Fara and Williamson against the Counterpart Theory
A few comments on Counterparts and Actuality by Michael Fara and Timothy Williamson (via Brian, of course).
Fara and Williamson argue that if Quantified Modal Logic is enriched by an "actually" operator, then given some further assumptions there is no correct translation scheme from QML to Counterpart Theory. Here, a correct translation scheme is one that translates theorems of QML into theorems of CT and non-theorems of QML into non-theorems of CT. (theorems of which QML? -- good question; read on.).
I'll focus on the examples not containing names, since F&W's treatment of names is so completely different from how Lewis says they should be treated ("CT&QML", pp.32f. in Papers) that it is unclear whether their examples with names work at all. The name-free example that plays the main role in §2 of the paper is:
12)x (ACT Fx
ACT ~Fx).
This, F&W say, is a logical falsehood, but on any plausible translation scheme for "ACT" ("actually"), it gets translated into a satisfiable sentence of CT. More specifically, its CT-translation will presumably be true iff there is a possible object that has no counterpart at our world. So this is one of the "further assumptions" mentioned above: that there is a possible object having no counterpart at our world. It's a reasonable assumption, and it's certainly endorsed by Lewis. It is in fact just the rejection of the Barcan formula, expressed in CT.
Let's accept the assumption, if only for the sake of the argument. So we assume there could be things that don't actually exist. Let P be a predicate that applies only to such things ("Cartesian soul" might be an example). Then what do we want to say about sentences like
i) There could be a P which is actually F,
where F is some ordinary predicate? I would say (i) is false: No possible P is actually F, simply bacause no possible P actually exists. Similarly, I'd say that
ii) There could be a P which is actually not F
is false: just as no possible P is actually married, so no possible P is actually unmarried. But together with the assumption that there could be a P, not-(i) and not-(ii) presumably entail
iii)
which in turn entails (12). That is, on the assumption that there could be non-actual objects, (12) looks reasonable. At any rate, on that assumption it is not at all clear that (12) should be regarded as a logical falsehood, as F&W suppose.
Assume on the other hand we accept the Barcan formula and intuit that there could not be non-actual objects. In this case it is reasonable to regard (12) as a logical falsehood. But in this case there is nothing wrong with the translations of (12) into CT. For the translations only come out true on the assumption that some things have no counterpart at our world, which is just the assumption we currently suppose is false.
So either way, there is no problem.
The situation is more or less the same for the other examples F&W discuss. Consider
24)
Again, F&W say this is a logical falsehood. This time, the problem is that its CT-translation presumably comes out true on the assumption that some thing at another world may have two counterparts at our world. But again, on this assumption it not at all clear that (24) should be accepted as a logical falsehood: If there could be a thing that is two things at another world, there is little reason why contradictory predicates can't be true of it (them?) at that other world. And again, if the assumption is rejected, there remains nothing objectionable about the CT-translation.
I guess it's true that if non-actual possibilia and multiple counterparts are rejected (and thus also (12) and (24)), CT loses much of its attraction. CT is both a translation scheme for modal statements and a theory of modality. The translation scheme still works for other theories of modality (aka QMLs), with the "postulates" of CT replaced by other postulates, e.g. by a postulate ruling out non-actual possibilia. But then the translations might get less useful in their application to philosophical problems. In extreme cases, if the theory is sufficiently weird -- as when one intuits that the Gödel-Löb provability axiom expresses a fundamental truth about metaphysical modality -- the translation probably becomes completely useless.
F&W show that given certain views on the logic of modality, CT can't deliver all the goods it's supposed to deliver. E.g. if one believes that the true logic of modality rules our (24), then CT presumably can't solve the problem of constitution. But we who do not share these views -- e.g. we who believe that there might well be things that don't actually exist --, we can still accept the solutions offered by CT, and we can still count them as advantages of CT. We knew all along that people with strange views on modality won't find our solutions acceptable. Fara and Williamson have told us that they are among those people.
> But again, on this assumption it not at all clear that (24) should be
> accepted as a logical falsehood: If there could be a thing that is two
> things at another world, there is little reason why contradictory
> predicates can't be true of it (them?) at that other world.
A better way to make the point _I think_ you intend here is: if there
could be a single thing that is two things _in this world_, there is
little reason why it can't, in that other world, actually be F and
actually be not-F. (Namely, because one of its counterparts in this
world is F and the other is not F.)
Is that what you intended?
Question: What should it take to actually be not-F? Is it enough to have _a_ counterpart in this world who isn't F? Or should it require you to have _no_ counterparts in this world who are F? Both translations seem defensible. But only the first provides for an object simultaneously being actually F and being actually not-F.