Saturday, June 19, 2010

Getting to S4

This is an outline with proofs omitted. Start with a notion of necessity L that satisfies the constraints of System T and take as an axiom the Natural Numbers Barcan Formula (NNBF):

  1. L(n)(Fn) iff (n)LFn,
where the quantification is over natural numbers only. I think NNBF is pretty plausible. It certainly avoids all of the implausibilities of the standard Barcan Formula. Basically, it just says that every world has the same natural numbers.

Now, we can bootstrap our way up to a logic satisfying S4 from the logic that uses L. Let Lnp be L...Lp with n Ls. Let L*p be (n)Lnp. I think the following is true, though it may take some work to prove it and will need for every p a predicate F such that Fn iff Lnp: if L satisfies System T and NNBF is an axiom, then L* satisfies S4. Moreover, intuitively, L* has every bit as much, and maybe more, right to be called "metaphysical necessity" as L does. So, given a modal logic that satisfies T and NNBF, both of which are pretty plausible, we can define a metaphysical necessity operator that satisfies S4. I think this makes it plausible that ordinary metaphsyical necessity satisfies S4.

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