Recursive enumerability and elementary frame definability in predicate modal logic

TL;DR

This paper explores the limits of elementary frame definability in predicate modal logic, demonstrating that some Kripke complete, recursively enumerable logics cannot be characterized by elementary classes of frames.

Contribution

It constructs a predicate modal logic that is Kripke complete and recursively enumerable but not elementary frame definable, answering an open question in the field.

Findings

Existence of Kripke complete, recursively enumerable logics not characterized by elementary classes.

Counterexample logic not complete with respect to elementary class of frames.

Some logics are complete with respect to non-rooted elementary frame classes.

Abstract

We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On the one hand, it is well-known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e., a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on `natural' propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.