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Dual-intuitionistic logic. (English) Zbl 0869.03008
Summary: The sequent system \({\mathbf L}{\mathbf D}{\mathbf J}\) is formulated using the same connectives as Gentzen’s intuitionistic sequent system \({\mathbf L}{\mathbf J}\), but is dual in the following sense: (i) whereas \({\mathbf L}{\mathbf J}\) is singular in the consequent, \({\mathbf L}{\mathbf D}{\mathbf J}\) is singular in the antecedent; (ii) whereas \({\mathbf L}{\mathbf J}\) has the same sentential counter-theorems as classical \({\mathbf L}{\mathbf K}\) but not the same theorems, \({\mathbf L}{\mathbf D}{\mathbf J}\) has the same sentential theorems as \({\mathbf L}{\mathbf K}\) but not the same counter-theorems. In particular, \({\mathbf L}{\mathbf D}{\mathbf J}\) does not reject all contradictions and is accordingly paraconsistent. To obtain a more precise mapping, both \({\mathbf L}{\mathbf J}\) and \({\mathbf L}{\mathbf D}{\mathbf J}\) are extended by adding a “pseudo-difference” operator \(\overset{.}-\) which is the dual of intuitionistic implication. Cut-elimination and decidability are proved for the extended systems \({\mathbf L}{\mathbf J}^{\overset{.}-}\) and \({\mathbf L}{\mathbf D}{\mathbf J}^{\overset{.}-}\), and a simply consistent but \(\omega\)-inconsistent Set Theory with Unrestricted Comprehension Schema based on \({\mathbf L}{\mathbf D}{\mathbf J}\) is sketched.

03B20 Subsystems of classical logic (including intuitionistic logic)
03F05 Cut-elimination and normal-form theorems
03E70 Nonclassical and second-order set theories
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