We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justi cations 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 de nition 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 de ned 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, de ned as a hypothesis that in some situation the truth of p is epistemically necessary. (shrink)

We consider a “polarized” version of bi-intuitionistic logic [5, 2, 6, 4] 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 [28, 29], whose categorical models are all partial orders by Crolard’s theorem [8]. We show that P.A. Melliès notion of chirality [21, 22] 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’s pragmatic interpretation of intuitionism as a logic of assertions [10] 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 logical systems that classify 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 their “pragmatic interpretation” a bona fide representation of intuitionism. We sketch a meaning-asuse interpretation of co-intuitionism that seems to fulfil the requirements of Dummett and Prawitz’s justificationist approach. We extend the Brouwer-Heyting-Kolmogorov interpretation to bi-intuitionism by regarding co-intuitionistic formulas as types of the evidence for them: if conclusive evidence is needed to justify assertions, only a scintilla of evidence suffices to justify hypotheses. (shrink)

We consider a class of graphs embedded in $R^2$ as noncommutative proof-nets with an explicit exchange rule. We give two characterization of such proof-nets, one representing proof-nets as CW-complexes in a two-dimensional disc, the other extending a characterization by Asperti. As a corollary, we obtain that the test of correctness in the case of planar graphs is linear in the size of the data. Braided proof-nets are proof-nets for multiplicative linear logic with Mix embedded in $R^3$ . In order to (...) prove the cut-elimination theorem, we consider proof-nets in $R^2$ as projections of braided proof-nets under regular isotopy. (shrink)