Definability and Interpolation in Non-Classical Logics

Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra.In this paper we present an overview of results on interpolation and definability in modal and positive logics,and also in extensions of Johansson's minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi.

[1]  Silvio Ghilardi,et al.  Sheaves, games, and model completions - a categorical approach to nonclassical propositional logics , 2011, Trends in logic.

[2]  L. L. Maksimova,et al.  ALGEBRAIC APPROACH TO NON-CLASSICAL LOGICS , 2008 .

[3]  L. L. Maksimova,et al.  Modal logics and varieties of modal algebras: The beth properties, interpolation, and amalgamation , 1992 .

[4]  Tomasz Kowalski,et al.  Splittings in the variety of residuated lattices , 2000 .

[5]  L. Maksimova.  The Projective Beth Property and Interpolation in Positive and Related Logics , 2006 .

[6]  A. V. Chagrov.  Undecidable properties of extensions of the logic of provability , 1990 .

[7]  Michael Zakharyaschev,et al.  Modal Logic , 1997, Oxford logic guides.

[8]  L. L. Maksimova.  Decidable properties of logical calculi and of varieties of algebras , 2006 .

[9]  J. Czelakowski.  Logical matrices and the amalgamation property , 1982 .

[10]  Larisa Maksimova,et al.  Complexity of interpolation and related problems in positive calculi , 2002, Journal of Symbolic Logic.

[11]  A. Tarski,et al.  Cylindric Algebras. Part II , 1988 .

[12]  Wolfgang Rautenberg,et al.  Klassische und nichtklassische Aussagenlogik , 1979 .

[13]  Larisa Maksimova,et al.  Complexity of Some Problems in Modal and Intuitionistic Calculi , 2003, CSL.

[14]  H. Rasiowa.  An Algebraic Approach To Non Classical Logics , 1974 .

[15]  G. Kreisel.  Reviews. Kurt Schütte. Beweistheorie . Springer-Verlag, Berlin-Göttingen-Heidelberg 1960, X + 355 pp. , 1960 .

[16]  L. Maksimova.  Interpolation theorems in modal logics and amalgamable varieties of topological Boolean algebras , 1979 .

[17]  L. Maksimova.  Interpolation theorems in modal logics. Sufficient conditions , 1980 .

[18]  Ildikó Sain,et al.  Beth's and Craig's properties via epimorphisms and amalgamation in algebraic logic , 1988, Algebraic Logic and Universal Algebra in Computer Science.

[19]  L. Maksimova,et al.  Definability in Normal Extensions of S4 , 2004 .

[20]  A. V. Chagrov.  Undecidable properties of extensions of a provability logic. II , 1990 .

[21]  Nobu-Yuki Suzuki,et al.  Halldén-Completeness in Super-Intuitionistic Predicate Logics , 2003, Stud Logica.

[22]  L. L. Maksimova,et al.  Projective beth properties in modal and superintuitionistic logics , 1999 .

[23]  L. L. Maksimova,et al.  Decidability of the Projective Beth Property in Varieties of Heyting Algebras , 2001 .

[24]  L. Maksimova.  Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras , 1977 .

[25]  Roger D. Maddux,et al.  Algebraic Logic and Universal Algebra in Computer Science , 1990, Lecture Notes in Computer Science.

[26]  Larisa Maksimova.  Strongly Decidable Properties of Modal and Intuitionistic Calculi , 2000, Log. J. IGPL.

[27]  L. L. Maksimova.  Absence of the interpolation property in the consistent normal modal extensions of the dummett logic , 1982 .

[28]  A. Wronski.  On a Form of Equational Interpolation Property , 1985 .

[29]  J. Czelakowski,et al.  Amalgamation and Interpolation in Abstract Algebraic Logic , 2003 .

[30]  H. Ono.  Interpolation and the Robinson property for logics not closed under the Boolean operations , 1986 .

[31]  L. L. Maksimova,et al.  Interpolation properties of superintuitionistic logics , 1979 .

[32]  H. Ono.  Substructural Logics and Residuated Lattices — an Introduction , 2003 .

[33]  A. Robinson.  A Result on Consistency and its Application to theTheory of Definition , 1956 .

[34]  L. Maksimova.  Interpolation and Definability in Extensions of the Minimal Logic , 2005 .

[35]  Larisa Maksimova.  Restricted Interpolation in Modal Logics , 2002, Advances in Modal Logic.

[36]  William Craig,et al.  Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory , 1957, Journal of Symbolic Logic.

[37]  L. L. Maksimova,et al.  An analog of Beth's theorem in normal extensions of the modal logic K4 , 1992 .

[38]  L. Maksimova.  Implicit Definability and Positive Logics , 2003 .

[39]  Dov M. Gabbay,et al.  Investigations in modal and tense logics with applications to problems in philosophy and linguistics , 1976 .

[40]  Larisa Maksimova,et al.  Intuitionistic logic and implicit definability , 2000, Ann. Pure Appl. Log..

[41]  Larisa Maksimova,et al.  Projective Beth Property in Extensions of Grzegorczyk Logic , 2006, Stud Logica.

[42]  L. L. Maksimova,et al.  Restricted Interpolation and the Projective Beth Property in Equational Logic , 2003 .

[43]  Evert W. Beth,et al.  On Padoa’s Method in the Theory of Definition , 1953 .

[44]  D. Gabbay,et al.  Interpolation and Definability: Modal and Intuitionistic Logic , 2005 .

[45]  Larisa Maksimova.  Projective Beth's Properties in Infinite Slice Extensions of the Modal Logic K4 , 2000, Advances in Modal Logic.

[46]  Dov M. Gabbay,et al.  Interpolation in Practical Formal Development , 2001, Log. J. IGPL.

[47]  D. Gabbay.  Semantical investigations in Heyting's intuitionistic logic , 1981 .

[48]  R. Sikorski,et al.  The mathematics of metamathematics , 1963 .

[49]  Larisa Maksimova,et al.  Amalgamation and interpolation in normal modal logics , 1991, Stud Logica.

[50]  E. Hoogland.  Definability and Interpolation: Model-theoretic investigations , 2001 .