## Saturday, May 16, 2015

### A quick route from mathematics to metaphysical necessity

The Peano Axioms are consistent. If not, mathematics (and the science resting on it) is overthrown. Moreover, it is absurd to suppose that they are merely contingently consistent: that in some other possible world a contradiction follows logically from them, but in the actual world no contradiction follows from them. So the Peano Axioms are necessarily consistent. But they aren't logically necessarily consistent: the consistency of the Peano Axioms cannot be proved (according to Goedel's second incompleteness theorem, not even if one helps oneself to the Peano Axioms in the proof, at least assuming they really are consistent). So we must suppose a necessity that isn't logical necessity, but is nonetheless very, very strong. We call it metaphysical necessity.

Mike Almeida said...

van Inwagen offers some useful advice here. Logical possibility is not a species of possibility (similarly for necessity, of course). It is not as though there are propositions that are logically possible but not metaphysically so. If the proposition is not metaphysically possible, it's not possible simpliciter. There are no worlds in which it is true, so it surely isn't logically possible either: there's no larger set of worlds in some of which it is true. It's not any-wise possible. Healthful advice, I think, http://www.andrewmbailey.com/pvi/Modal_Epistemology.pdf.

Alexander R Pruss said...

I am inclined to agree that logical possibility isn't a species of possibility. It's more like "conjunction-elimination possibility", where a proposition is conjunction-elimination possible iff one cannot derive a contradiction from it by applying conjunction-elimination. I.e., logical possibility is possibility with respect to a subset of the real rules.

Mike Almeida said...

logical possibility is possibility with respect to a subset of the real rules.

That makes it sound like p is logically possible only if it is consistent with the real rules. But that's false. Lot's of 'logically possible' propositions are not consistent with any set of rules.

Michael Gonzalez said...

If their non-contingency is based on no contradiction following from them in any world, as you say, then that just IS logical necessity of consistency, no? To prove X is usually just to show a contradictory entailment in denying X, right?

Mike Almeida said...

Let L be a logically possible, but metaphysically impossible proposition. Suppose we understand L is logically possible iff. we cannot derive a contradiction from L in S. But then L is either a theorem or contingent. Either way, L is consistent with the rules of S. There has to be some model on which L comes out true in some world. But it is false that L is consistent with the rules of S, since L is metaphysically impossible. So, it can't be right that L is logically possible iff. we cannot derive a contradiction from L in S.

JesseM said...

For something to be logically necessary, must it be provable in a computable way, using a finite number of steps? Or can we expand the notion of logical proof to include things that could be proved by hypothetical machines that could perform an infinite number of steps and draw conclusions about the results (a form of hypercomputation), for example verifying the Goldbach conjecture by showing explicitly that for each integer N, there exist two primes that add to N?