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In the 1951 Gibbs lecture, Godel asserted his famous dichotomy, where the notion of informal proof is at work. G. Priest developed an argument, grounded on the notion of naive proof, to the effect that Godel’s first incompleteness theorem suggests the presence of dialetheias. In this paper, we adopt a plausible ideal notion of naive proof, in agreement with Godel’s conception, superseding the criticisms against the usual notion of naive proof used by real working mathematicians. We explore the connection between Godel’s theorem and naive proof so understood, both from a classical and a dialetheic perspective.

The goal of [3] is to sketch the construction of a syntactic categorical model of the bi-intuitionistic logic of assertions and hypotheses AH , axiomatized in a sequent calculus AH-G1 , and to show that such a model has a chirality-like structure inspired by the notion of dialogue chirality by P-A. Mellies [8]. A chirality consists of a pair of adjoint functors L ⊣ R , with L : A → B, R : B → A, and of a functor (.)* : A → B op (0 , 1) satisfying certain conditions. The definition of the logic AH in [3] needs to be modified so that our categories A and B are actually dual. With this modification, a more complex structure emerges.

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. Mellies 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.