An argument against neo-conventionalism in modality

The neo-conventionalist account of necessity holds that necessity is just a messy property accidentally created by our conventions. We historically happened to distinguish a certain family N of true sentences. For instance, N might include the mathematical truths, the truths about the identities of natural kinds (e.g., "water = H₂O"), the truths about the scope of composition, etc. Then we said that a sentence is necessarily true if and only if it is a member of the closure C(N) of N under some logical deduction rules. (Alternately, one might do this in terms of propositions.)

Here is a criterion of adequacy for a theory of modality. That theory must yield the following obvious, uncontroversial and innocuous-looking fact:

1. Necessarily, some sentence is not necessary.

Some things just have to be possible. (Note: In System T, if p is any tautology, then necessarily ~p is not necessary.)

A neo-conventionalist proposal consists of a family N of true sentences and a closure operator C. For any neo-conventionalist proposal, we then can raise the question whether it satisfies condition (1). Formulating this condition precisely within neo-conventionalism takes a bit of work, but basically it'll say something like this:

2. "Some sentence is not a member of C(N)" is a member of C(N).

There is a more intuitive way of thinking about the above condition. A family A of sentences is such that C(A) is all sentences if and only if the family A is C-inconsistent, i.e., inconsistent with respect to the rules defining C. (This is actually a fairly normal way to define inconsistency in a wide range of logics.) So (2) basically says:

3. "N is C-consistent" is C-provable from N.

Put that way, we see that our innocuously weak assumption (1) is actually a pretty strong condition on a neo-conventionalist proposal. It is certainly not guaranteed to be satisfied. For instance, a neo-conventionalist proposal where N is a finite set of axioms and C is a formal system (with the axioms and formal system sufficient for the operations in C) will fail to satisfy (3) by Goedel's Second Incompleteness Theorem.

This last observation shows that the question of whether a neo-conventionalist proposal satisfies (3) can be far from trivial. Now, in practice nobody espouses a neo-conventionalist proposal with a finite set of axioms. All the proposals in the literature that I've seen just throw all mathematical truths in, so Goedel's Second Incompleteness Theorem is not applicable.

But even if it's not applicable, it shows that the question is far from trivial. And that is unsatisfactory. For (1) is obviously true. Yet on a neo-conventionalist proposal it becomes a very difficult question. That by itself is a reason to be suspicious of neo-conventionalism. In fact, we might say: We know (1) to be true; but if neo-conventionalism is true, we do not know (1) to be true; hence, neo-conventionalism is not true.

Now, one can probably craft neo-conventionalist proposals that satisfy our constraint. For instance, if N is just the set of mathematical truths (considered broadly enough to include truths about what sentences are C-provable from what) then "N is C-consistent" will be true, and hence a member of N, and hence C-provable from N. But of course that's just another proposal that nobody endorses: there are more necessities than the mathematical ones.

And here's the nub. The neo-conventionalist isn't just trying to craft some proposal or other that satisfies (1). She is proposing to let N be those truths that we have conventionally distinguished (she may not be making an analogous move about C; she could let C be closure under provability in the One True Logic). But we did not historically craft our choice of distinguished truths so as to ensure (3). Consider the following curious definition of an even number:

4. A number is even if and only if it has the same parity as the number of words in my previous blog post.

This account might in fact get things right—if we are lucky enough that the number of words in my previous post is divisible in two. But I did not choose my wording in that post with that divisibility in mind. I chose the wording for completely different reasons. We don't have reason to think, without actually counting, that (4) is correct. And even if it is correct, it is only my luck that I happened to choose an even number of words, and we don't want a theory to rest on luck like that.