Saturday, November 25, 2006

When identity isn't identity, part 2

It just occurred to me that in the previous post on identity in intuitionistic logic, I made a bit of an error (one that survived the suggested correction). The problems with transworld identity are only problems in the context of modal logic. At least, I'm not sure of any problems that arise with identity of individuals across possible worlds in Kripke semantics for intuitionistic logic. There are enough structural similarities between the box and the universal quantifier that I imagine some formula could be cooked up. No clue what a version of Chisholm's problem with transworld identity would look like in terms of witnesses for intuitionistic formulas though. This is a roundabout way of saying that the conclusion of the previous post (about the apparent possibility of problems in classical logic) doesn't pan out. Since classical logic is validated on Kripke frames with only one world, there won't be transworld identity cause there is no transworld to speak of. Alas.

Thursday, November 23, 2006

Citing my sources, of sorts

Some people asked about where the inference in the previous post came from. The two most recent instances I've encountered were in conversation with another student and in the comments on another student's blog. I'm not sure where (whether?) I've seen it in an article or book. However, I have asked my roommate, another grad student, who says he has encountered similar ideas before; it sounded like it was in conversation with other people. I will be tickled pink if I find something like it mentioned in print. If that happens I will straight away put up a citation.

Thursday, November 16, 2006

How does one make this inference?

In the last week or two I've encountered something in several places that strikes me as strange. It is the claim that liking or using formalisms commits one to certain views about the mind. To broaden this a little, we could say commits one to views about anything. The idea seems to be that liking/using FOL or other formal tools commits one to viewing the mind as a Turing machine. I'm not sure how this goes, since it strikes me as both wrong and unsupported. My guess is that the ideas goes:
Turing machines are formal tools;
people that like formal tools tend to like Turing machines;
so, this will commit them to certain views on the mind.
This is clearly not a good line of thought. At best, liking Turing machines might bias you towards one line of thought, although there are at least a few logicians who are quite adept at using Turing machine arguments that think the mind is not one. It would be interesting to see if one finds a greater percentage of those that use formal tools often view the mind as a Turing machine than those that don't use formal tools as much. My guess is no. I'd bet that among computer scientists, one finds a great tendency towards viewing the mind in that way, but I'm not sure if that is relevant.

Monday, November 06, 2006

When is identity not identity?

In Kripke model semantics for intuitionistic first-order logic, it turns out that you can't interpret '=' as the identity relation between objects in the domains. You can interpret it as a mapping from domains to domains.
[Edit: as Aidan pointed out, leaving open the possibility of ~(x=y OR ~x=y), what I said previously, doesn't make sense. The following is what I meant to say.]
The reason is that you need to leave open the possibility that for some x and y, at node a x=y is not true (a doesn't force x=y) but at some node b>=a, x=y is true at b (b forces x=y), which means that a doesn't force ~x=y.
If you want decidable equality you don't have to worry about this. If you're dealing with classical logic, you get excluded middle, so I think you can use the identity relation and get all the puzzles about trans-world identity. However, for Kripke models of classical logic, there is only one world, so only one domain. I'm not sure how the puzzles creep in exactly.