Classical Logic as Intuitionistic Logic with Duality
For logicians and philosophers of logic, this offers a new perspective on the relationship between classical and intuitionistic logic, addressing foundational challenges in proof-theoretic semantics.
The paper proposes a new proof-theoretic semantics for classical logic that extends the semantics for intuitionistic logic by operating over literals with a primitive duality, showing classical logic can be seen as intuitionistic logic plus duality.
The field of proof-theoretic semantics (P-tS) offers an alternative approach to meaning in logic that is based on inference and argument (rather than truth in a model). It has been successfully developed for various logics; in particular, Sandqvist has developed such semantics for both classical and intuitionistic logic. In the case of classical logic, P-tS provides a conception of consequence that avoids an a priori commitment to the principle of bivalence, addressing what Dummett identified as a significant foundational challenge in logic. In this paper, we propose an alternative P-tS for classical logic, which essentially extends the P-tS for intuitionistic logic by operating over literals rather than atomic propositions. Importantly, literals are atomic and not defined by negation but are related by a primitive duality encoded inferentially at the atomic level. This semantics illustrates the perspective that classical logic can be understood as intuitionistic logic supplemented by a principle of duality, offering fresh insights into the relationship between these two systems.