Proof Theory for First Order Łukasiewicz Logic

Authors:
Matthias Baaz;George Metcalfe
Affiliations:
Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria;Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240, USA
Venue:
TABLEAUX '07 Proceedings of the 16th international conference on Automated Reasoning with Analytic Tableaux and Related Methods
Year:
2007

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Abstract

An approximate Herbrand theorem is proved and used to establish Skolemization for first-order 茂戮驴ukasiewicz logic. Proof systems are then defined in the framework of hypersequents. In particular, extending a hypersequent calculus for propositional 茂戮驴ukasiewicz 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.