Wednesday, February 24, 2010

Another Point About Lying Curries

[Note: For the sake of continuing a thread of the last discussion, I'm pushing back the already-written-and-scheduled post on the use of the negation-revenge distinction in discussions about revenge problems until Monday.]

A couple of days ago, I talked a bit about the overlap between the Liar and Curry Paradoxes. The standard dialetheist line is that Liar sentences (sentences that, in one form or another, deny their own truth) are both true and false, whereas Curry sentences (sentences that say of themselves that if they are true, some other claim P is true) that they are (just) false. After all, if such sentences were either (just) true or both true and false, given that "P" can be anything, triviality would ensue.

Of course, the problem with Curry sentences is that even if one says that they're false, if one acknowledges that (a) their truth conditions are given by the relevant instances of the T-Schema, and (b) that contraction--the inference from "if P, then if P, then Q" to "if P, then Q"--is valid, it follows that such sentences are true. Since a dialetheist (who argues for dialetheism on the basis of the Liar Paradox) is hardly likely to accept that the T-Schema ever fails (given its role in arguments from contradictions from standard Liars), so, unsurprisingly, they pick (b).

Now, contraction certainly sounds valid, but it's a bit obscure--on first consideration, we might think that it's not something that comes up a lot in important non-paradoxical contexts. (Of course, given the assumption that conditionals are truth-functional and that all sentences are true, false or both, it's hard to get around contraction's validity, but we have other good reasons to reject the idea that conditionals are entirely truth-functional anyway.)

Whether or not all of this is plausible, notice that in order for contraction to fail, either Modus Ponens or conditionalization must fail as well, and conditionalization is one of the most basic rules related to the conditional that there is, one that one might be forgiven for describing as basic to the very notion of a conditional. Assume the antecedent. Prove the consequent. Conclude that if the antecedent is true, so is the consequent. Intuitively, this an Modus Ponens are two sides of the same coin--Modus Ponens tells us that, given "if P, then Q," P entails Q, whereas conditionalization tells us that, given that P entails Q, "if P, then Q." This intuitive symmetry is routinely honored in introductory logic textbooks that refer to Modus Ponens as "conditional-elimination" and conditionalization as "conditional-introduction," and by philosophers and logicians who think nothing of casually talking about the "rule form" and "conditional form" of various rules.*

...all of which is to say the intuitive cost is considerable, but at least the paradox is blocked.

Now, in the last post, I raised the question of "lying Curries," paradoxical sentences that combine features of traditional Liar and Curry sentences, like LC, below.

LC: "If LC is true, then LC is false."

...I suggested that, even given the ways in which standard dialetheist solutions to Curry restrict the power of conditionals, standard Liar reasoning might get us the result that LC is both true and false. Thus, the dialetheist might be deprived of their ability to have a uniform policy whereby Curry sentences are always (just) false.

In the discussion in the comment thread, I was convinced that the reasoning I employed relied on contraction at a crucial step. Thus, contrary to my original claim, the standard dialetheist Curry-solver does have the formal resources for blocking a contradiction about the LC.

Of course, I still think that the dialetheist who says that Curry sentences are invariably false has to recognize at least one glutty Curry, namely the Truth-Telling Curry (TTC), below.

TTC: "If this sentence is true, this sentence is true."

Given the law of identity (still universally validated with the "suitable conditionals" favored by dialetheists like Priest and Beall), and Capture (the rule that says that we can infer "'A' is true" from A), TTC must be true. Thus, if the dialetheist Curry-solver says that all Curry sentences are false, they have to acknowledge that this is both true and false. If they abandon it in the face of this example, the TTC is still (just) true. One way or the other, they aren't entitled to a unified policy of regarding Curry sentences as (just) false. How big a deal this is depends on how compelling you find the sorts of symmetry considerations that lead people to seek out unified policies.

All that said, I think the LC itself nicely demonstrates quite a different objection to standard dialetheist solutions to Curry. Pre-philosophically, "this sentence is false" and "if this sentence is true, it's false" look just the same. They seem confusing in the same way, they seem to be saying almost exactly the same thing, and the same steps--the assumption that they must be either true or false, the realization that either answer generates the other and so on--seem to be the same, even if, once one sits down with the kind of formal logic one learns in an introductory class and works out the derivations of the contradictions, one proof is a bit longer than the other.

Given all of this, one can imagine someone in this position--no familiarity with the higher-level literature on paradoxes, but quick on their feet and equipped with good enough proficiency in (orthodox) formal logic to follow the moves--having the following conversation with a dialetheist paradox-solver.

(We'll call the pre-philsophical Worrier About Paradox WAP and the dialetheist paradox-solver DPS.)

WAP: "I'm really worried by these paradoxical sentences."

DPS: "Which ones?"

WAP: "This sentence is false" and "if this sentence is true, it's false."

DPS: "Well, for that second sentence you mentioned, we can actually solve the paradox by weakening the inferential power of our conditionals. See, there's this rule called contraction..."

DPS goes on to explain about contraction and "suitable conditionals" and the rest.

WAP: "Cool! So which rule about conditionals do I have to reject to get around the first sentence?"

DPS: "Oh, you can't solve that one by doing anything with the conditional. That one actually forces us to acknowledge that some sentences are both true and false."

WAP: "What?!? Really?"

DPS: "Yep."

WAP: "Hold on. What about this rule that we used to get a contradiction out of that first sentence, the one that says that from two conditionals with the same consequent and the disjunction of their antecedents, you can derive the consequent? If conditionals aren't truth-functional, like you explained to me when we were going over that second sentence, and we're messing around with what we can infer from them to get around these paradoxes anyway, why don't we reject this rule, too, just like we rejected that contraction rule? That way, we could still say that if first sentence is true, it's false, and we could still say that if it's false, it's true, and we could still say that the first sentence was just false, just like we got to say that the second sentence was just false when we got rid of contraction. And, just like you explained to me when we talked about contraction, we still get to keep the most basic rules about conditionals, like Modus Ponens and identity. Wouldn't my way be better than just accepting the contradiction?"

DPS: "No, that would be ad hoc."

WAP: "Huh?"

*Note too that standard explanations of the conditionals crucial for the derivation of a contradiction from the Liar Paradox look suspiciously like instances of conditionalization. After all, if someone asks why they should believe that the Liar is true iff it's false, a standard explanation would start with "well, look, let's say that it's false. Well, that's what it says, so it must be true. Alternately, let's say that it's true..."

Of course, this point doesn't quite show that no one who rejects conditionalization is entitled to argue for dialetheism from the Liar Paradox, since they can get the relevant conditionals straight from the T-Schema. Still, in terms of the intuitive justification, if someone asked why they should accept the relevant T-Schema instance rather than taking the paradox to show that some instances of the T-Schema are wrong, the obvious way to explain it to them would be the one just mentioned.

This still doesn't quite show that conditionalization-rejecters are deprived of the argument for dialetheism from the Liar Paradox, since they could go about justifying their adherence to T-Schema absolutism in other ways, but it does show that, without conditionalization, the argument from the Liar to contradiction loses a good bit of its original intuitive force.

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