A pragmatic dialogic interpretation of bi-intuitionism

We consider a " polarized " version of bi-intuitionistic logic [9, 7, 10, 11] as a logic of assertions and hypotheses and show that it supports a " rich proof theory " and an interesting categorical interpretation, unlike the standard approach of C. Rauszer's Heyting-Brouwer logic [48, 49], whose categorical models are all partial orders by Crolard's theorem [19]. We show that P. A.Mellì es notion of chirality [42, 43] appears as the right mathematical representation of the mirror symmetry between the intuitionistic and co-intuitionistc sides of polarized bi-intuitionism. Philosophically, we extend Dalla Pozza and Garola pragmatic interpretation of in-tuitionism as a logic of assertions [21] to bi-intuitionism as a logic of assertions and hypotheses. We focus on the logical role of illocutionary forces and justification conditions in order to provide " intended interpretations " of logic systems that classifies inferential uses in natural language and remain acceptable from an intuitionistic point of view. Although Dalla Pozza and Garola originally provide a constructive interpretation of intuitionism in a classical setting, we claim that some conceptual refinements suffice to make the " pragmatic interpretation " a bona fide representation of intuitionism. For co-intuitionism we sketch a meaning-as-use interpretation that appears as able to fulfill the requirements of Dummett and Prawitz's justificationist approach. We extend the Brouwer-Heyting-Kolmogorov interpretation of intuition-ism by regarding co-intuitionistic formulas as types of the evidence for them. Next, assuming a notion of duality between assertions and hypotheses, we give a " dialogic interpretation " of multiplicative linear polarized bi-intuitionistic logic which can be regarded as a translation into intuitionistic multiplicative linear logic with products. Mathematically, the interplay between evidence for and evidence against assertions and hypotheses is inspired by Chu's construction [8], usually regarded as an abstract form of the " game semantics " for linear logic. 1 Introduction. A mathematical prelude.