Abstract
We consider the equationally orderable quasivarieties and associate with them deductive systems defined using the order. The method of definition of these deductive systems encompasses the definition of logics preserving degrees of truth we find in the research areas of substructural logics and mathematical fuzzy logic. We prove several general results, for example that the deductive systems so defined are finitary and that the ones associated with equationally orderable varieties are congruential.
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References
Babyonishev S. V.: Fully Fregean logics. Reports on Mathematical Logic 37, 59–78 (2003)
Bergman J., Blok W. J.: Algebras defined from ordered sets and the varieties they generate. Order 23, 65–88 (2006)
Bou F., Esteva F., Font J. M., Gil A., Godo L., Torrens A., Verdú V.: Logics preserving degrees of truth from varieties of residuated lattices. Journal of Logic and Computation 19, 1031–1069 (2009)
Celani C., Jansana R: A new semantics for positive modal logic. Notre Dame Journal of Formal Logic 38, 1–18 (1997)
Celani S., Jansana R.: A closer look at some subintuitionistic logics. Notre Dame Journal of Formal Logic 42, 225–255 (2001)
Czelakowski J.: Protoalgebraic Logics. Kluwer Academic Publishers, Dordrecht (2001)
Czelakowski, J., The Suszko Operator. Part I, Studia Logica, Special Issue on Algebraic Logic II 74:181–231, 2003.
Dunn J. M.: Positive modal logic. Studia Logica 55, 301–317 (1995)
Figallo A. Jr., Ramón G., Saad S.: A note on Hilbert algebras with infimum. Mathematica Contemporânea 24, 23–37 (2003)
Font J. M.: On substructural logics preserving degrees of truth. Bulletin of the Section of Logic 36, 117–129 (2009)
Font J. M.: Taking degrees of truth seriously. Studia Logica 91, 383–406 (2009)
Font, J. M., and R. Jansana, A General Algebraic Semantics for Sentential Logics, Lecture Notes in Logic, Vol. 7, Springer, 1996, Second revised edition by ASL 2009, available at http://projecteuclid.org/euclid.lnl/1235416965.
Font, J. M., R. Jansana, and D. Pigozzi, A survey of abstract algebraic logic, Studia Logica, Special Issue on Algebraic Logic 74:13–97, 2003.
Font J. M., Rodríguez G.: Algebraic study of two deductive systems of relevance logic. Notre Dame Journal of Formal Logic 35, 369–397 (1994)
Font, J. M., and V. Verdú, Algebraic logic for classical conjunction and disjunction, Studia Logica, Special Issue on Algebraic Logic 50:391–419, 1991.
Gehrke, M., R. Jansana, and A. Palmigiano, Canonical extensions for congruential logics with the deduction theorem, Annals of Pure and Applied Logic 161:1502–1519, 2010.
Jansana R.: Selfextensional logics with a conjunction. Studia Logica 84, 63–104 (2006)
Jansana, R., Selfextensional logics with implication, in J.-Y. Béziau (ed.), Universal Logic, Birkhäuser, Basel, (2007).
Pigozzi, D., Partially ordered varieties and quasivarieties, Manuscript available at http://orion.math.iastate.edu/dpigozzi/, 2004.
Pynko A.: Definitional equivalence and algebraizability of generalized logical systems. Annals of Pure and Applied Logic 98, 1–68 (1999)
Raftery J. G.: Order algebraizable logics. Annals of Pure and Applied Logic 164, 251–283 (2013)
Rasiowa H.: An Algebraic Approach to Non-classical Logics. North-Holland Publishing Company, Amsterdam (1974)
Wójcicki, R., Logical matrices strongly adequate for structural sentential calculi, Bulletin de l’Académie Polonaise des Sciences Classe III XVII:333–335, 1969.
Wójcicki, R. Theory of Logical Calculi. Basic Theory of Consequence Operations (Synthese Library, vol. 199), Reidel, Dordrecht, 1988.
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Jansana, R. On the Deductive System of the Order of an Equationally Orderable Quasivariety. Stud Logica 104, 547–566 (2016). https://doi.org/10.1007/s11225-016-9650-7
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DOI: https://doi.org/10.1007/s11225-016-9650-7