(a) violates the very-weak-Brouwer (VWB) axiom, or
(b) is not strong enough to make arithmetical facts be necessary, or
(c) makes every proposition necessarily true.
And of course a modality like that is clearly not what we mean by "necessity/possibility" or even "logical necessity/possibility". This post is an expansion of the brief argument here.
Now it's time to argue for my claim.
By VWB, I mean the following thesis:
- At least one proposition is necessarily possible.
According to the Brouwer axiom (and, a fortiori, given S5), every true proposition is necessarily possible. But VWB is much weaker than that. (In fact, it follows from Axioms T and Necessitation: given T, for any tautology p, Possibly(p) is a theorem; but by Necessitation, Necessarily(Possibly(p)) follows.) It is about as uncontroversial a modal axiom as one can get.
Now to prove my claim.
A proposition p is narrowly logically possible provided that a contradiction cannot be proved from p while p is narrowly logically necessary provided that p can be proved (within the logical system that defines narrowly logical modality).
Now suppose VWB is true for our narrow logical necessity. Then it is necessarily true that p is possible for some p. I.e., it can be proved that no contradiction can be proved from p. But if no contradiction can be proved from p, then our logical system is consistent: in an inconsistent classical system, a contradiction can be proved from every claim. And this conditional can be proved.
Hence, given VWB, it follows that one can prove in the system that the system in question is consistent. It follows by Goedel's Second Incompleteness Theorem that either (i) the system is inconsistent or (ii) the system is not strong enough to contain arithmetic. In case (i), every proposition can be proved (this is classical logic, so we have explosion) and hence we have (c) and in case (ii) we have (b). So if VWB is true, we have (b) or (c). Thus, in general, we have either (a) or (b) or (c).