Representation theorems and the semantics of (semi)lattice-based logics

This paper gives a unified presentation of various non-classical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, distributive lattices and semilattices) allows to establish a relationship between algebraic models and Kripke-style models, and illustrate the ideas on several examples. Based on this, we present a method for automated theorem proving by resolution for such logics. Other representation theorems, as algebras of sets or as algebras of relations, as well as relational models are also mentioned.

[1]  A. Tarski,et al.  Boolean Algebras with Operators , 1952 .

[2]  Ewa Orlowska,et al.  Relational proof system for relevant logics , 1992, Journal of Symbolic Logic.

[3]  Ivo Düntsch,et al.  Structures with many-valued information and their relational proof theory , 2000, Proceedings 30th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2000).

[4]  Viorica Sofronie-Stokkermans,et al.  Automated Theorem Proving by Resolution for Finitely-Valued Logics Based on Distributive Lattices with Operators , 2001 .

[5]  Richard Routley,et al.  Relevant logics and their rivals , 1982 .

[6]  Chrysafis Hartonas,et al.  Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics , 1997, Stud Logica.

[7]  Szabolcs Mikulás,et al.  Lambek Calculus and its relational semantics: Completeness and incompleteness , 1994, J. Log. Lang. Inf..

[8]  Viorica Sofronie-Stokkermans.  Representation Theorems and Automated Theorem Proving in Non-Classical Logics , 1999 .

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

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

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

[12]  Robert Goldblatt,et al.  Varieties of Complex Algebras , 1989, Ann. Pure Appl. Log..

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

[14]  Harald Ganzinger,et al.  A Resolution-Based Decision Procedure for Extensions of K4 , 1998, Advances in Modal Logic.

[15]  R. Goldblatt.  Mathematics of modality , 1993 .

[16]  Jan Łukasiewicz,et al.  Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls (micro) , 1930 .

[17]  Viorica Sofronie-Stokkermans,et al.  Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of Non-Classical Logics I , 2000, Stud Logica.

[18]  George Georgescu,et al.  Lukasiewicz-Moisil algebras , 2012, Annals of discrete mathematics.

[19]  J. Michael Dunn,et al.  Positive modal logic , 1995, Stud Logica.

[20]  Marcelo F. Frias,et al.  A proof system for fork algebras and its applications to reasoning in logics based on intuitionism , 1995 .

[21]  Alasdair Urquhart,et al.  Duality for algebras of relevant logics , 1996, Stud Logica.

[22]  Wendy MacCaull.  A note on Kripke semantics for residuated logic , 1996, Fuzzy Sets Syst..

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

[24]  Stanley Burris,et al.  The Elements of Universal Algebra , 1981 .

[25]  Ewa Orlowska,et al.  Post relation algebras and their proof system , 1991, [1991] Proceedings of the Twenty-First International Symposium on Multiple-Valued Logic.

[26]  Alasdair Urquhart,et al.  A topological representation theory for lattices , 1978 .

[27]  Wendy MacCaull.  Relational Proof System for Linear and Other Substructural Logics , 1997, Log. J. IGPL.

[28]  Hiroakira Ono,et al.  Logics without the contraction rule , 1985, Journal of Symbolic Logic.

[29]  Wojciech Buszkowski.  Completeness Results for Lambek Syntactic Calculus , 1986, Math. Log. Q..

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

[31]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[32]  Wendy MacCaull,et al.  A Proof System for Dependencies for Information Relations , 2000, Fundam. Informaticae.

[33]  Néstor G. Martínez.  A topological duality for some lattice ordered algebraic structures including ℓ-groups , 1994 .

[34]  M. Stone.  The theory of representations for Boolean algebras , 1936 .

[35]  Viorica Sofronie-Stokkermans.  Priestley Duality for SHn-algebras and Applications to the Study of Kripke-style Models for SHn-logics , 2000 .

[36]  Hans Jürgen Ohlbach,et al.  Translation Methods for Non-Classical Logics: An Overview , 1993, Log. J. IGPL.

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

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

[39]  Wojciech Buszkowski,et al.  Representation of Residuated Semigroups in Some Algebras of Relations (The Method of Canonical Models) , 1997, Fundam. Informaticae.

[40]  Viorica Sofronie-Stokkermans,et al.  On the Universal Theory of Varieties of Distributive Lattices with Operators: Some Decidability and Complexity Results , 1999, CADE.

[41]  Luisa Iturrioz.  Modal operators on symmetrical Heyting algebras , 1982 .

[42]  Wendy MacCaull,et al.  Relational semantics and a relational proof system for full Lambek calculus , 1998, The Journal of Symbolic Logic.

[43]  Luisa Iturrioz.  Symmetrical Heyting Algebras with Operators , 1983, Math. Log. Q..

[44]  David M. Clark,et al.  Natural Dualities for the Working Algebraist , 1998 .

[45]  Chrysafis Hartonas,et al.  Stone duality for lattices , 1997 .

[46]  Ullrich Hustadt,et al.  Resolution-Based Methods for Modal Logics , 2000, Log. J. IGPL.