but I’m pretty sure it’s not true of the logic we use day-to-day.
I’m curious—can you give an example?
Are you thinking of e.g. the set {”there are a finite number of spoons and at least n spoons” for all n}? For any finite subset of those sentences you can imagine a possible world that satisfies them, but there isn’t a possible world that satisfies them all.
Right. In order to satisfy all of those statements about Jim simultaneously, you need a nonstandard model of number theory. And generally we want our statements to be interpreted according to the standard model of number theory (which sort of makes numbers part of the “logic” rather than the thing the logic is operating on).
I was assuming we’ve specified the behaviour of the numbers sufficiently in the background ;) But the difficulty of doing that is part and parcel of the FOL weirdness: you don’t get the Lowenheim Skolem theorem in SOL! I was going to say that you need Compactness to prove it, but now that I think about it I don’t know that you need it, although it’s usually used in the proof of the upwards component.
I’m curious—can you give an example?
Are you thinking of e.g. the set {”there are a finite number of spoons and at least n spoons” for all n}? For any finite subset of those sentences you can imagine a possible world that satisfies them, but there isn’t a possible world that satisfies them all.
That would do. Also: {”Jim has non-zero height and Jim is less than 1/n metres tall”}. It generally screws up talk about numbers in this way.
Right. In order to satisfy all of those statements about Jim simultaneously, you need a nonstandard model of number theory. And generally we want our statements to be interpreted according to the standard model of number theory (which sort of makes numbers part of the “logic” rather than the thing the logic is operating on).
I was assuming we’ve specified the behaviour of the numbers sufficiently in the background ;) But the difficulty of doing that is part and parcel of the FOL weirdness: you don’t get the Lowenheim Skolem theorem in SOL! I was going to say that you need Compactness to prove it, but now that I think about it I don’t know that you need it, although it’s usually used in the proof of the upwards component.