A few months ago I wrote a short argument against Stephen Yablo’s definition of a priority as truth that could be recognised in virtue of understanding. Less wordilly, S is a priori iff understanding S is sufficient for recognising S as true. The objection turned on Pierce’s Law:

(1) ((p -> q) -> p) -> p

I said that Yablo couldn’t have all the following things. First, this is a logical truth. Second, all logical truths are a priori. Third, his account of what it is to be a priori. Fourth, that the meaning of the -> is its introduction/elimination rules in a single-conclusion natural-deduction system. He can’t have them all because the rules for -> alone don’t let you prove (1), you also need the rules for ~.

You might think I’m being fussy here in saying that he needs rules for some other connective. But actually that amount of fussiness was completely in the spirit of Yablo’s definition. The definition was designed to get the result that the a priori is not closed under logical consequence, for just the reason that A might be a priori, and If A then B might be a priori in part in virtue of the meanings of the terms in A, while B is not a priori. It might be, that is, that once you understand the terms in A you will realise that B is true, but just understanding the terms in B might not be enough to show you that B is true. The non-closure of the a priori would be VERY BAD NEWS for we small band of two-dimensionalists, so I thought I was making a small blow for freedom with this little argument.

Partial disclosure: Just how much of a two-dimensionalist I am is not entirely clear. I’m at least a fellow-traveller, but saying more would require me feeling less one-thirty-in-the-morningish.

Second partial disclosure: It wasn’t very clear in the original post just what the argument was. It was partially an ad hominem, since I suspected Yablo believed the other three horns of the quadrilemma. And it was partially a plausibility argument against his account of the a priori, because I suspected most readers would think his account of the a priori was the weakest of the four claims. But that was sort of in bad faith, because it isn’t what I believe. Precisely what I believed may become clear below.

Anyway, I just realised going over the argument again tonight was that it seems to cause just as many problems for views I hold as it does for Yablo’s position. (Think of this as the philosophical logical version of blowback.) Roughly, I want to hold all of the following theses.

(2) All logical truths are analytic.
(3) All analytic truths are true in virtue of the meanings of their terms.
(4) If (2) and (3) are true, then if all instances of a particular schema are logical truths, they must all be true in virtue of the meanings of the logical connectives.
(5) The meaning of the -> is given by the introduction/elimination rules for it in a single-conclusion natural-deduction system (i.e. the deduction theorem and modus ponens).

Accepting (2) to (5) means giving up that (1) is a logical truth. That’s OK I think. I actually think (2) through (5) constitute a pretty plausible argument for the claim that (1) is not a logical truth. The problems start arising soon after that.

(6) The meaning of ~ is given by its introduction/elimination rules in a classical single-conclusion natural-deduction system (i.e. reductio ad absurdum and double negation elimination).
(7) If A is true in virtue of the meanings of its logical connectives, then it is a logical truth.
(8) If If A then B is a logical truth, and A is a logical truth, then B is a logical truth.

And here things start to go badly wrong for me. Because (6) and (7) commit me to (9) and (10) being logical truths, so (8) commits me to (1) also being a logical truth.

(9) ~~[((p -> q) -> p) -> p]
(10) [~~[((p -> q) -> p) -> p]] -> [((p -> q) -> p) -> p]

And now I’m stuck in a contradiction. I have to give up one of (2) through (8). (Ever been in a heptalemma before? It’s kinda neat in a screwy way.) There’s little to recommend giving up (4) or (7), which seem fairly unobtrusive steps in the argument. And (3) looks definitional.

I could sort of understand giving up (2). I think it’s the position that most naturally fits with what Yablo says about the a priori. And I’m somewhat tempted by it actually. Say the intuitionists are sort of right about analyticity, so (1) is not analytic. But they are wrong about logical truth. Logical truth is not truth in virtue of logical constants, its being a consequence of something(s) true in virtue of logical constants. So (1) is a logical truth because (9) and (10) are, not because it is true in virtue of its logical constants.

Dummett, of course, recommends giving up (6). Deep down, I suspect that’s the most sane option available. But even if I’m right, in philosophy that is not always the most telling of considerations.

Stephen Read in some recent work has recommended giving up (5). The meaning of -> is given by its in/elim rules in multiple-conclusion natural-deduction systems. In those systems (1) can be deduced from the meaning of -> alone. I’m going to be in print sort of endorsing this position soon, which may not be wise. (It’s a throwaway comment in the Problem of the Many paper. And strictly I say only that this is the best response to certain arguments of Dummett’s, of which (2) to (5) is a loose paraphrase. But it’s probably still too strong.)

Anyway, the bold late night conjecture here is that (8) could be the culprit. Perhaps logical truth is not closed under modus ponens. That would be exciting!

There’s a few take-home lessons from this. I’ll just mention two. First, the argument for intuitionism from (2) to (5) is a little less compelling than it first appears, because accepting it requires giving up (6), (7) or (8). This is probably obvious to everyone who thinks about the matter for more than two seconds, but it was only working through Yablo’s definition of a priority that made me realise it. Secondly, once you start giving up closure for various things, life gets pretty interesting…

What triggered all this off was that Steve Yablo was at Brown today to give a talk on two-dimensionalism. It covered a lot of ground and there’s no way I could do justice to the points made in this format, or at this hour. But when Beyond Rigidification appears on a website near you, I highly recommend checking it out.