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Proof Theory for First Order Łukasiewicz Logic

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Book cover Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2007)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4548))

Abstract

An approximate Herbrand theorem is proved and used to establish Skolemization for first-order Łukasiewicz logic. Proof systems are then defined in the framework of hypersequents. In particular, extending a hypersequent calculus for propositional Łukasiewicz logic with usual Gentzen quantifier rules gives a calculus that is complete with respect to interpretations in safe MV-algebras, but lacks cut-elimination. Adding an infinitary rule to the cut-free version of this calculus gives a system that is complete for the full logic. Finally, a cut-free calculus with finitary rules is obtained for the one-variable fragment by relaxing the eigenvariable condition for quantifier rules.

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Author information

Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria

    Matthias Baaz

  2. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240, USA

    George Metcalfe

Authors
  1. Matthias Baaz
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  2. George Metcalfe
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Nicola Olivetti

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Baaz, M., Metcalfe, G. (2007). Proof Theory for First Order Łukasiewicz Logic. In: Olivetti, N. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2007. Lecture Notes in Computer Science(), vol 4548. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73099-6_5

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  • DOI: https://doi.org/10.1007/978-3-540-73099-6_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73098-9

  • Online ISBN: 978-3-540-73099-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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