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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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
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