Possibility Frames and Forcing for Modal Logic

TL;DR

This paper introduces a new model theory for normal modal logics using possibility semantics based on posets, extending classical semantics and developing foundational duality, definability, and completeness results.

Contribution

It develops the foundational theory of possibility frames for modal logic, including duality, correspondence, and completeness, offering an alternative to traditional Kripke semantics.

Findings

Possibility frames generalize Kripke frames with posets and regular open sets.

Standard modal frames are a special case with discrete posets.

Initial duality and completeness results are established for possibility semantics.

Abstract

This paper develops the model theory of normal modal logics based on partial "possibilities" instead of total "worlds," following Humberstone (1981) instead of Kripke (1963). Possibility semantics can be seen as extending to modal logic the semantics for classical logic used in weak forcing in set theory, or as semanticizing a negative translation of classical modal logic into intuitionistic modal logic. Thus, possibility frames are based on posets with accessibility relations, like intuitionistic modal frames, but with the constraint that the interpretation of every formula is a regular open set in the Alexandrov topology on the poset. The standard world frames for modal logic are the special case of possibility frames wherein the poset is discrete. We develop the beginnings of duality theory, definability/correspondence theory, and completeness theory for possibility frames.