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Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic ${\cal L}$ is associated with a matrix semantics $Mo{d^{\rm{*}}}{\cal L}$ . This article is a contribution to the systematic study of the so-called truth sets of the matrices in $Mo{d^{\rm{*}}}{\cal L}$ . In particular, we show that the fact that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of ${\cal L}$ . This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ are implicitly definable if and only if the Leibniz operator is injective on deductive filters of ${\cal L}$ over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of ${\cal L}$ to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in $Mo{d^{\rm{*}}}{\cal L}$ that corresponds to the order-reflection of the Leibniz operator.

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[1]Albuquerque, H., Font, J. M., Jansana, R., & Moraschini, T. (2018). Assertional logics, truth-equational logics, and the hierarchies of abstract algebraic logic. In Czelakowski, J., editor. Don Pigozzi on Abstract Algebraic Logic and Universal Algebra. Outstanding Contributions, Vol. 16. Berlin: Springer-Verlag, pp. 5379.
[2]Babyonyshev, S. V. (2003). Fully Fregean logics. Reports on Mathematical Logic, 37, 5977.
[3]Blok, W. J. & Pigozzi, D. (1986). Protoalgebraic logics. Studia Logica, 45, 337369.
[4]Blok, W. J. & Pigozzi, D. (1989). Algebraizable Logics. Memoirs of the American Mathematical Society, Vol. 396. Providence: American Mathematical Society.
[5]Blok, W. J. & Pigozzi, D. (1992). Algebraic semantics for universal Horn logic without equality. In Romanowska, A. and Smith, J. D. H., editors. Universal Algebra and Quasigroup Theory. Berlin: Heldermann, pp. 156.
[6]Blok, W. J. & Rebagliato, J. (2003). Algebraic semantics for deductive systems. Studia Logica, Special Issue on Abstract Algebraic Logic, Part II, 74(5), 153180.
[7]Bou, F. & Rivieccio, U. (2011). The logic of distributive bilattices. Logic Journal of the Interest Group in Pure and Applied Logics, 19(1), 183216.
[8]Czelakowski, J. (1985). Algebraic aspects of deduction theorems. Studia Logica, 44, 369387.
[9]Czelakowski, J. (1986). Local deductions theorems. Studia Logica, 45, 377391.
[10]Czelakowski, J. (2001). Protoalgebraic Logics. Trends in Logic—Studia Logica Library, Vol. 10. Dordrecht: Kluwer Academic Publishers.
[11]Czelakowski, J. (2003). The Suszko operator. Part I. Studia Logica, Special Issue on Abstract Algebraic Logic, Part II, 74(5), 181231.
[12]Czelakowski, J. & Jansana, R. (2000). Weakly algebraizable logics. The Journal of Symbolic Logic, 65(2), 641668.
[13]Czelakowski, J. & Pigozzi, D. (2004). Fregean logics. Annals of Pure and Applied Logic, 127(1–3), 1776.
[14]Czelakowski, J. & Pigozzi, D. (2004). Fregean logics with the multiterm deduction theorem and their algebraization. Studia Logica, 78(1–2), 171212.
[15]Descalço, L. & Martins, M. A. (2005). On the injectivity of the Leibniz operator. Bulletin of the Section of Logic, 34(4), 203211.
[16]Font, J. M. (2016). Abstract Algebraic Logic - An Introductory Textbook. Studies in Logic - Mathematical Logic and Foundations, Vol. 60. London: College Publications.
[17]Font, J. M., Guzmán, F., & Verdú, V. (1991). Characterization of the reduced matrices for the -fragment of classical logic. Bulletin of the Section of Logic, 20, 124128.
[18]Font, J. M. & Jansana, R. (2009). A General Algebraic Semantics for Sentential Logics (second edition 2017). Lecture Notes in Logic, Vol. 7. Cambridge: Cambridge University Press.
[19]Font, J. M., Jansana, R., & Pigozzi, D. (2003). A survey on abstract algebraic logic. Studia Logica, Special Issue on Abstract Algebraic Logic, Part II, 74(1–2), 13–97. With an “Update” in 91(2009), 125130.
[20]Font, J. M. & Moraschini, T. (2014). Logics of varieties, logics of semilattices, and conjunction. Logic Journal of the Interest Group in Pure and Applied Logics, 22, 818843.
[21]Font, J. M. & Moraschini, T. (2014). A note on congruences of semilattices with sectionally finite height. Algebra Universalis, 72(3), 287293.
[22]Font, J. M. & Verdú, V. (1991). Algebraic logic for classical conjunction and disjunction. Studia Logica, Special Issue on Algebraic Logic, 50, 391419.
[23]Ginsberg, M. L. (1988). Multivalued logics: A uniform approach to inference in artificial intelligence. Computational Intelligence, 4, 265316.
[24]Herrmann, B. (1993). Algebraizability and Beth’s theorem for equivalential logics. Bulletin of the Section of Logic, 22(2), 8588.
[25]Herrmann, B. (1993). Equivalential Logics and Definability of Truth. Ph.D. Thesis, Berlin, Freie Universität.
[26]Herrmann, B. (1997). Characterizing equivalential and algebraizable logics by the Leibniz operator. Studia Logica, 58, 305323.
[27]Jansana, R. & Moraschini, T. (2017). Advances in the Theory of the Leibniz Hierarchy. Manuscript.
[28]Moraschini, T. (2016). Investigations in the Role of Translations in Abstract Algebraic Logic. Ph.D. Thesis, University of Barcelona.
[29]Moraschini, T. (2016). On the Complexity of the Leibniz Hierarchy. Submitted manuscript.
[30]Moraschini, T. (2018). A computational glimpse at the Leibniz and Frege hierarchies. Annals of Pure and Applied Logic, 169(1), 120.
[31]Noguera, C. & Cintula, P. (2013). The proof by cases property and its variants in structural consequence relations. Studia Logica, 101, 713747.
[32]Raftery, J. G. (2006). The equational definability of truth predicates. Reports on Mathematical Logic, 41, 95149.
[33]Raftery, J. G. (2011). A perspective on the algebra of logic. Quaestiones Mathematicae, 34, 275325.
[34]Rautenberg, W. (1993). On reduced matrices. Studia Logica, 52, 6372.
[35]Rivieccio, U. (2010). An Algebraic Study of Bilattice-based Logics. Ph.D. Dissertation, University of Barcelona.
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