Omitting uncountable types and the strength of [0, 1]-valued logics

Abstract We study a class of [ 0 , 1 ] -valued logics. The main result of the paper is a maximality theorem that characterizes these logics in terms of a model-theoretic property, namely, an extension of the omitting types theorem to uncountable languages.

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

[2]  H. Jerome Keisler,et al.  Continuous Model Theory , 1966 .

[3]  V. Novák.  Fuzzy sets and their applications , 1989 .

[4]  Jan Pavelka,et al.  On Fuzzy Logic II. Enriched residuated lattices and semantics of propositional calculi , 1979, Math. Log. Q..

[5]  P. Hájek.  Fuzzy logic and arithmetical hierarchy , 1995 .

[6]  Xavier Caicedo,et al.  The Abstract Compactness Theorem Revisited , 1999 .

[7]  Jon Barwise,et al.  Model-Theoretic Logics , 2016 .

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

[9]  Jouko Väänänen,et al.  Abstract model theory as a framework for universal logic , 2005 .

[10]  Xavier Caicedo,et al.  Lindström's Theorem for Positive Logics, a Topological View , 2015, Logic Without Borders.

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

[12]  Leonard Gillman,et al.  Rings of continuous functions , 1961 .

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

[14]  V. Novak,et al.  A new proof of completeness of fuzzy logic and some conclusions for approximate reasoning , 1995, Proceedings of 1995 IEEE International Conference on Fuzzy Systems..

[15]  Lotfi A. Zadeh,et al.  Fuzzy Logic , 2009, Encyclopedia of Complexity and Systems Science.

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

[17]  Xavier Caicedo,et al.  Continuous Operations on Spaces of Structures , 1995 .

[18]  Petr Hájek,et al.  Rational Pavelka predicate logic is a conservative extension of Łukasiewicz predicate logic , 2000, Journal of Symbolic Logic.

[19]  Alexander Usvyatsov,et al.  On d-finiteness in continuous structures , 2007 .

[20]  Anand Pillay,et al.  Model theory with applications to algebra and analysis , 2008 .

[21]  H. Jerome Keisler,et al.  Categoricity without equality , 2001 .

[22]  Petr Hájek.  Fuzzy Logic and Arithmetical Hierarchy III , 2001, Stud Logica.

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

[24]  Per Lindström.  Omitting uncountable types and extensions of Elementary logic , 2008 .

[25]  Xavier Caicedo.  Compactness and Normality in Abstract Logics , 1993, Ann. Pure Appl. Log..

[26]  Per Lindström,et al.  On Extensions of Elementary Logic , 2008 .

[27]  C. Ward Henson,et al.  Model Theory with Applications to Algebra and Analysis: Model theory for metric structures , 2008 .

[28]  Alexander Usvyatsov,et al.  CONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY , 2008, 0801.4303.

[29]  José Iovino.  On The Maximality of Logics with Approximations , 2001, J. Symb. Log..

[30]  Jan Pavelka,et al.  On Fuzzy Logic III. Semantical completeness of some many-valued propositional calculi , 1979, Math. Log. Q..

[31]  M. García-Matos,et al.  A Framework for Maximality and Interpolation in Abstract Logics with and without Negation , 2004 .