Abstract
Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a ⊙ b ≤ a for all b, then a ⊙ a* ⊙ a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on a Stone style representation theorem for complete lattices. A Rasiowa/Sikorski style semantic tableau system is also presented with the property that if all branches of a tableau are closed, then the formula in question is a theorem of Gelfand Logic. An open branch in a completed tableaux guarantees the existence of an Kripke model in which the formula is not valid; hence it is not a theorem of Gelfand Logic.
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References
Alan Ross Anderson and Nuel D. Belnap, Jr., Entailment: The Logic of Relevance and Necessity, volume I, Princeton University Press, 1975.
Alan Ross Anderson, Nuel D. Belnap Jr., and J. Michael Dunn, Entailment: The Logic of Relevance and Necessity, volume II, Princeton University Press, 1992.
Gerard Allwein and J. Michael Dunn, 'Kripke models for linear logic', Journal of Symbolic Logic 58: 514-545, 1993.
Gerard Allwein, The Duality of Algebraic and Kripke Models for Linear Logic, PhD thesis, Indiana University, 1992.
Gerard Allwein and Wendy MacCaull, 'Duality for complete lattices', Technical report, Indiana University, 1998.
Carolyn Brown and Douglas Gurr, 'A representation theorem for quantales', Journal of Pure and Applied Algebra 85: 27-42, 1993.
Brian F. Chellas, Modal Logic, an Introduction, Cambridge University Press, 1980.
J. Michael Dunn, 'Algebraic completeness results for r-mingle and its extensions', Journal of Symbolic Logic 35:1-13, 1970.
J. Michael Dunn, 'Gaggle theory: An abstraction of galois connections and residuation with applications to negation and various logical operations', in Logics in AI, Proceedings European Workshop JELIA, LNCS 478, Springer-Verlag, 1990.
J. Michael Dunn, 'Partial gaggle theory applied to logics with restricted structural rules', in K. Došen and P. Schroeder-Heister, editors, Substructural Logics, 1993.
Dov Gabbay, 'Fibered semantics and the weaving of logics. Part 1: Modal and intuitionistic logics', Journal of Symbolic Logic 61:1055-1119, 1996.
C.A.R. Hoare and He Jifeng, 'A weakest pre-specification', Inform. Process. Lett. 24: 127-132, 1987.
Wendy MacCaull, 'Relational proof theory for linear and other substructural logics', Journal of the Interest Group of Pure and Applied Logic 5:673-697, 1997.
Wendy MacCaull, 'Relational semantics and a relational proof system for full Lambek calculus', Journal of Symbolic Logic 63:623-637, 1998.
C. J. Mulvey and M. Nawaz, 'Quantales: Quantal sets', in Ulrich Hoehle and Erich Peter Klement, editors, Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer Academic Publishers, 1995.
Chris Mulvey and Joan Wick Pelletier, 'The quantization of the calculus of relations', in “Canadian Mathematical Society Conference Proceedings”, American Mathematical Society, Providence, volume 13, pages 345-360, 1992.
Chris Mulvey and Joan Wick Pelletier, 'The quantization of points', to appear in Journal of Pure and Applied Algebra, 2000.
Chris Mulvey, &, Suppl. Rend. Circ. Mat. Palermo Ser. II 12:99-104, 1986.
Hiroakira Ono and Yuichi Komuri, 'Logics without the contraction rule', Journal of Symbolic Logic 50:169-201, 1985.
Hiroakira Ono, 'Semantics for substructural logics', in K. Došen and P. Schroeder-Heister, editors, Substructural Logics, pages 259-291, 1993.
Ewa Or9łowska, 'Relational proof system for relevant logics', Journal of Symbolic Logic 57:1425-1440, 1992.
Ewa Or9łowska, 'Relational semantics for logics: formulas are relations', in Jan Wolenski, editor, Philosophical Logic in Poland, pages 167-186, Kluwer, 1994.
Ewa Or9łowska, 'Relational environment for semigroup logics', in André Fuhrmann and Hans Rott, editors, Logic, Action, and Information, Essays on Logic in Philosophy and Artificial Intelligence, pages 351-391, Walter de Gruyter, (Berlin, New York), 1996.
J. Wick Pelletier and J. RosickÝ, 'Simple involutive quantales', Journal of Algebra 195:367-386, 1997.
H. A. Priestley, 'Ordered topological spaces and the representation of distributive lattices', in Proceedings of the London Mathematical Society, volume 24, pages 507-530, 1972.
Richard Routley and Robert K. Meyer, 'The semantics of entailment', in H. Leblanc, editor, Truth, Syntax, and Modality, pages 194-243, North Holland, 1973.
Kimmo Rosenthal, Quantales and their applications, Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, Copublished in the United States with John Wiley and Sons, Inc., New York, 1990.
Helena Rasiowa and Roman Sikorski, The Mathematics of Metamathematics, Polish Scientific Publishers, 1963.
M.H. Stone, 'The theory of representation for Boolean algebras', Transactions of the American Mathematical Society 40:37-111, 1936.
Alasdair Urquhart, 'A topological representation theory for lattices', Algebra Universalis 8:45-58, 1978.
Johan van Benthem, 'Correspondence theory', in Handbook of Philosophical Logic, Vol. II, Synthese Lib. 165, pages 167-247, D. Reidel Publishing Company, 1984.
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Allwein, G., MacCaull, W. A Kripke Semantics for the Logic of Gelfand Quantales. Studia Logica 68, 173–228 (2001). https://doi.org/10.1023/A:1012495106338
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DOI: https://doi.org/10.1023/A:1012495106338