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

*(English)*Zbl 1181.03063The aim of the present paper is to propose an order-theoretic and categorial framework for various constructions and concepts connected with the study of logical consequence relations. 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 [Algebraizable logics. Mem. Am. Math. Soc. 396 (1989; Zbl 0664.03042)] 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 [“Equivalence of consequence operations”, Stud. Log. 83, No. 1–3, 91–110 (2006; Zbl 1106.03059)]. Other authors have extended this result to the cases of \(k\)-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. The main result of this paper subsumes all existing results in the literature and reveals their common character. The proofs are order-theoretic and categorial in nature.

At the beginning, the authors review the case of an algebraizable consequence relation and give an equivalent formulation of the definition in terms of translators, which extends to the situation of consequence relations over sequents. Then they characterize the extensions of these maps to powersets and provide the necessary intuition leading to the definition of a module in the more general setting of complete lattices. They show that consequence relations on a given lattice are in bijective correspondence with closure operators on it. Next, they discuss the appropriate notion of substitution invariance for both consequence relations and closure operators in the setting where the lattice is endowed with the additional structure of a module. This provides the required background for formulating the definition of equivalence of two consequence relations and to prove the main theorem of this paper. The authors use the residuation setting to give characterizations of the notions of similarity and equivalence of consequence relations. They show that every similarity between structural closure operators induced by translators is structural and then put things into a categorial setting by identifying the modules for which equivalences (or structural representations) are induced by translators with the projective modules in the appropriate category. At the same time, cyclic and cyclic projective modules are characterized, while the consequence relations on the set of formulas are shown to be particular cases of cycling projective modules. Finally, it is shown that the assumption of finitarity can be safely added to the preceding study. By working in the appropriate ‘finitary’ subcategory it is proved that the inducing translators can be taken to be finitary if all the other objects involved are assumed finitary. This involves the identification of the notion of regular modules, which are shown to be projective now in the said subcategory.

At the beginning, the authors review the case of an algebraizable consequence relation and give an equivalent formulation of the definition in terms of translators, which extends to the situation of consequence relations over sequents. Then they characterize the extensions of these maps to powersets and provide the necessary intuition leading to the definition of a module in the more general setting of complete lattices. They show that consequence relations on a given lattice are in bijective correspondence with closure operators on it. Next, they discuss the appropriate notion of substitution invariance for both consequence relations and closure operators in the setting where the lattice is endowed with the additional structure of a module. This provides the required background for formulating the definition of equivalence of two consequence relations and to prove the main theorem of this paper. The authors use the residuation setting to give characterizations of the notions of similarity and equivalence of consequence relations. They show that every similarity between structural closure operators induced by translators is structural and then put things into a categorial setting by identifying the modules for which equivalences (or structural representations) are induced by translators with the projective modules in the appropriate category. At the same time, cyclic and cyclic projective modules are characterized, while the consequence relations on the set of formulas are shown to be particular cases of cycling projective modules. Finally, it is shown that the assumption of finitarity can be safely added to the preceding study. By working in the appropriate ‘finitary’ subcategory it is proved that the inducing translators can be taken to be finitary if all the other objects involved are assumed finitary. This involves the identification of the notion of regular modules, which are shown to be projective now in the said subcategory.

Reviewer: Florentina Chirteş (Craiova)

##### MSC:

03G27 | Abstract algebraic logic |

03B22 | Abstract deductive systems |

03G30 | Categorical logic, topoi |

06A15 | Galois correspondences, closure operators (in relation to ordered sets) |

06F05 | Ordered semigroups and monoids |

##### Keywords:

consequence relation; closure operator; algebraizable logic; residuated lattice; module; projective
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\textit{N. Galatos} and \textit{C. Tsinakis}, J. Symb. Log. 74, No. 3, 780--810 (2009; Zbl 1181.03063)

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