Monoidal t-norm based logic: towards a logic for left-continuous t-norms

Hajek's BL logic is the fuzzy logic capturing the tautologies of continuous t-norms and their residua. In this paper we investigate a weaker logic, MTL, which is intended to cope with the tautologies of left-continuous t-norms and their residua. The corresponding algebraic structures, MTL-algebras, are defined and completeness of MTL with respect to linearly ordered MTL-algebras is proved. Besides, several schematic extensions of MTL are also considered as well as their corresponding predicate calculi.

[1]  Morgan Ward,et al.  The Closure Operators of a Lattice , 1942 .

[2]  Siegfried Gottwald.  Axiomatizing t-norm based logic , 1999, EUSFLAT-ESTYLF Joint Conf..

[3]  Agata Ciabattoni.  On Urquhart's C logic , 2000, Proceedings 30th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2000).

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

[5]  J. Fodor.  Nilpotent minimum and related connectives for fuzzy logic , 1995, Proceedings of 1995 IEEE International Conference on Fuzzy Systems..

[6]  Lluis Godo,et al.  QBL: towards a logic for left-continuous t-norms , 1999, EUSFLAT-ESTYLF Joint Conf..

[7]  Alasdair Urquhart.  Many-valued Logic , 1986 .

[8]  Sándor Jenei,et al.  New family of triangular norms via contrapositive symmetrization of residuated implications , 2000, Fuzzy Sets Syst..

[9]  E. Trillas.  Sobre funciones de negación en la teoría de conjuntos difusos. , 1979 .

[10]  S. Gottwald.  A Treatise on Many-Valued Logics , 2001 .

[11]  A. Troelstra.  Lectures on linear logic , 1992 .

[12]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[13]  Dov M. Gabbay,et al.  Handbook of Philosophical Logic , 2002 .

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

[15]  Lluis Godo,et al.  Basic Fuzzy Logic is the logic of continuous t-norms and their residua , 2000, Soft Comput..

[16]  Petr Hájek,et al.  Basic fuzzy logic and BL-algebras II , 1998, Soft Comput..

[17]  Ulrich Höhle,et al.  Non-classical logics and their applications to fuzzy subsets : a handbook of the mathematical foundations of fuzzy set theory , 1995 .

[18]  D. Mundici,et al.  Algebraic Foundations of Many-Valued Reasoning , 1999 .

[19]  Petr Hájek,et al.  Residuated fuzzy logics with an involutive negation , 2000, Arch. Math. Log..

[20]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.