Standard Completeness for Extensions of MTL: An Automated Approach

We provide general conditions on hypersequent calculi that guarantee standard completeness for the formalized logics. These conditions are implemented in the PROLOG system AxiomCalc that takes as input any suitable axiomatic extension of Monoidal T-norm Logic MTL and outputs a hypersequent calculus for the logic and the result of the check. Our approach subsumes many existing results and allows for the computerized discovery of new fuzzy logics.

[1]  Rostislav Horcík.  Alternative Proof of Standard Completeness Theorem for MTL , 2007, Soft Comput..

[2]  Hiroakira Ono,et al.  Logics without the contraction rule , 1985, Journal of Symbolic Logic.

[3]  Alex K. Simpson,et al.  Computational Adequacy in an Elementary Topos , 1998, CSL.

[4]  Agata Ciabattoni,et al.  Density elimination , 2008, Theor. Comput. Sci..

[5]  A. Avron.  The method of hypersequents in the proof theory of propositional non-classical logics , 1996 .

[6]  D. Gabbay,et al.  Proof Theory for Fuzzy Logics , 2008 .

[7]  Peter Jipsen,et al.  Residuated lattices: An algebraic glimpse at sub-structural logics , 2007 .

[8]  Lluis Godo,et al.  Monoidal t-norm based logic: towards a logic for left-continuous t-norms , 2001, Fuzzy Sets Syst..

[9]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[10]  Franco Montagna,et al.  A Proof of Standard Completeness for Esteva and Godo's Logic MTL , 2002, Stud Logica.

[11]  Satoko Titani,et al.  Intuitionistic fuzzy logic and intuitionistic fuzzy set theory , 1984, Journal of Symbolic Logic.

[12]  Franco Montagna,et al.  Kripke Semantics, Undecidability and Standard Completeness for Esteva and Godo's Logic MTL∀ , 2002, Stud Logica.

[13]  Kazushige Terui,et al.  MacNeille completions of FL-algebras , 2011 .

[14]  Matthias Baaz,et al.  Hypersequent and the Proof Theory of Intuitionistic Fuzzy Logic , 2000, CSL.

[15]  Franco Montagna,et al.  Substructural fuzzy logics , 2007, Journal of Symbolic Logic.

[16]  Franco Montagna,et al.  On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic , 2002, Stud Logica.

[17]  Kazushige Terui,et al.  From Axioms to Analytic Rules in Nonclassical Logics , 2008, 2008 23rd Annual IEEE Symposium on Logic in Computer Science.