don't have the right relationship with each other for either "if P, then Q" or "if P, then ~Q" to be true. Fair enough. The problem is that in this case, P=Q, so by the law of identity (for any sentence P, "if P, then P" is true), "if P, then Q" must be true, from which, as we've seen, "if P, then ~Q" follows, and "if P, then ~Q" simply is the content of LC.

In every discussion I've seen by a dialetheist about what a suitable conditional is for their system, universally satisfying identity is always part of their criteria, and, given identity, (II) must be the right choice, and (I) follows from (II) in any case, and the fact that (I) entails that LC is both true and false is the uninterestingly obvious part.

Now, to be honest, I'm not sure how big a problem this is for the dialetheist. Nothing I've said shows that they can't continue to say that most Curry sentences--i.e. sentences of the format "if this sentence is true, then P"--are (just) false. It does, however, show that any dialetheist who bases their dialetheism on standard Liar reasoning can't insist that all such conditionals are (just) false. They can't, in other words, have a unified policy on Curry sentences.

How much of a problem that is, of course, depends on just what on what you take the virtue of a unified policy to be. For example, in In Contradiction, Graham Priest admits that the Truth-Teller intuitively looks like an excellent status for "gap" status (unlike, he thinks, the more obviously glutty Liar), but argues that both are gluts on the basis of "symmetry" considerations. If one finds such considerations compelling, the issue about lying Curries looks like a bit of an embarrassment.

*This is often put by unsympathetic critics as the claim that "in classical logic, the conditional is 'either not-p or q," but in the absence of a persuasive argument for identifying meaning with truth conditions, this strikes me as a considerable over-statement.***

**One might think that we can construct a lying Curry for which we get (2) on the cheap by specifying that the conditional is a classical one. The problem is that, even if one keeps around the classical conditional in a dialetheist logic along with a more suitable conditional, that conditional won't detach--i.e. Modus Ponens will fail for it. When it comes to Curry sentences constructed with a classical conditional in a dialetheic logical setting, calling them true doesn't commit you to embracing their consequents.

***Interestingly enough, in "Spandrels...", Beall both explicitly rejects--on deflationist grounds--the identification of meaning with truth conditions, and claims that "the hook" (i.e. the conditional of classical logic) just is (~P v Q). Go figure.


Colin said...

Ben, I've been meaning to get back to the other thread of discussion we were having, I just haven't had the time recently. Similarly I don't have time to pore over every detail of this post, but it caught my attention so I just want to toss out a quick comment.

I assume you want (1) to say that if LC is true, then LC is false. Why think so? Well we could reason as usual from the T-schema or from intersubstitution rules for T as follows (letting > be the conditional used in formulating the sentence LC): T(LC) iff T(LC) > F(LC). Presumably we are using the conditional in the metalanguage that we are discussing in the object language so we should be able to infer from this: T(LC) > (T(LC) > F(LC)). But now in order to actually infer the result you want in (1) this would have to be a contracting conditional and that is something Priest, Field, and Beall all reject. Anyway that's kind of quick and maybe off base, if so I apologize. I'm going to try to continue the other thread of discussion later because it was quite interesting stuff.

Ben said...

Hi Colin,

I don't think I need contraction for (1), just (a single instance of) Modus Ponens. Assume that LC is true, apply Modus Ponens, we derive the result that it's false. For the entailment to guarantee the truth of the conditional, we need conditionalization, which might or might not be off the table for certain "suitable conditionals," but the conditional phrasing is irrelevant to the case. (1) could have just as easily been re-phrased as...

"(1) Say that LC is true. It obviously follows that it's false, so it's both."

...which is clearer in any case, so probably how I should have put it initially.

OTOH, now that I think about it, I'm a bit worried that (2) really might only go through given contraction, since the key move is the inference from (II) to LC being true. (To get from (I) to the LC being true, you just need a single application of intersubstitutivity. The more interesting question is (II), especially because (II), being an instance of identity, is going to be true in any case, even for Priest/Beall/Field-type conditionals.) Certainly, the easiest way to get from (II) to Tr is through contraction, but I'm not sure if that's the only way.

Certainly, if we have conditionalization, then there's an easy way that doesn't obviously rely on Contraction, but I'm still not clear on how Contraction can fail without either Modus Ponens or conditionalization also failing. (We talked about this briefly in New York, but that bit of the conversation was unsatisfying in exactly the way that all verbal coversations about formal details are unsatisfying if there isn't a chalkboard in the room. My sense was that what you were saying relied on the sort of relevance logic proof machinery that wouldn't be available to, e.g. Priest, given his endorsement of validity-as-truth-preservation in Doubt Truth To Be A Liar, although it might work for Beall.) So...I'm not sure.

I do still note that even if (2) fails and thus LC gets to be (just) false, if one regards Curry sentences as uniformly false, but still maintains the law of identity, and Capture, then those three things will generate at least one (not terribly interesting) glutty Curry, which we can call the Truth-Telling Curry:

TTC: "If TTC is true, then it's true."

Anyway, on the other discussion, take your time. It's been busy over here too...just busy in a way that happily coincides with generating lots of blog material, since I've pretty much been in the library 24/7 working up my last-ditch dissertation edits. I'm supposed to defend on March 8th, so I'm trying to work up the last minute edits in line with Otavio and Hartry's suggestions on earlier drafts to them before this weekend.

In any case, I think that I'm going to try to write up the criticism of Beall's (tentatively endorsed) view on uniform gluttiness of the conditional-free/ttruth-ineliminable fragment of his language (and how the hole might be plugged within the resources of his view) as a short paper and try to shop it around. If you want to look at that, that might be a good jumping off-point for continuing the earlier discussion.


Colin said...

Sorry I don't have longer on this, but yeah I can verify that Priest, Field, and Beall all reject conditionalisation so that might be an issue. Good luck prepping for your defense, and definitely send on that paper if you put it together.

Ben said...


OK. That's what I always figured--on the face of it, contraction is just Modus Ponens + conditionalization--but it's not a point that tends to be spelled out explicitly in discussions of rejecting conditionalization, and it (initially sounded like) you were saying otherwise in our discussion in NY. Anyway, good to hear, since the section on Curry in the existing draft of my dissertation stops making sense if standard dialetheist solutions don't require us to reject conditionalization, so I'd hate to have to revise or qualify that bit this late in the game.

Anyway, like I said, I don't think I need contraction or conditionalization for (1), but it looks like things are different for (2). As such, I think standard non-classical Curry-solvers might have a good way around LC after all. I still think TTC might be a problem, although how much of a problem it is depends on how much weight one gives to symmetry considerations.