Friday, May 11, 2007

The moving parts of modal logic

Here's another post on something that Karen Bennett said, which will hopefully be more substantive than the last one. In one of the discussions, Bennett said that Lewis's semantics for modal logic runs into trouble when talking about the "moving parts of his system". This means, roughly, when it starts talking about matters metalinguistic, like the domain of worlds, truth, domains of individuals, etc. Her particular example was: necessarily there are many worlds. Apparently Lewis's semantics mess this up. I don't know the details since I don't know what sort of language Lewis was using and the particular semantics. That's not really the point. It got me wondering how prevalent this sort of thing is. When do semantics go astray for things that are naturally situated in the metalangauge? The truth predicate can cause some problems. Does the reference predicate? I'm not really sure I've read anything about logics that include an object language reference predicate.

The idea dovetailed nicely with some stuff I was working on for Nuel Belnap's modal logic class. Aldo Bressan's quantified modal logic includes a construction of terms that represent worlds. This construction works at least as long as the number of worlds is countable. I'm not sure what happens when there are uncountably many worlds since that requires me to get much clearer on a few other things. This allows the construction of nominal-like things and quantification over them. The language lets you say that for any arbitrary number of worlds, necessarily there are at least that many worlds. The neat point is that the semantics works out correctly. [The following might need to be changed because I'm worried that I glossed over something important in the handwaving. I think the basic point is still correct.] For example, (finessing and handwaving some details for the post) the sentence "necessarily there are at least 39 worlds" will be true iff for all worlds, the sentence "there are at least 39 worlds" is true, which will be just in case there are at least 39 worlds. This is because you can prove that there is a one-one correspondence between worlds and the terms that represent worlds. Bressan uses S5. The way "worlds" is defined uses a diamond, so the specific world at which we are evaluating does not really matter. So, the semantics gets it right. Of course, this doesn't say that there are only models with at least 39 worlds. If that is what we want to say, we can't. It does say that within a model, it's true at all worlds that there are at least 39 worlds, which is all the object language necessity says. This gives us a way to talk about a bit of the metalanguage in the object language, albeit in a slightly restricted way.

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