In thinking about an idea that I had earlier about just what entities we might be committed to, I was rereading Quine’s essay “On What There Is”. On going through it again though, I realize that he seems to endorse another idea I once had about a form of structuralism more extreme than the standard one. He seems to shy away from saying that we’re committed to the existence of certain entities, instead saying that we’re committed to the existence of entities satisfying certain properties. Thus, he seems to be advocating a much more linguistically-driven picture of the world than one might have supposed otherwise. Although this paper is often considered the one that restored metaphysics to philosophy, he seems to restore it only in an extremely deflated way. He suggests that we use science and philosophy (and the rest of our means of explaining the world) to come up with the best theory of the world, and that we merely endorse that theory. All that it means for us to say something exists is for an existential claim of appropriate form to appear in that theory. Thus, we aren’t in a sense talking about the thing itself (because any object could play the same role in an appropriate model of the theory), and we can’t even say anything about how many objects of a certain type there are unless there are finitely many. So in a sense it can’t make sense on this picture to say there are countably many natural numbers and uncountably many reals. We can just say there is no function pairing them up.
Also, for this same project I’m working on (about analyzing whether a Quinean view like this can even say what sorts of things there are) I’d like to use a many-sorted logic. I basically know how this sort of logic works (and can refer to my notes from my model theory class in case i forget), but I’d like to have some sort of reference I can cite. Does anyone know of a logic book that talks about many sorted logic as well as traditional one-sorted first-order logic?