On Gentzen Relations Associated with Finite-valued Logics Preserving Degrees of Truth

Studia Logica 101 (4):749-781 (2013)
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Abstract

When considering m-sequents, it is always possible to obtain an m-sequent calculus VL for every m-valued logic (defined from an arbitrary finite algebra L of cardinality m) following for instance the works of the Vienna Group for Multiple-valued Logics. The Gentzen relations associated with the calculi VL are always finitely equivalential but might not be algebraizable. In this paper we associate an algebraizable 2-Gentzen relation with every sequent calculus VL in a uniform way, provided the original algebra L has a reduct that is a distributive lattice or a pseudocomplemented distributive lattice. We also show that the sentential logic naturally associated with the provable sequents of this algebraizable Gentzen relation is the logic that preserves degrees of truth with respect to the original algebra (in contrast with the more common logic that merely preserves truth). Finally, for some particular logics we obtain 2-sequent calculi that axiomatize the algebraizable Gentzen relations obtained so far.

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References found in this work

Proof Theory of Finite-valued Logics.Richard Zach - 1993 - Dissertation, Technische Universität Wien
Correspondences between Gentzen and Hilbert Systems.J. G. Raftery - 2006 - Journal of Symbolic Logic 71 (3):903 - 957.
Elimination of Cuts in First-order Finite-valued Logics.Matthias Baaz, Christian G. Fermüller & Richard Zach - 1993 - Journal of Information Processing and Cybernetics EIK 29 (6):333-355.
Gentzen-type systems, resolution and tableaux.Arnon Avron - 1993 - Journal of Automated Reasoning 10:265-281.

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