Speakers
- Bradley Shubert
Abstract
This paper proposes a novel truth-functional semantics for modal logic by replacing Kripke-style possible worlds with many-valued matrix systems. Each finite model with worlds is represented by a many-valued logic with truth-values, encoded as binary strings corresponding to world-indexed truth assignments. Accessibility relations are captured through truth-functional operations, yielding a unique modal matrix for each frame. This framework enables a fully algebraic treatment of modal operators, handling frame-dependent validity in a purely truth-functional way. While Dugundji’s theorem is standardly seen as precluding matrix semantics of this kind, we demonstrate how the approach here sidesteps those results by indexing semantics to individual frames. We establish several technical results: a general procedure for generating matrices from Kripke frames, injectivity of the frame-to-matrix mapping, and frame recoverability given a modal matrix. We outline matrix-theoretic completeness results for the modal logic , and define bisimulation in the matrix setting. Expressivity is examined and infinite extensions are explored. Finally, we introduce the concept of structural bivalence, which underwrites the constructions developed here, alongside broader technical and philosophical implications.
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