## Sunday, June 2, 2013

### Salmon's argument against S4

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:

1. 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).

Brian Cutter said...

This objection looks right on target. Salmon's argument looks suspiciously like the following (bad) sorites argument:

1. If x is bald with n hairs on his head, the x couldn't be bald with n+100,000 hairs on his head.
2. If x is bald with n hairs on his head, the x could have been bald with n+1 hairs on his head.

Given (1) and (2), of course, we have to abandon S4. But the right response here (the analogue to your response to Salmon's argument) is to point out that no one should accept (2) who is not also prepared to accept:

2*. If x could have been bald with n hairs on his head, then x could have been bald with n+1 hairs on his head.

And of course, given the T axiom, 2* contradicts 1.

Anonymous said...

I'm curious: would you reject 1 as well, or not?

Alexander R Pruss said...

Brian: