Dear Matthew, (01)
I agree that some concept or mechanism such as possible worlds
is necessary to apply a set-based approach to hypothetical
reasoning or any reasoning about plans for the future. (02)
But as I said, using the term 'extensional' when talking about
possible worlds is highly misleading. In fact, Montague called
his version of logic that talked about sets of possible worlds
and sets of things in them IL -- Intensional Logic. (03)
MW> The good news for me is that having taken on board possible
> worlds, I get a very good return for my ontological commitment. (04)
I agree that the method can work. But I just want to clarify
the point that what makes it work are the *intensional* aspects. (05)
MW> ... if I want to talk about 2 eyed sheep and 4 legged sheep
> and determine if they are necessarily the same, then I need
> only use sets that that go across all possible worlds. (06)
But those sets are purely imaginary. Your operations are
logical manipulations based on the axioms and descriptions
of those possible worlds. You are doing the same kind of
reasoning that Montague and others call 'intensional'. (07)
MW> I can see that these sets are... (08)
The word 'see' in that sentence is metaphor. Nobody can
actually observe an imaginary world or anything in it. (09)
MW> I am able to deal with the situations that Pat and John say
> will cause me trouble without any trouble at all, I am still
> only using set theory with extension as the basis for identity. (010)
Montague built up all his mathematical constructions for possible
worlds and intensional logic out of sets, and he used the same
identity conditions that you do. But he made it clear that IL
was an *intensional logic*. (011)
In summary, I believe that your method of reasoning is basically
sound. But as Pat and Ingvar were pointing out, your terminology
can lead to some confusions. Making a clear distinction between
actual sets and hypothetical or imaginary sets would be useful. (012)
John (013)
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