Notre Dame Journal of Formal Logic

Dual-Intuitionistic Logic

Igor Urbas

Abstract

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

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

Dates
First available in Project Euclid: 14 December 2002

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1039886520

Digital Object Identifier
doi:10.1305/ndjfl/1039886520

Mathematical Reviews number (MathSciNet)
MR1434429

Zentralblatt MATH identifier
0869.03008

Subjects
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

Citation

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


Export citation

References

  • [1] Curry, H. B., Foundations of Mathematical Logic, Dover, New York, 1976.
  • [2] Czermak, J., ``A remark on Gentzen's calculus of sequents,'' Notre Dame Journal of Formal Logic, vol. 18 (1977), pp. 471--474.
  • [3] Gentzen, G., ``Investigations into logical deduction,'' The Collected Papers of Gerhard Gentzen, edited by M. E. Szabo, North-Holland, Amsterdam, 1969.
  • [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.
  • [5] Goodman, N. D., ``The logic of contradiction,'' Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 27 (1981), pp. 119--126.