Proof Theory for First Order Łukasiewicz Logic

  • Matthias Baaz
  • George Metcalfe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4548)


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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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Matthias Baaz
    • 1
  • George Metcalfe
    • 2
  1. 1.Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040 WienAustria
  2. 2.Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240USA

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