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This is a ’latest changes’, but for the Café rather than the backroom! Can David C (or someone) fix the link that does not work to Steve Awodey’s paper (It should be http://www.andrew.cmu.edu/user/awodey/preprints/FoS4.phil.pdf).
Thanks for the alert. I have fixed it now.
Thanks. You constructed (most of) those pages on algebraic and geometric models for modal logic, Tim. I wonder what can be said about the relation between them. There’s the idea of converting a Kripke frame into a topology, using the Alexandroff topology.
I had a word with Steve Awodey, who didn’t sound overly convinced by the coalgebraic approach. It’s a pretty big movement though. A lot of people would have to be wrong if it’s not going anywhere important.
My feeling when looking around in that area I got the feeling that what was missing was the intuitionist or constructivist viewpoint, as everything used BAOs. There was also my ’intuition’ that there should be a higher dimensional viewpoint that would really be needed here… depending on the applications. But these comments are a bit off the top of my head so may not be that pertinent. (Again on the off chance that it may be useful, don’t forget Eric Goubault and his brother wrote on the interconnections amongst these sorts of models for S4 logics, and there is the Leeds group that tries to use them to develop a spatial language (in some sense).) I must dash.
I have an inkling that ionads may be good for modelling first-order modal logic. I’m having difficulty, though, picturing the interior operator $Set^X \to Set^X$. Is there a nice simple example to keep in mind?
I am only an amateur in the modal stuff. I will try to look at ionads (and really like the name). I always felt that the treatment in Kracht’s book was one that should be expanded greatly exactly in the sort of directions that you need. I would also repeat my point about the paper listed here from HHA (J. Goubault-Larrecq and É. Goubault. On the Geometry of Intuitionistic S4 Proofs. Homology, Homotopy and Applications 5(2), pages 137-209, 2003, available here) . The use of the Decalage type operator to model the Box in S4 seemed to me to be excellent and the general philosophical and methodological points in that paper are also very important. (I liked the style as well, since to give a DCPO and topological approach only to say ’ that is not really what is going on and here is the secret’ is great.)
Although their approach is not coalgebraic and so does not directly give you an answer, and it is trying to do something different, but I think it may be more higher dimensional, more n-lab related and in fact side step some of the problems with the coalgebraic approach (once pushed a bit further and adapted towards the first order context). As I said, I am not expert in the modal side, but on the simplicial side, that paper is very pleasing to me.
There is a set of lecture notes (I only just saw them so have not read them) at here. There is a fibred category look to this and the coalgebraic viewpoint does not seem rich enough to handle that (uninformed opinion <- not to be given too much weight!!!!:-))
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