Between model theory and category theory broadly conceived: not anything really compelling, because a category, on its own, does not stand as an interpretation for anything.
Between model theory and categorical logic, however: yes, I think the overlap is large.
A spot of history: the man most deserving, in my opinion, of being called the father of model theory is Alfred Tarski, who came from a Polish school of logic that, I understand, was very much within the algebraic school. His model theory was more in the vein of a reworking of the Polish-style algebraic logic (this is not, in anway, to talk down his achievement).
Blackburn &al (2001, pp40-41) talk of a might-have-been for the Jónsson-Tarski representation theorem:
...while modal algebras were useful tools, they seemed of little help in guiding logical intuitions. The [theorem] should have swept this apparent shortcoming away for good, for in essence they showed how to represent modal algebras as the structures we now call models! In fact, they did a lot more that this. Their representation technique is essentially a model building technique, hence their work gave the technical tools needed to prove the completeness result that dominated [work on modal logic before Kripke].
They go on to present a nice anecdote showing how Tarski did not seem to think this algebraic approach provided a semantics for modal logic, even after Kripke stressed how important it was to Kripke semantics. It seems that sometimes algebraic logic and model theory are more similar than they appear.
Like model theory, categorical logic can seem to be a special way of doing algebraic logic. And with some theories, model theory and algebraic logic sometimes seem to differ only in trivialities; with categorical logic I am more hesitant in making sweeping judgements, but it sometimes feels that way to me too.
Ref: Blackburn, de Rijke, & Venema (2001) Modal Logic, CUP.
No comments:
Post a Comment