Wednesday, October 13, 2010

A Simplified Version of My Revenge Paradox for Paracomplete Solutions to the Liar Paradox

I've played with versions of this sort of thing here a couple of times before, but right now this new version of the sentence seems best to me (in terms of denying the paracomplete theorist escape routes that may have been afforded by earlier, sloppier versions):

Sentence S: An omniscient and ideally rational being would not accept this sentence.


If you take S to be true, you're committed to the uncomfortable claim that an ideally rational being would reject a sentence that they knew to be true. If you take S to be false, then you're faced with an unattractive choice between being committed to saying that (a) it's also true, or that (b) sometimes an ideally rational being would accept a sentence they knew to be false. If you take S to be one of those sentences about which a good paracompletist recommends rejecting the relevant instance of the Law of the Excluded Middle, then you are committed to saying that some such sentences are also true, which would seem to defeat the whole point of the paracomplete manuever of switching from making claims about truth-values to making claims about acceptance and rejection.

Moreover, S also seems to pose a big problem for dialetheists who lean on acceptance/rejection talk to get around their difficulties with the notion of "just false." (The problem, of course, being that for whatever formulation you use to differentiate false statements that are also true from other false statements, you can always construct a sentence that says of itself that it is in the latter category--e.g. "this sentence is just false and not true.") At one time, Graham Priest was in this category--in the first edition of In Contradiction, he claims that dialetheists, despite their rejection of Disjunctive Syllogism, are still entitled to a Disjunctive Syllogism-like rule of reasoning expressed in acceptance-and-rejection talk rather than negation-talk. In the second edition, he renounces this claim, and in Doubt Truth To Be A Liar he has an extended discussion where he says that good dialetheists should accept that the ground for rational acceptance might sometimes overlap with the ground for rational rejection. JC Beall, on the other hand, still seems to think that he can avoid this conclusion. In Spandrels of Truth, he uses acceptance and rejection talk almost as much as Field does in Saving Truth From Paradox and for many of the same sorts of purposes, for example saying that he thinks that Curry sentences should all be rejected. (Basically, it looks to me paracompletists want to use "accept", "reject" and "neither accept nor reject" pretty much exactly the way truth-value-gap theorists use "true," false," and "neither true nor false," but without running into the revenge paradoxes, and that dialetheists like Beall or the previous version of Priest want to use "accept" and "reject" as stand-ins for the classical behavior of "true" and "false." In either case, the psychological concepts are substituted for the semantic ones, so that one can have one's cake and eat it too in terms of one's preferred non-classical solution to the paradoxes.) These purposes, it seems to me, are pretty well foiled if it turns out that (given his committment to thinking that paradoxical sentences are meaningful, have their apparent truth conditions and so on) he has no choice but to admit that either (a) some sentences both should and shouldn't be accepted, or (b) some true sentences shouldn't be accepted, or (c) some sentences should be accepted despite failing to be true.

1 comment:

Glowing Face Man said...

These posts are very interesting. It gives me the idea that people in philosophy departments (the various -ists you mentioned, not you yourself) are very confused.

I think 99% of the confusion around liar's paradox seems to come from "true" having two different senses, which are confused endlessly.

Given a language with an unary predicate T (and optionally an unary predicate F), and a model of that language, one says that a formula phi is "true" if the Godel number of phi is an element of the model's interpretation of T. One says that phi "holds in the model", or that phi is "satisfied", or whatever, if (insert inductive definition of a model satisfying a formula, here). All the confusion and "paradoxes" seem to arise from confusing these notions, as if they were equivalent.

(In "mainstream" logic there is no danger of using "true" to mean "holds in the model", it's only when the language has that added truth predicate that trouble arises)

Now as for the interesting paradox you introduced in this post... there are three ways I can think of to formalize it:

1. Introduce predicates about which (Godel numbers of) formulas the omniscient being accepts. Say, A for accept, R for reject. Omniscience might be interpreted axiomatically as "phi iff A(phi)" and "~phi iff R(phi)" [I abuse notation and write phi for the Godel number of phi inside predicates]. Sadly, these axioms are inconsistent (e.g., by the plain vanilla liar's paradox), no model exists, and everything follows vacuously.

2. Instead, introduce modal operators A and R. Then liar's paradoxes don't arise because of the lack of free variables needed to invoke Godel. But then, neither can you formalize the revenge paradox.

Or 3. Use 2nd order logic, and formalize it as something like "for every set X, if X contains precisely the Godel numbers of formulas which hold, then X doesn't contain the Godel number of this sentence". Unfortunately I don't know enough about 2nd order logic to know whether you can actually use fixed point lemmas to pull this off...