Monday, April 6, 2015

More against neo-conventionalism about necessity

Assume the background here. So, there is a privileged set N of true sentences from some language L, and N includes, among other things, all mathematical truths. There is also a provability-closure operator C on sets of L-sentences. And, according to our neo-conventionalist, a sentence p of L is necessarily true just in case pC(N).

Moreover, this is supposed to be an account of necessity. Thus, N cannot contain sentences with necessity operators and C must have the property that applying C to a set of sentences without necessity operators does not yield any sentence of the form Lp, where L is the necessity operator (It may be OK to yield tautologies like "Lp or ~Lp" or conjunctions of tautologies like that with sentences in the input set, etc.) If these conditions are not met, then we have an account of necessity that presupposes a prior understanding of necessity.

Now consider an objection. Then not only is L(1=1) true, but it is necessarily true. But now we have a problem. For C(N) by the conditions in the previous paragraph contains no Lp sentences. Hence it doesn't contain the sentence "L(1=1)".

But this was far too quick. For the neo-conventionalist can say that "L(1=1)" is short for something like "'1=1'∈C(N)". And the constraints on absence of necessity operators is compatible with the sentence "'1=1'∈C(N)" itself being a member of C(N).

This means that the language L must contain a name for N, say "N", or some more complex rigidly designating term for it (say a term expressing the union of some sets). Let's suppose that "N" is in L, then. Now, sentences are mathematical objects—finite sequences of symbols in some alphabet. (Or at least that seems the best way to model them for formal purposes.) We can then show (cf. this) that there is a mathematically definable predicate D such that D(y) holds if and only if y is the following sentence:

  • "For all x, if D(x), then ~(xN)."
But if y is this sentence, then y is a mathematical claim. If this mathematical claim isn't true, then y is a member of N. But then y is true. On the other hand, if y is true, then being a mathematical claim it is a member of N, and hence y is false. (This is, of course, structurally like the Liar. But it is legitimate to deploy a version of the Liar against a formal theory whose assumptions enable that deployment. That's what Goedel's incompleteness theorems do.)

To recap. We have an initial difficulty with neo-conventionalism in that no sentences with a necessity operator ends up necessary. That difficulty can be overcome by replacing sentences with a necessity operator with their neo-conventionalist analyses. But doing that gets us into contradiction.

(It's perhaps formally a bit nicer to formulate the above in terms of Goedel numbers. Then we replace Lp with nC*(N*) where n is the Goedel number of p, and C* and N* are the Goedel-number analogues of C and N. Diagonalization then yields a contradiction.)

One place where I imagine pushback is my assumption that C doesn't generate Lp sentences. One might think that C embodies the rule of necessitation, and hence in particular it yields Lp for any theorem p. But I think necessitation presupposes necessity, and so it is illegitimate to use rules that include necessitation to definite necessity. However, this is a part of the argument that I am not deeply confident of.

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