An Analytic Calculus for Quantified Propositional Gödel Logic

We define a hypersequent calculus for Godel logic enhanced with (fuzzy) quantifiers over propositional variables. We prove soundness, completeness and cut-elimination for this calculus and provide a detailed investigation of the so-called Takeuti-Titani rule which expresses density of the ordering of truth values. Since this rule is critical from the point of view of proof search we characterize a fragment of the logic for which it can be eliminated.

[1]  Mitio Takano,et al.  Another proof of the strong completeness of the intuitionistic fuzzy logic , 1987 .

[2]  Roy Dyckhoff.  Automated Reasoning with Analytic Tableaux and Related Methods , 2000, Lecture Notes in Computer Science.

[3]  Alfred Horn,et al.  Logic with truth values in A linearly ordered heyting algebra , 1969, Journal of Symbolic Logic.

[4]  Helmut Veith,et al.  Proof Theory of Fuzzy Logics: Urquhart's C and Related Logics , 1998, MFCS.

[5]  Michael Dummett,et al.  A propositional calculus with denumerable matrix , 1959, Journal of Symbolic Logic (JSL).

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

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

[8]  Jozef Gruska,et al.  Mathematical Foundations of Computer Science 1998 , 1998, Lecture Notes in Computer Science.

[9]  J. Dunn,et al.  Algebraic Completeness Results for Dummett's LC and Its Extensions , 1971 .

[10]  Helmut Veith,et al.  Interpolation in fuzzy logic , 1999, Arch. Math. Log..

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

[12]  Helmut Veith,et al.  On the Undecidability of some Sub-Classical First-Order Logics , 1999, FSTTCS.

[13]  Helmut Veith,et al.  Quantifier Elimination in Fuzzy Logic , 1998, CSL.

[14]  M. Baaz.  Infinite-valued Gödel logics with $0$-$1$-projections and relativizations , 1996 .

[15]  Albert Visser,et al.  On the completenes principle: A study of provability in heyting's arithmetic and extensions , 1982, Ann. Math. Log..

[16]  Christian G. Fermüller,et al.  Analytic Calculi for Projective Logics , 1999, TABLEAUX.

[17]  Helmut Veith,et al.  An axiomatization of quantified proposition Gödel logic using the Takeuti-Titani rule , 2000 .

[18]  Roy Dyckhoff,et al.  A Deterministic Terminating Sequent Calculus for Gödel-Dummett logic , 1999, Log. J. IGPL.

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