Salmon's argument against S4

Start with:

1. If x originates from chunk α of matter and β is a non-overlapping chunk of matter, then x couldn't have originated from β.
2. If x originates from chunk α of matter and α' is a chunk of matter that almost completely overlaps α, then x could have originated from α'.

Iterating (2), and assuming a finite sequence of almost completely overlapping chunks between α and β, we conclude that an object x that originates from α possibly possibly ... possibly (with a finite number of possiblys) originates from β. By S4, we conclude that x could have originated from β, contrary to (1). Nathan Salmon uses this as an argument against S4.

But this is a mistaken line of thought. For (2) is not significantly more plausible than:

3. If x could have originated from chunk α of matter and α' is a chunk of matter that almost completely overlaps α, then x could have originated from α'.

Both (2) and (3) embody the same prima facie plausible small-variation intuition. If one thought (3) was false, one would have little reason to think (2) is true.

But given (3), Salmon's argument can be run without S4—all we need is T (what is actually true is possible). Iterating uses of (3) and modus ponens, we conclude that (1) is false. In other words, we cannot hold both (1) and (3). And since (2) has little plausibility apart from (3), we shouldn't hold both (1) and (2). Thus, Salmon's argument is not an argument against S4, but an argument against the conjunction of (1) and (2). And I say we should reject (2).

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.

S4

Here is an argument for S4. We want metaphysical necessity to be the strongest kind of necessity without arbitrary restrictions. If one responds that conceptual or strictly logical necessity are stronger, the answer is that they are, nonetheless, arbitrarily restricted, being dependent on a particular set of rules of inference and axioms. (The only non-arbitrary way to specify which which axioms are permitted is to say that it is all the fundamental metaphysically necessary propositions that are axioms, and then we presumably get metaphysical necessity.) Now, if L is a necessity operator, then LL is also a necessity operator. If LL is not equivalent to L, then LL is a stronger necessity operator. If LL counts as arbitrarily restricted, then we have reason to think that so does L, since L is even more restricted than LL, and it seems arbitrary to work with L instead of LL or LLL. And if LL doesn't count as arbitrarily restricted, then L is not the strongest non-arbitrarily restricted necessity operator. So if L is metaphysical necessity, L and LL are equivalent.

The dual of this argument is that metaphysical possibility is the most fundamental sort of possibility. But if M is metaphysical possibility, and MM is not equivalent to M, then MM will be a more fundamental possibility. So, if M is metaphysical possibility, M and MM are equivalent.