A pragmatic framework for intuitionistic modalities: Classical logic and Lax logic

We reconsider Dalla Pozza and Garola pragmatic interpretation of intuitionistic logic [13] where sentences and proofs formalize assertions and their justifications and revise it so that the costruction is done within an intuitionistic metatheory. We reconsider also the extension of Dalla Pozza and Garola’a approach to cointuitionistic logic, seen as a logic of hypotheses [5, 9, 4] and the duality between assertions and hypotheses represented by two negations, the assertive and the hypothetical ones. By adding illocutionary forces of conjecture, defined as a hypothesis that an assertion is justified and of expectation, an assertion that a hypothesis is justified we obtain pragmatic counterparts of the modalities of classical S4, but also a framework for different interpretations of intuitionistic modalities necessity and possibility. We consider two applications: one is typing Parigot’s λμ calculus in a bi-intuitionistic logic of expectations. The second is an interpretation of Fairtlough and Mendler’s Propositional Lax Logic as an extension of intuitionistic logic with a co-intuitionistic operator of empirical possibility.