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In the SEP article on Model Theory by Wilfrid Hodges (here), he writes:

Particular kinds of model theory use particular kinds of structure; for example mathematical model theory tends to use so-called first-order structures, model theory of modal logics uses Kripke structures, and so on.

I take it that by a "Kripke structure" here, he simply means a Kripke model--- taken to be an ordered quadruple $\langle D, W, R, I \rangle$ where $D$ is a non-empty set of elements (the "domain of discourse"), $W$ is the set of "worlds", $R$ is the "accessibility relation", and $I$ is the interpretation function which assigns to every constant a member of $D$ and assigns to each world-predicate pair $\langle w, P\rangle$ an ordered n-tuple of $D$ (where n is determined by the arity of $P$).

I also take it that by a "first-order structure" he simply means the usual notion of a model in first-order model theory--- taken to be an ordered pair $\langle D, I \rangle$ where $D$ is a non-empty set of elements (the "domain of discourse") and $I$ is an interpretation function which assigns to every constant in the language an element of $D$, assigns to every n-ary predicate an ordered n-tuple of elements of $D$, and so on.

Is the only difference between these two sorts of structures the presence of an accessibility relation and a set of worlds in Kripke structures?

Could you do "mathematical model theory" with Kripke structures instead of first-order structures (that is, could it been done in any interesting sense, not simply by having a set of worlds and an accessibility relation that plays no role in the theory) ?

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Yes -- model theory with Kripke structures is the standard way to exhibit a semantics of the usual first-order connectives for which intuitionistic logic is a sound and complete proof system.

The Wikipedia article has some details.

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  • $\begingroup$ Good call. I was aware that intuitionistic logic can be interpreted as an S4 modal logic so this should have been an obvious step. An accessibility relation makes sense in intuitionistic logic, though, because of what the wiki calls the "persistency condition" and the sense in which statements in intuitionistic logic are indexed to a time. If you wanted to keep things classical, though, would such an approach still be feasible? Could you simply make accessibility an equivalence relation and work with an S5 modal logic? $\endgroup$ – Dennis Mar 21 '13 at 2:41

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