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What are categorical models of logics good for?

by Greg Restall <Greg.Restall@cisr.anu.edu.au>

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Greg Restall

Uday S Reddy

I've read with interest the discussion of categorical models of linear logics. I've wrapped my brain around the recent results, and I've enjoyed it immensely. I do have a question about categorical models, however, and I think that this list is as good a place as any to have them answered. My question is this.

What are categorical models of logics good for?

1. They help us understand our term assignment systems for these logics.
2. They help us prove results about our logics, as we can use category theory.
3. We can show that our favourite logic corresponds to some already-found structure in category theory, and hence, our logic gets brownie-points for being `natural.'
4. The category (or class of categories) we are interested in is the intended model for our logic. That is, we are using our logic to model some phenomenon, which we have already presented categorically. As we have shown that this category (or class of categories) is a model of our logic, we thereby show that our logic models nicely the phenomenon of interest.

Are any of these answers correct? If so, which of them? Or are they missing the point? Are categorical models interesting for some other reason? I'd be very interested if there are any results like those foreshadowed in answer `4' actually appear in the literature on linear logic. The most I can find is answers in the ballpark of 1, 2 and maybe 3.

Replies can be sent to the list, or to me. Either is fine.