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Indefinite divisibility

with 2 comments

If you’re interested, I’ve written a short paper on my nominalistic indefinite extensibility arguments. (This is also my way of making good on my offer in the comments to discuss a sort of consistency result for supergunk—it’s in the appendix.)

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Written by Jeff

February 17, 2009 at 8:49 pm

2 Responses

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  1. Hi Jeff,

    This looks very interesting – I’ll have to have a proper read later.

    One thing that I’m puzzled about (which comes up in footnote 14 and the appendix among other places) is why these kinds of consistency proofs are substantive?

    For example, why can’t I apply the same kinds of arguments to show that \exists y \forall x x\not= y or even \forall x Fx \wedge \neg \forall x Fx are consistent? (So, I’d just say they become consistent when I add the appropriate subscripts. E.g. \exists_1 y \forall_2 x x\not= y.) Is there supposed to be some kind of “penumbral” connection between certain occurences of quantifiers which prevents me from doing this?

    Andrew

    February 18, 2009 at 10:37 am

  2. That’s a good question, and I wish I had a better answer. Clearly more has to be said about how logic with “ambiguous quantifiers” is supposed to work: presumably, certain kinds of statements “trigger” expansions of the domain, or something along those lines. (Maybe this can be relegated to a kind of “logical pragmatics”, but we still need a serious story.) I didn’t want to get into the details of that in the paper, since it’s a big problem and I don’t have a whole lot to say about it.

    But I still think my “model” show something substantive, in the way I described at the beginning of the appendix: if you let the plural quantifiers (or range of the schemes) be restricted to sets, then you get something consistent. Whatever story we give about the logic of extensibility, it should handle the core case of set theory with comprehension, and my cases fall out of that—so even though I don’t know what the story is, (or why the resulting logic is strong enough to rule out really bad cases) I know the story renders my cases consistent.

    Jeff

    February 20, 2009 at 8:49 am


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