First-Order Possibility Models and Finitary Completeness Proofs

Abstract

This article builds on Humberstone’s idea of defining models of propositional modal logic where total possible worlds are replaced by partial possibilities. We follow a suggestion of Humberstone by introducing possibility models for quantified modal logic. We show that a simple quantified modal logic is sound and complete for our semantics. Although Holliday showed that for many propositional modal logics, it is possible to give a completeness proof using a canonical model construction where every possibility consists of finitely many formulas, we show that this is impossible to do in the first-order case. However, one can still construct a canonical model where every possibility consists of a computable set of formulas and thus still of finitely much information.

Building models of strongly minimal theories

“…For example, it follows that a first-order theory is equivalent to a Π in n -sentence if and only if it has a ∀ n -axiomatization. This is related to Keisler's result [Ke65] that was recently reproved by Harrison-Trainor and Kretschmer [HK23] that if a first-order formula is definable by an infinitary Π in n -formula, then it must already be definable by a ∀ n -formula. In fact, our result applied to a single formula implies this via an easy application of compactness.…”

“…We observe that a special case of Theorem 4.9 implies a case of a theorem of Keisler [Ke65, Corollary 3.4], recently reproved by Harrison-Trainor and Kretschmer [HK23].…”

Results in modal correspondence theory for possibility semantics

“…It follows from this and other results [21] that more normal modal logics are sound and complete with respect to some class of possibility frames than with respect to some class of Kripke frames. For other recent results on possibility semantics and mention related work, see [4,5,17,16].…”