Journal of Symbolic Logic

Equivalence of consequence relations: an order-theoretic and categorical perspective

Nikolaos Galatos and Constantine Tsinakis

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Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be defined syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in [4] that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in [3]. Other authors have extended this result to the cases of k-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature.

Article information

J. Symbolic Logic, Volume 74, Issue 3 (2009), 780-810.

First available in Project Euclid: 16 June 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03G10: Lattices and related structures [See also 06Bxx]
Secondary: 06F05: Ordered semigroups and monoids [See also 20Mxx]

Consequence relation closure operator equivalence algebraizable residuated lattice module projective


Galatos, Nikolaos; Tsinakis, Constantine. Equivalence of consequence relations: an order-theoretic and categorical perspective. J. Symbolic Logic 74 (2009), no. 3, 780--810. doi:10.2178/jsl/1245158085.

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