Natural Deduction and Term Assignment for Co-heyting Algebras in Polarized Bi-intuitionistic Logic

We reconsider Rauszer’s bi-intuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of polarized bi-intuitionistic logic (PBL) consists of two fragments, positive intuitionistic logic LJ and its dual LJ rg , extended with two negations partially internalizing the duality between LJ and LJ . Modal interpretations and Kripke’s semantics over bimodal preordered frames are considered and a Natural Deduction system PBN is sketched for the whole system. A stricter interpretation of the duality and a simpler natural deduction system is obtained when polarized bi-intuitionistic logic is interpreted over S4 rather than bi-modal S4 (a logic called intuitionistic logic for pragmatics of assertions and conjectures ILPAC). The term assignment for the conjectural fragment LJ rg exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The duality is extended from formulas to proofs and it is shown that every computation in our calculus is isomorphic to a computation in the simply typed λ-calculus. §1. Preface. We present a natural deduction system for propositional polarized bi-intuitionistic logic PBL, (a variant of) intuitionistic logic extended with a connective of subtraction A r B, read as “A but not B”, which is dual to implication. The logic PBL is polarized in the sense that its expressions are regarded as expressing acts of assertion or of conjecture; implications and conjunctions are assertive, subtractions and disjunctions are conjectural. Assertions and conjectures are regarded as dual; moreover there are two negations, transforming assertions into conjectures and viceversa, in some sense internalizing the duality. Our notion of polarity isn’t just a technical device: it is rooted in an analysis of the structure of speech-acts, following the viewpoint of the We thank Stefano Berardi, Tristan Crolard, Arnaud Fleury, Nicola Gambino, Maria Emilia Maietti and Graham White for their help at various stages of this project.