## Monday, March 24, 2008

Sorry about the lag between posts.....

Meanwhile, here's something that I've been thinking about. Here's a simple form of Curry's Paradox:

"If this sentence is true, then β."

Plugging it into the T-Schema, we get the result that that sentence is true if and only if, if it's true, then B, or formally:

Tr<α> ↔ (Tr<α>→β)

The logical principle of absorption (or contraction) says that any time we have something of the form α→(α→β), we can infer straight from there to α→β. Or, if you want to do the same thing more slowly, you can just do a conditional proof--all you'll need is Modus Ponens a couple of times, and a logic that let's you use the same premise more than once--and, one way or the other, you get the result Tr<α>→β. But from this and the right-to-left version of the biconditional above, we can infer Tr<α>. Here, of course, we now have in our possession Tr<α> and Tr<α>→β, so we can just plug in one last instance of Modus Ponens and get β, for any and every arbitrary β. Explosion without even having to get a contradiction on the way.

Just as the ordinary semantic paradoxes, like the Liar, are prima facie sound arguments for dialetheism (the position that some, but not necessarily all, contradictions are true) from intuitively plausible premises via intuitively reasonable steps, so that to show that they are unsound you have an uphill battle to explain why the premises are wrong or what's wrong with the reasoning, Curry represents a prima facie sound argument for trivialism (the position that everythign is true.) Non-trivialist dialetheists will want to avoid this at all costs--the whole project of carving out a plausible-sounding version of dialetheism is to show how some but not all contradictions can be true--and, from my point of view, the interesting thing is that their options here aren't that different from the options confronting a defender of the Law of Non-Contradiction when defusing an ordinary semantic paradox like the Liar. It seems to me that there are three ways a dialetheist (or any one else, of course) could deal with the Curry paradox:

(1) They could deny that the original sentence was a truth-bearer, e.g. on Kripke's grounds that sentences ascribing truth to other sentences are meaningful if and only if the series eventually grounds out in a sentence that's actually about external extra-semantic reality in some way, or of course on whatever other grounds.

(2) They could institute some sort of formal rules a la Tarski to ban the expression of the sentence in the first place.

(3) They could deny that the T-Schema holds universally, and make an exception for Curry.

(4) They could tinker with the logical rules that get us from Tr<α> ↔ (Tr<α>→β) to β.

Although I think Graham Priest actually goes with (4) in In Contradiction and elsewhere, I think this is probably the least plausible response. After all, on the face of it, the logical rules in question still look universally truth-preserving whether or not propositions can be simultaneously true and false, so the dialetheist has no special right (given their assumptions) to change them that anyone else does, and I don't see why anyone does. One can simply declare that "my conditional is not the conditional of classical logic, and given that, you can't make the inference from α→(α→β) to α→β with it," but (a) I'm deeply skeptical that this can be explained in any way that blunts its radically counter-intuitive edge, and (b) it looks like this is a "solution" bought via the loss of expressive power, since the → in this logic simply won't capture the notion of "if, then" in even the minimal way that → does in classical logic. Worse yet, if the whole motivation for this artifical restriction of the conditional is the avoidance of Curry problems, then it looks to me like the dialetheist who picks this option is engaging in an ad hoc manuever and begging the question against the trivialist.

The other options, however, look even less promising. Priest has been savage in his criticisms of "ad hoc exceptions to the T-Schema" for the Liar and other semantic paradoxes, and no one has been clearer in explaining why (2) doesn't solve or explain anything but merely represents a decision not to talk about it. That leaves us with (1). I think that this is the most promising option, since it represents a more than purely formal solution, and, if the independent grounding given is good enough, the one that looks least like it's assuming what needs to be proved.

The problem, of course, is that the alleged meaningfulness of ordinary paradox-producing sentences like the Liar would be an almost inevitable casualty of any explanation of why the Curry sentence wasn't meaningful, so the dialetheist who took option (1) would be sacrificing a huge part of the positive case for dialetheism.