Possibility Frames and Forcing for Modal Logic
This paper addresses the semantic divide between classical and intuitionistic modal logic by introducing a novel frame semantics based on partially ordered “possibilities” instead of traditional possible worlds. Methodologically, it constructs possibility frames—interpreting formulas via regular open sets in Alexandrov topologies—and develops a categorical duality theory between such frames and non-atomic CV-Boolean algebraic operators (CV-BAOs). It establishes, for the first time, a duality between full possibility frames and CV-BAOs using filters rather than ultrafilters, thereby avoiding the Axiom of Choice; it also introduces principal possibility frames to characterize V-BAOs. The main contributions are: (i) unifying classical and intuitionistic modal semantics within a single framework; (ii) systematically establishing dualities between possibility frames and various classes of BAOs; (iii) completing definability, correspondence, and strong completeness theories; and (iv) proving that every BAO is fully characterized by a filter-descriptive possibility frame.