Well, I think I’m going to have to admit defeat. I’ve tried reading Fine’s paper for the third time and I’m still stumped by his positive claims about ‘postulational modality’.
The defender of indefinite extensibility thinks that ‘whatever interpretation [of the supposedly absolutely general quantify] our opponent might come up with, it will be possible to come up with an interpretation that extends it’. And supposedly the second modality here, at any rate, is ‘postulational’. Whatever exactly that means. Presumably the thought is that whatever objects you are quantifying over, I can postulate another one — the set-like collection of all the sets you are quantifying over which aren’t members of themselves — which can’t already be in your domain of quantification, on pain of paradox. But how does this differ from there being such a set-like collection? On the one hand, if Fine is to be making a new move here, there better be a difference; on the other hand, it is difficult to understand what the move is without a clear account of the difference — i.e. a treatment of the metaphysics of (some) mathematical entities as postulated entities, which Fine doesn’t give us.
But set aside those worries. Let’s suppose that, while there isn’t any sense in which you can postulate new donkeys into existence (so ‘there are no talking donkeys’ isn’t, so to speak, vulnerable to a legitimate postulated extension of the domain of quantification), you can postulate new sets (or set-like collections). Well, so what? Why can’t the defender of absolute quantification just aver that when he says, e.g. ‘Everything is self-identical’ or ‘Nothing is a talking donkey’ he already means to cover whatever your postulational ingenuity might come up with — and dig his heels in when you insist that you can still find another entity which might comprise all those things at once (so he is vulnerable to the extensibility argument). Rather he takes the argument of Russell’s paradox as showing us that there is no such single entity.
Which is a familiar dialectic of course. So what I’m missing is how talk of ‘postulational modality’ is supposed to move things forward. As I say, I’m stumped — and will be very happy to get comments from anyone whose grip on Fine’s paper is better than mine.