Order algebraizable logics

Annals of Pure and Applied Logic 164 (3):251-283 (2013)
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Abstract

This paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic . Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: it is injective, it is isotonic, it commutes with the inverse image operator of any algebraic homomorphism, and it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties –

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Citations of this work

Admissible Rules and the Leibniz Hierarchy.James G. Raftery - 2016 - Notre Dame Journal of Formal Logic 57 (4):569-606.
The Poset of All Logics II: Leibniz Classes and Hierarchy.R. Jansana & T. Moraschini - 2023 - Journal of Symbolic Logic 88 (1):324-362.
On the complexity of the Leibniz hierarchy.Tommaso Moraschini - 2019 - Annals of Pure and Applied Logic 170 (7):805-824.

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References found in this work

Algebraizable Logics.W. J. Blok & Don Pigozzi - 2022 - Advanced Reasoning Forum.
Modal logic.Alexander Chagrov - 1997 - New York: Oxford University Press. Edited by Michael Zakharyaschev.
Modal Logic.Yde Venema, Alexander Chagrov & Michael Zakharyaschev - 2000 - Philosophical Review 109 (2):286.
A survey of abstract algebraic logic.J. M. Font, R. Jansana & D. Pigozzi - 2003 - Studia Logica 74 (1-2):13 - 97.
Protoalgebraic logics.W. J. Blok & Don Pigozzi - 1986 - Studia Logica 45 (4):337 - 369.

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