On an intuitionistic logic for pragmatics

Abstract

We reconsider the pragmatic interpretation of intuitionistic logic [21] 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 [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the S4 modal translation, we give a definition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism 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 is defined and a probabilistic interpretation of linear co-intuitionism is given as in [9]. 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.

On assertion and denial in the logic for pragmatics

“…In the logical system called Logic for Pragmatics (LP), Dalla Pozza and Garola [9] provided a formal treatment of assertion by introducing some pragmatic connectives and the sign of force, which are required to formulate a pragmatic interpretation of intuitionistic propositional logic as a logic of assertions. 2 Assertions are intended as "purely logical entities ... without making reference to the speaker's intention or beliefs" (Dalla Pozza and Garola [9], 83). LP is composed of two sets of formulas: radical and sentential.…”

Assertion and hypothesis: a logical framework for their opposition relations:

“…In addition, a formula δ is pragmatically valid or p-valid (respectively, invalid or p-invalid) if, for every π and σ , the formula δ is justified (respectively, δ is unjustified). 3 In any case, no principle analogous to the truth-functionality principle for classical connectives holds for the pragmatic connectives in LP, since pragmatic connectives are partial functions of justification. Moreover, a function () * mapping the set of sfs into an extension of the set of rfs obtained by means of the modal operator (proved) i.e.…”

“…by H5, when the following condition holds: The logic of hypothesis HLP resembles a dual-intuitionistic logic with an implication-like connective, while the fragments ILP plus HLP* shows some features of an (illocutionary) bi-intuitionistic logic. 10 A meta-theoretical duality between assertions and hypotheses can be guaranteed once we deal with assertions that have a positive content and hypotheses with a negative content, as proposed by Bellin et al [3]. A variant of this logic can be a logic where the content of hypotheses is positive, while the content of assertions is negative.…”

“…Recent developments of LP are pointed out in[2][3][4]6]. 6 In other works on pragmatic logic, hypothetical negation has a slightly different meaning 7.…”

A proof-theoretic study of bi-intuitionistic propositional sequent calculus

“…For example, in line with Curien-Herbelin's study [8] of dualities in classical sequent calculus, Crolard [7] gave an interpretation of bi-intuitionistic exclusion as a type for coroutines (a restricted form of continuations). With philosophical motivations, Bellin and colleagues [1,2] considered a different symmetrisation of intuitionistic logic (also called bi-intuitionistic) from the point of view of pragmatics, a logic of assertions and hypotheses and, in the same vein, Wansing [37] combined verificationism and falsificationism.…”