On an intuitionistic logic for pragmatics

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.Acomputational 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 of p 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 that p, defined as a hypothesis that in some situation the truth of p is epistemically necessary.