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INVERSION BY DEFINITIONAL REFLECTION AND THE ADMISSIBILITY OF LOGICAL RULES

  • WAGNER DE CAMPOS SANZ (a1) and THOMAS PIECHA (a2)
Abstract

The inversion principle for logical rules expresses a relationship between introduction and elimination rules for logical constants. Hallnäs & Schroeder-Heister (1990, 1991) proposed the principle of definitional reflection, which embodies basic ideas of inversion in the more general context of clausal definitions. For the context of admissibility statements, this has been further elaborated by Schroeder-Heister (2007). Using the framework of definitional reflection and its admissibility interpretation, we show that, in the sequent calculus of minimal propositional logic, the left introduction rules are admissible when the right introduction rules are taken as the definitions of the logical constants and vice versa. This generalizes the well-known relationship between introduction and elimination rules in natural deduction to the framework of the sequent calculus.

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*FACULDADE DE FILOSOFIA, UNIVERSIDADE FEDERAL DE GOIÁS, CEP 74001-970, GOIÂNIA, GO, BRAZIL E-mail:sanz@fchf.ufg.br
WILHELM-SCHICKARD-INSTITUT, UNIVERSITÄT TÜBINGEN, SAND 13, 72076 TÜBINGEN, GERMANY E-mail:piecha@informatik.uni-tuebingen.de
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G Gentzen . (1935). Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39, 176210, 405–431. English translation (1969) in M. E. Szabo , editor. The Collected Papers of Gerhard Gentzen. Studies in Logic and the Foundations of Mathematics. Amsterdam, The Netherlands: North-Holland, pp. 68–131.

L. Hallnäs , & P Schroeder-Heister . (1990). A proof-theoretic approach to logic programming. I. Clauses as rules. Journal of Logic and Computation, 1, 261283.

L. Hallnäs , & P Schroeder-Heister . (1991). A proof-theoretic approach to logic programming. II. Programs as definitions. Journal of Logic and Computation, 1, 635660.

L Hallnäs . (1991). Partial inductive definitions. Theoretical Computer Science, 87, 115142.

R. Iemhoff , & G Metcalfe . (2009). Proof theory for admissible rules. Annals of Pure and Applied Logic, 159, 171186.

E Jeřábek . (2008). Independent bases of admissible rules. Logic Journal of the IGPL, 16, 249267.

P Lorenzen . (1955). Einführung in die operative Logik und Mathematik (second edition 1969). Berlin: Springer.

E. Moriconi , & L Tesconi . (2008). On inversion principles. History and Philosophy of Logic, 29, 103113.

J von Plato . (2001). Natural deduction with general elimination rules. Archive for Mathematical Logic, 40, 541567.

D Prawitz . (1979). Proofs and the meaning and completeness of the logical constants. In J. Hintikka , I. Niiniluoto , and E. Saarinen , editors. Essays on Mathematical and Philosophical Logic. Dordrecht, The Netherlands: Reidel, pp. 2540. Revised and extended German translation (1982): Beweise und die Bedeutung und Vollständigkeit der logischen Konstanten. Conceptus, XVI, 3–44.

V. V Rybakov . (1997). Admissibility of Logical Inference Rules. Studies in Logic and the Foundations of Mathematics 136. Amsterdam, The Netherlands: North-Holland.

P Schroeder-Heister . (1993). Rules of definitional reflection. In Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science (Montreal June 19–23, 1993). Los Alamitos, CA: IEEE Computer Society, pp. 222232.

P Schroeder-Heister . (2007). Generalized definitional reflection and the inversion principle. Logica Universalis, 1, 355376.

P Schroeder-Heister . (2008). Lorenzen’s operative justification of intuitionistic logic. In M. van Atten , P. Boldini , M. Bourdeau , and G. Heinzmann , editors. One Hundred Years of Intuitionism (1907-2007): The Cerisy Conference. Basel, Switzerland: Birkhäuser, pp. 214240.

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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
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