Symmetric Generalized Galois Logics

Symmetric generalized Galois logics (i.e., symmetric gGls) are distributive gGls that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGls by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGls. We generalize the weak distributivity laws between fusion and fission to interactions of certain monotone operations within distributive supergGls. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGls and classes of structures for them.

[1]  Joachim Lambek,et al.  On the Calculus of Syntactic Types , 1961 .

[2]  Alonzo Church,et al.  Introduction to Mathematical Logic , 1991 .

[3]  N. Belnap,et al.  Entailment. The Logic of Relevance and Necessity. Volume I , 1978 .

[4]  Katalin Bimbó,et al.  Dual Gaggle Semantics for Entailment , 2009, Notre Dame J. Formal Log..

[5]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[6]  Katalin Bimbó,et al.  Two Extensions of the Structurally Free Logic LC* , 1998, Log. J. IGPL.

[7]  Willard Van Orman Quine,et al.  Philosophy of Logic. , 1988 .

[8]  Alasdair Urquhart.  Katalin Bimbó and J. Michael Dunn. Relational semantics of nonclassical logical calculi. CSLI Lecture Notes, no. 188. CSLI Publications, Stanford University, 2008, x + 382 pp. , 2010, The Bulletin of Symbolic Logic.

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

[10]  Michael Moortgat,et al.  Symmetries in Natural Language Syntax and Semantics: The Lambek-Grishin Calculus , 2007, WoLLIC.

[11]  Hilary A. Priestley,et al.  Representation of Distributive Lattices by means of ordered Stone Spaces , 1970 .

[12]  J. Michael Dunn,et al.  Relevance Logic and Entailment , 1986 .

[13]  M. Stone.  Topological representations of distributive lattices and Brouwerian logics , 1938 .

[14]  Gerard Allwein,et al.  Kripke models for linear logic , 1993, Journal of Symbolic Logic.

[15]  Greg Restall,et al.  An Introduction to Substructural Logics , 2000 .

[16]  Katalin Bimbó,et al.  Generalized Galois Logics: Relational Semantics of Nonclassical Logical Calculi , 2008 .

[17]  J. Dunn.  Partial-Gaggles Applied to Logics with Restricted Structural Rules , 1991 .

[18]  J. Michael Dunn,et al.  Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation, Implication, and Various Logical Operations , 1990, JELIA.

[19]  B. Davey,et al.  Introduction to Lattices and Order: Appendix B: further reading , 2002 .

[20]  Nuel D. Belnap,et al.  Entailment : the logic of relevance and necessity , 1975 .

[21]  J. Lambek.  The Mathematics of Sentence Structure , 1958 .

[22]  Katalin Bimbó.  Semantics for Structurally Free Logics LC+ , 2001, Log. J. IGPL.

[23]  Donald M. Borchert.  Encyclopedia of Philosophy , 1967 .

[24]  Roman Jakobson,et al.  Structure of Language and Its Mathematical Aspects , 1961 .

[25]  R. Meyer,et al.  Algebraic analysis of entailment I , 1972 .

[26]  Robert K. Meyer,et al.  Combinators and Structurally Free Logic , 1997, Log. J. IGPL.

[27]  Gary M. Hardegree,et al.  Algebraic Methods in Philosophical Logic , 2001 .

[28]  Katalin Bimbó,et al.  Relational semantics of nonclassical logical calculi. CSLI Lecture Notes, no. 188 , 2010 .