Elements of model theory in higher-order fuzzy logic

In this paper, we turn our attention to model theory of higher-order fuzzy logic (fuzzy type theory). This theory generalizes model theory of predicate logic but has some interesting specificities. We will introduce few basic concepts related to homomorphism, isomorphism, submodel, etc. and show some properties of them.

[1]  V. Novák,et al.  Mathematical Principles of Fuzzy Logic , 1999 .

[2]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[3]  Franco Montagna,et al.  Distinguished algebraic semantics for t-norm based fuzzy logics: Methods and algebraic equivalencies , 2009, Ann. Pure Appl. Log..

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

[5]  Petr Hájek,et al.  Formal systems of fuzzy logic and their fragments , 2007, Ann. Pure Appl. Log..

[6]  Vilém Novák,et al.  On good EQ-algebras , 2011, Fuzzy Sets Syst..

[7]  Leon Henkin,et al.  Completeness in the theory of types , 1950, Journal of Symbolic Logic.

[8]  Pilar Dellunde.  Preserving mappings in fuzzy predicate logics , 2012, J. Log. Comput..

[9]  Johan van Benthem,et al.  Higher-Order Logic , 1989 .

[10]  William M. Farmer,et al.  The seven virtues of simple type theory , 2008, J. Appl. Log..

[11]  Peter B. Andrews.  General Models, Descriptions, and Choice in Type Theory , 1972, J. Symb. Log..

[12]  Bernard De Baets,et al.  EQ-algebras , 2009, Fuzzy Sets Syst..

[13]  Petr Hájek,et al.  On theories and models in fuzzy predicate logics , 2006, Journal of Symbolic Logic.

[14]  Vilém Novák,et al.  On fuzzy type theory , 2005, Fuzzy Sets Syst..

[15]  Vilém Novák,et al.  Omitting types in fuzzy logic with evaluated syntax , 2006, Math. Log. Q..

[16]  V. Novák.  On Virtues of Many-Valued (Fuzzy) Type Theories , 2010 .

[17]  B. Russell.  Mathematical Logic as Based on the Theory of Types , 1908 .

[18]  Francesc Esteva,et al.  On elementary equivalence in fuzzy predicate logics , 2012, Archive for Mathematical Logic.

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

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

[21]  C. Toffalori,et al.  A guide to classical and modern model theory , 2003 .

[22]  Peter B. Andrews.  An introduction to mathematical logic and type theory - to truth through proof , 1986, Computer science and applied mathematics.

[23]  Vilém Novák,et al.  EQ-algebra-based fuzzy type theory and its extensions , 2011, Log. J. IGPL.

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

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