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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.

##### MSC:
 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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##### References:
 [1] Curry, H. B., Foundations of Mathematical Logic , Dover, New York, 1976. · Zbl 0396.03001 [2] Czermak, J., “A remark on Gentzen’s calculus of sequents,” Notre Dame Journal of Formal Logic , vol. 18 (1977), pp. 471–474. · Zbl 0314.02026 [3] Gentzen, G., “Investigations into logical deduction,” The Collected Papers of Gerhard Gentzen , edited by M. E. Szabo, North-Holland, Amsterdam, 1969. · Zbl 0209.30001 [4] Glivenko, V., “Sur quelques points de la logique de M. Brouwer,” Académie Royale de Belgique, Bulletins de la classe des sciences , ser. 5, vol. 15 (1929), pp. 183–188. · JFM 55.0030.05 [5] Goodman, N. D., “The logic of contradiction,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik , vol. 27 (1981), pp. 119–126. · Zbl 0467.03019
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