On an intuitionistic logic for pragmatics

摘要

We reconsider Dalla Pozza and Garola's pragmatic interpretation of intuitionistic logic regarded as a logic of assertions and their justifications and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication developed especially by K. Ranalter. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses in work by G Bellin and others and on polarized bi-intuitionistic logic as a logic of assertions and hypotheses: looking at the S4 modal translation, we consider variants of the system AHL of bi-intuitionistic logic to represent the duality between the intuitionistic and the co-intuitionistic fragments, correcting and improving on the previous treatment. A computational interpretation of co-intuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction. Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines which was used by Bellin to give a categorical model of co-intuitionistic linear logic, and also a probabilistic interpretation of linear co-intuitionism. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth ofp is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture thatp, defined as a hypothesis that in some situation the truth ofp is epistemically necessary.