Monday, May 31, 2010

Another Revenge Paradox For Beall

In Spandrels of Truth, JC Beall abandons his previous belief in the existence of truth-value gaps, now declaring that "negation is exhaustive."

Intuitively, this means that there are now three possible statuses that statements can have on the new version of his account (unlike the old version, where there were four):

(1) (Just) true
(2) (Just) false
(3) Both true and false

In Spandrels..., Beall treads carefully around the "just" terminology, since he recognizes that for any such formulation you like, you can always devise a Liar sentence for it, e.g.

"This sentence is just false."

He does, however, sometimes express the distinction between (1)/(2) on the one hand and (3) on the other by talking about "treating" statements "classically." For example, in section 2.3.2, in his discussion of the status of non-paradoxical sentences that (like Liars) ineliminably refer to their own truth-or-falsehood status, like....

"This sentence is true."

....he says that he is open to "an asymmetric treatment of such sentences (e.g. treating gluts, some classically." (p. 15)

In section 5.4, to express his view that true contradictions only arise as by-products of the introduction of notions like "truth" into our language, and that the portion of the language that's free of such terms is also free of true contradictions, he says "our base classical." (p. 126)

And so on.

An important note about all this:

Given his rejection of gaps, it looks like "glutty" and "treated classically" are jointly exhaustive of the conceptual options. To really harp on the point, we can put this precisely as the Joint Exhaustion Principle (JEP).

JEP: Every statement is either a glut or classical.

Given Beall's account, I don't see how he could reject the JEP. He's a tpaints to explicitly reject the possibility of any sort of "paracomplete" break-downs of Excluded Middle, he goes out of his way to commit himself to a classical account of vagueness problems, etc. If someone who knows Beall's work can come up with a third option that he has room to adopt, I'd love to hear it in the comments, but right now, it sure doesn't look like he has the resources to deny the JEP.

So, given that all non-glutty statements are to be treated classically, what does that mean?

Presumably "is classical", "treated classically", etc., means at the very least that reasoning about it according to all the rules of classical logic (including those which dialetheists take to be invalid in inconsistent contexts) is appropriate when it comes to such statements.

(Note that, since Beall and I agree about the equivalence of "P is false" and "~P," I'll use them interchangeably in what follows.)

This language, however, seems to be just as prone to revenge paradoxes as the avoided language of "just false," etc. After all, what can he do with the following sentence?

# The (whole) sentence marked with the number sign is false and should be treated classically.

Like any conjunction, we have four options here:

(1) Both conjuncts are true, so the whole thing is true.
(3) The first conjunct is true, but the second conjunct is false, so the whole thing is false.
(3) The second conjunct is true, but the first conjunct is false, so the whole thing is false.
(4) Both conjuncts are false, so the whole thing is false.

(1) would entail triviality in a straightforward way. Given the assumed equivalence of "P is false" with "~P," noted above, a statement that's (a) both true and false, and (b) treated classically, is a statement from which triviality can be derived.

(2) would mean that # was not classical, which, given the JEP, would entail that # is a glut, and hence entail triviality.

(3) would entail triviality just like (1) would. If the first conjunct is false, it's also true. If the second conjunct is true, triviality is entailed by the contradiction.

(4) entails triviality in the same way. If the first conjunct is false, it's also true. If the second conjunct is false, the whole thing is (by the JEP) a glut, which makes the second conjunct true, which in turn means that we can derive triviality from the contradiction.

So....unless anyone can see a way out of the JEP on Beall's account...this looks like a pretty serious problem.

Of course, there's at least one way out:

Just as dialetheists (including Beall) standardly weaken the inferential power of their conditionals to get around Curry, Beall could weaken his even more in order to get around the Sentence # Paradox. He could claim that "P or Q", "if P, then R" and "if Q, then R" could all be true without R being true. If he made this move, however, then he would have no remaining motivation for believing in true contradictions. After all, he doesn't believe in "base language gluts," and given the sort of conditional-weakening proposal under consideration, no contradictions would follow from standard versions of any semantic paradox.

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