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:

- 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:

- 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:

- 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.