Notre Dame Journal of Formal Logic

Dual-Intuitionistic Logic

Igor Urbas


The sequent system LDJ is formulated using the same connectives as Gentzen's intuitionistic sequent system LJ, but is dual in the following sense: (i) whereas LJ is singular in the consequent, LDJ is singular in the antecedent; (ii) whereas LJ has the same sentential counter-theorems as classical LK but not the same theorems, LDJ has the same sentential theorems as LK but not the same counter-theorems. In particular, LDJ does not reject all contradictions and is accordingly paraconsistent. To obtain a more precise mapping, both LJ and LDJ are extended by adding a "pseudo-difference" operator $\raisebox{5pt}{.}\kern-6.25pt{-}$ which is the dual of intuitionistic implication. Cut-elimination and decidability are proved for the extended systems ${\bf LJ}^{\kern2pt{\raisebox{3.5pt}{.}}\kern-4.8pt{-}}$ and ${\bf LDJ}^{\kern2pt{\raisebox{3.5pt}{.}}\kern-4.8pt{-}}$, and a simply consistent but $\omega$-inconsistent Set Theory with Unrestricted Comprehension Schema based on LDJ is sketched.

Article information

Notre Dame J. Formal Logic Volume 37, Number 3 (1996), 440-451.

First available in Project Euclid: 14 December 2002

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Zentralblatt MATH identifier

Primary: 03B53: Paraconsistent logics
Secondary: 03B20: Subsystems of classical logic (including intuitionistic logic) 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10] 03B55: Intermediate logics 03E70: Nonclassical and second-order set theories 03F05: Cut-elimination and normal-form theorems 03F55: Intuitionistic mathematics


Urbas, Igor. Dual-Intuitionistic Logic. Notre Dame J. Formal Logic 37 (1996), no. 3, 440--451. doi:10.1305/ndjfl/1039886520.

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