Thursday, March 15, 2018

Logical closure accounts of necessity

A family of views of necessity (e.g., Peacocke, Sider, Swinburne, and maybe Chalmers) identifies a family F of special true statements that get counted as necessary—say, statements giving the facts about the constitution of natural kinds, the axioms of mathematics, etc.—and then says that a statement is necessary if and only if it can be proved from F. Call these “logical closure accounts of necessity”. There are two importantly different variants: on one “F” is a definite description of the family and on the other “F” is a name for the family.

Here is a problem. Consider:

  1. Statement (1) cannot be proved from F.

If you are worried about the explicit self-reference in (1), I should be able to get rid of it by a technique similar to the diagonal lemma in Goedel’s incompleteness theorem. Now, either (1) is true or it’s false. If it’s false, then it can be proved from F. Since F is a family of truths, it follows that a falsehood can be proved from truths, and that would be the end of the world. So it’s true. Thus it cannot be proved from F. But if it cannot be proved from F, then it is contingently true.

Thus (1) is true but there is a possible world w where (1) is false. In that world, (1) can be proved from F, and hence in that world (1) is necessary. Hence, (1) is false but possibly necessary, in violation of the Brouwer Axiom of modal logic (and hence of S5). Thus:

  1. Logical closure accounts of necessity require the denial of the Brouwer Axiom and S5.

But things get even worse for logical closure accounts. For an account of necessity had better itself not be a contingent truth. Thus, a logical closure account of necessity if true in the actual world will also be true in w. Now in w run the earlier argument showing that (1) is true. Thus, (1) is true in w. But (1) was false in w. Contradiction! So:

  1. Logical closure accounts of necessity can at best be contingently true.

Objection: This is basically the Liar Paradox.

Response: This is indeed my main worry about the argument. I am hoping, however, that it is more like Goedel’s Incompleteness Theorems than like the Liar Paradox.

Here's how I think the hope can be satisfied. The Liar Paradox and its relatives arise from unbounded application of semantic predicates like “is (not) true”. By “unbounded”, I mean that one is free to apply the semantic predicates to any sentence one wishes. Now, if F is a name for a family of statements, then it seems that (1) (or its definite description variant akin to that produced by the diagonal lemma) has no semantic vocabulary in it at all. If F is a description of a family of statements, there might be some semantic predicates there. For instance, it could be that F is explicitly said to include “all true mathematical claims” (Chalmers will do that). But then it seems that the semantic predicates are bounded—they need only be applied in the special kinds of cases that come up within F. It is a central feature of logical closure accounts of necessity that the statements in F be a limited class of statements.

Well, not quite. There is still a possible hitch. It may be that there is semantic vocabulary built into “proved”. Perhaps there are rules of proof that involve semantic vocabulary, such as Tarski’s T-schema, and perhaps these rules involve unbounded application of a semantic predicate. But if so, then the notion of “proof” involved in the account is a pretty problematic one and liable to license Liar Paradoxes.

One might also worry that my argument that (1) is true explicitly used semantic vocabulary. Yes: but that argument is in the metalanguage.


Martin Cooke said...

If you are right about this being a problem, then there is an obvious solution.
The threat is to necessary truths being all and only those things provable from F,
so why not have the necessary truths being anything provable from F, or (1), and so on, and only such things? The difference is minimal (my "and so on" is for "(2) Statement 2 cannot be proved from F and (1)" and so on), but necessary, if you are right.
(Given that this is essentially Liar territory, I do not think that you are.)

Alexander R Pruss said...

The strategy iterates. Let F' be the union of F with (1), (2), etc. Then form (1') which is just like (1) but with F' in place of F. No matter how much you add to F, the problem iterates.

This point is analogous to the observation that you can't get rid of incompleteness just by adding the Goedel sentences to the axiom set.

Martin Cooke said...

Oh yes, I see.
Still, Liar-style sentences prove (in ordinary logic) that "is true" can be a vague predicate in just such abnormal circumstances as (1), so that there is really no reason why (1) should be either true or else false.
(I don't know about Godel, but I'd guess that that was more formal?

Martin Cooke said...

I like your argument, it's got more to it than Patrick Grim's 2000 argument against omniscience, for example; but, if there was a being that knew everything, all about His creation, all about your future, and mine, and so forth ... except that it did not, because it could not, know Grim's (4):
God doesn't believe that Grim's (4) is true
then surely that being would, really, be omniscient. I mean, technically He wouldn't be, but really, if He knew everything else, except Grim's (4) and maybe infinitely many related things? And I feel the same way about your argument.

Logical closure accounts of necessity cannot be false because of your (1).
Your argument is really all about the Liar paradox.

The assumption that propositions, assertions, must be either true or else not true is plausible, of course, because to want the truth is to want things to be made clear. It is to want vagueness to be eliminated. But, the Liar paradox arises with sentences that are not normal sentences, and so the question is whether there are exceptional circumstances in which it would be highly implausible were ‘is true’ not behaving like a vague predicate; and indeed there are.

Suppose that @ is originally an apple but has its molecules replaced, one by one, with molecules of beetroot. Suppose that the question ‘What is @?’ is asked, after each replacement, and that the answer ‘It is an apple’ is always given. Originally it is a correct answer, because originally it is true that @ is an apple, but eventually it is not. If the proposition that @ is an apple must be true or else not, then an apple can be turned into something else (presumably a mixture of apple and beetroot) by replacing just one of its original molecules with a molecule of beetroot. And that is highly implausible. It is surely possible, since much more plausible, that @ would be, at such a stage, no less an apple than apple/beetroot mix, that it would be as much an apple as not, so that the proposition that @ is an apple would be as true as not.

Given that possibility, the following is a proof by reductio ad absurdum that ‘is true’ is indeed a vague predicate. I call the following assertion ‘L’:
The assertion that you are currently considering, this assertion, is not true.
L is clearly an assertion; it is the assertion that L is not true. So if L is true, then L is not true, but L cannot be true and not true, so L cannot be true. But, if L is not true, then L is true, and L must be either true or else not true. Contradiction; but, were ‘is true’ vague, L would be true insofar as L was not true. It would follow only that L was as true as not. No contradiction.

Martin Cooke said...

Without that argument, I would be tempted to conclude that Grim's (4) was nonsense from the assumption that God was omniscient. That would deal with my "technically" above.

There are other reasons to think of (4) as nonsense, of course (such as would defeat my argument above, via L being nonsense), all of which indicates that your (1) is also nonsense. I imagine that a fan of logical closure accounts would feel that way.

Perhaps you would agree that in the case of omniscience it would be reasonable? There is not the same almighty authority behind the closure accounts, but I wonder if you can use a kind of argument there that you would find it reasonable to reject in such a blunt way elsewhere.

William said...

"Thus (1) is true but there is a possible world w where (1) is false."

(1) is contingent on the post having been made. So the opposite of "(1) is true" could also be

"(1) is true (when asserted) but there is a possible world where (1) does not exist."

Alexander R Pruss said...


That's a good point, but it depends on what and how "(1)" designates. If it rigidly designates the sentence type or the proposition, we don't have the contingency problem.

Anyway, once (1) is rewritten syntactically, as in the diagonal lemma, the contingency problem disappears completely.