Adding truth-constants to logics of continuous t-norms: Axiomatization and completeness results

In this paper we study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for Lukasiewicz logic and more recent results for Godel and Product logics. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants {r@?|r@?C}, for a suitable countable C@?[0,1], and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of Lukasiewicz, Godel and Product components, (ii) the set of truth-constants covers all the unit interval in the sense that each component of the t-norm contains at least one value of C different from the bounds of the component, and (iii) the truth-constants in Lukasiewicz components behave as rational numbers.

[1]  Vilém Novák.  On the syntactico-semantical completeness of first-order fuzzy logic. II. Main results , 1990, Kybernetika.

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

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

[4]  P. Aglianò,et al.  Varieties of BL- algebras I: General properties. , 2003 .

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

[6]  Daniele Mundici,et al.  Partial isomorphisms on totally ordered abelian groups and Hajek completeness theorem for basic logic , 2001 .

[7]  P. Mostert,et al.  On the Structure of Semigroups on a Compact Manifold With Boundary , 1957 .

[8]  J. Czelakowski,et al.  Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class , 1990 .

[9]  Peter Vojtás,et al.  Fuzzy logic programming , 2001, Fuzzy Sets Syst..

[10]  E. Trillas,et al.  in Fuzzy Logic , 2002 .

[11]  Franco Montagna,et al.  Equational Characterization of the Subvarieties of BL Generated by t-norm Algebras , 2004, Stud Logica.

[12]  Petr Hájek,et al.  A complete many-valued logic with product-conjunction , 1996, Arch. Math. Log..

[13]  Turid Rustad,et al.  Acknowledgements , 1996, Schizophrenia Research.

[14]  Petr Cintula,et al.  Product Ł ukasiewicz Logic , 2004, Arch. Math. Log..

[15]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[16]  Lluis Godo,et al.  On Product Logic with Truth-constants , 2006, J. Log. Comput..

[17]  Petr Cintula,et al.  From fuzzy logic to fuzzy mathematics: A methodological manifesto , 2006, Fuzzy Sets Syst..

[18]  Louise Schmir Hay,et al.  Axiomatization of the infinite-valued predicate calculus , 1963, Journal of Symbolic Logic.

[19]  I. Ferreirim.  On varieties and quasivarieties of hoops and their reducts. , 1992 .

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

[21]  Jan Pavelka,et al.  On Fuzzy Logic I Many-valued rules of inference , 1979, Math. Log. Q..

[22]  A. Nola.  MV Algebras in the Treatment of Uncertainty , 1993 .

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

[24]  Franco Montagna,et al.  On Weakly Cancellative Fuzzy Logics , 2006, J. Log. Comput..

[25]  Petr Cintula.  Short note: on the redundancy of axiom (A3) in BL and MTL , 2005, Soft Comput..

[26]  Vilém Novák.  On the syntactico-semantical completeness of first-order fuzzy logic. I. Syntax and semantics , 1990, Kybernetika.