Resolution Matrix Semantics for Deontic Logic

This paper introduces Resolution Matrix Semantics (RMS) [Kuznetsov, 2025] as a novel framework for formalizing deontic logic, inspired by Y. Ivlev’s quasi-matrix approach [Ivlev, 1997]. Unlike traditional Kripkean semantics, RMS employs a truth-value-based approach with normative values—mandatory (m), indifferent (i), and forbidden (b)—to evaluate acts, and true (t) or false (f) for formulas, ensuring a clear distinction between normative and propositional domains. This truth value semantics is augmented with “undetermined” truth values (e.g. m/i, “either m, or i”), and RMS uses an interpretation and sub-interpretation mechanism to resolve these, enabling precise normative evaluations. We develop SDLm, a deontic system that mirrors Standard Deontic Logic (SDL) in its axiomatic structure but avoids well-known paradoxes, such as the Chisholm and Gentle Murderer Paradoxes. We prove SDLm’s soundness and completeness and provide tableau rules for automated theorem verification. By prioritizing normative status over existential considerations, SDLm offers a robust, paradox-free framework for reasoning about obligations, permissions, and prohibitions, with potential applications in ethics, law, and artificial intelligence.