Monday, January 4, 2010

How Not To Solve Curry

A thought inspired by a conversation at the Smoker at the APA:

Curry is, on the face of it, the same kind of problem for dialetheists that the Strengthened Liar is gap theorists. If you try to get around the Simple Liar ("this sentence is false") by saying that it's neither true nor false, you're going to have a hard time convincingly applying that solution to the Strengthened Liar ("this sentence is not true"), since, after all, if it's neither true nor false, it's not true, so it is true.

Similarly, when the dialetheist solves the Stregthened Liar by biting the bullet, embracing the contradiction and using paraconsistent logic to limit the damage, that solution is going to be utterly inapplicable to Curry sentences like

Sentence C: "If sentence C is true, the earth is flat."

Plugging this into the T-Schema, we get:

1. Tr(C) ↔ (Tr(C) →E)

Applying the definition of ↔, we get:

2. [Tr(C) → (Tr(C) → E)] & [(Tr(C) → E) → Tr(C)]

By Conjunction-Elimination, we get:

3. Tr(C) → (Tr(C) → E)

Now, from here, an easy conditional proof, using only Modus Ponens (MP), will get us to Tr(C) → E. Start by assuming the antecedent:

4. Tr(C)

Plug 3 and 4 into Modus Ponens to get:

5. Tr(C) → E

With one last application of Modus Ponens, to 4 and 5, we've got the consequent:

6. E

As such, by conditional proof, we have:

7. Tr(C) → E

Now, going back to 2, we apply Conjunction-Elimination a second time to get:

8. (Tr(C) → E) → Tr(C)

From 7, 8 and Modus Ponens, we get:

9. Tr(C)

Finally, from 7, 8 and Modus Ponens, we get:

10. E

So, regardless of whether you take C to be true, false, both, neither, tralse, frue or whatever, the mere statement of its truth conditions entails that the earth is flat. (Or, more to the point, entails everything, since the consequent could be anything.) For the dialetheist whose program is to embrace a few contradictions here and there while retaining the ability to reason non-trivially, this is every bit as much of a problem as it is for everyone else. Moreover, it's quite obviously a Liar-like semantic paradox, and the logical rules we appealed to (centrally Modus Ponens and Conditional Proof, which together seem like the most basic, intuitively obvious rules about conditionals that you could ever hope for) aren't ones whose justification relies in any way on the assumption that there are no true contradictions. (Contrast to the proof that contradictions entail everything, which relies on Disjunctive Syllogism, which in turn does rest on that assumption, since if there are true contradictions, then it's possible for P to be true and hence for (P v Q) to be true, and for ~P to be true as well, but for Q to be (just) false.*) As such, one might be forgiven for wondering if Curry shows about dialethetic approaches to the semantic paradoxes (even more specifically, the semantic paradoxes revolving around self-referential truth-talk) exactly what the Strengthened Liar shows about the (simplest versions of) gappist solutions to the same paradoxes: that they're simply a non-starter, because there are clear examples of the relevant category of paradoxes relative to which they don't help.

Of course, technically, someone could doggedly insist that the Simple Liar was neither true nor false, but diagnose the Strengthened Liar in some completely different way, but at the very least, they would owe us a really good explanation of the disconnect. Similarly here.***

Now, keeping in mind that any dialetheist's favored logic had better be paraconsistent (if they don't want every contradiction they believe in to entail everything), one paraconsistent option is relevance logic, where validity-as-mere-truth-preservation is rejected and a tighter sense of "follows from" is insisted on. Moreover, the proof machinery of some relevance logics--requirements about discharging certain premises, multiple uses counting the same as others, etc.--might invalidate the above proof, without forcing the dialetheist relevance logician to bite the considerable intuitive bullet of giving up on Conditional Proof or Modus Ponens.

That won't work.

The point, after all, of the more tightly disciplined inference rules of that relevance machinery is precisely that truth-preservation isn't enough, that inference needs to be held to a higher standard. Now, there's an interesting and respectable case to be made for that claim, but it doesn't strike me as being very relevant to why Curry is a problem.

If the dialetheist relevance logician grants that the statement of the truth conditions for Curry is true--Tr ↔ (Tr →E)--and, to be clear, they'd damn well better accept that, given that T-Schema-absolutism is central to the argument for true contradictions from the Liar, and the dialetheist can hardly start claiming when it comes to Curry that problematic sentences are meaningless or non-truth-evaluable--and they grant that all of the steps in the proof are truth-preserving, whether they take them to be valid just seems to be beside the point. The relevance logic proof machinery might stop you from having to come to every conclusion, but the proof still shows us that every conclusion is true.

As such, if you want to be entitled to reject the substantive thesis of trivialism--everything is true--any solution to Curry in terms of relevance requirements for validity simply won't cut it.

*Technically, given the logical equivalence of P→Q with (~P v Q) in classical logic**, one could argue that Modus Ponens--being logically equivalent to Disjunctive Syllogism--does rest on the assumption that there are no true contradictions. I'm not aware, however, of any dialetheists who take this route, and choose to live without the inerential power of Modus Ponens, rather than simply arguing against the equivalence of P→Q with (~P v Q).

**Sometimes critics say things about this like "in classical logic, the conditional is (~P v Q)" or "classical logic claims that P→Q means (~P v Q)," but I tend to think that this runs rough-shod over a rich problem area of philosophical debate about meaning. Intuitively, after all, two things can be true under all the same circumstances but still mean quite different things. ("Consider "the sun has a shape" and "the sun has a size.") There are philosophical theories of meaning that deny this intuition, but it is, at any rate, a contentious issue.

***Graham Priest thinks he has just such an explanation: the Incosure Schema. The Liar fits, the set-theoretic paradoxes fit, Curry doesn't, so it's OK that his solution to the first two doesn't apply to the third. I find this fairly unconvincing, for a variety of reasons, not least of which is that the subject matter of the first and third seem to have a lot more in common with each other than the first does with the second. A fuller critique of the IS is a subject for another day--and, actually, a paper I'm working on--but I'd say for now simply that if Curry doesn't fit, that looks like a pretty good counter-example to the claim that the IS captures the relevant cluster of paradoxes, given that Curry is a classic semantic paradox involving a self-referential truth-value ascription.

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