The Poset of All Logics III: Finitely Presentable Logics
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The Poset of All Logics III: Finitely Presentable Logics
Abstract. A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite rules. It is proved that binary non-indexed products of logics that are both finitely presentable and finitely equivalential are essentially finitely presentable. This result does not extend to binary non-indexed products of arbitrary finitely presentable logics, as shown by a counterexample. Finitely presentable logics are then exploited to introduce finitely presentable Leibniz classes, and to draw a parallel between the Leibniz and the Maltsev hierarchies. Keywords: Abstract algebraic logic, Algebraizable logic, Leibniz hierarchy, Maltsev conditions, Finitely presentable logic.
1.
Introduction
The interpretability relation introduced in [18] is a preorder on the class of all (propositional) logics. Its associated partially ordered collection Log, consisting of equivalence classes of equi-interpretable logics, was investigated in [18–20] under the name of the poset of all logics. We shall exploit this formalism to draw a precise relation between the Leibniz and Maltsev hierarchies studied, respectively, in abstract algebraic logic and universal algebra. The Maltsev hierarchy is a taxonomy of varieties of algebras in terms of syntactic principles describing the structure of congruence lattices [17,21, 28,32,34]. The Leibniz hierarchy [19] plays a similar role in algebraic logic, providing a classification of logics in terms of sequences of rule schemata that govern the interplay between lattices of logical theories and congruence lattices [2,3,8,10,22,29]. Even though the apparent analogy between the Maltsev and Leibniz hierarchies was fairly well known [30] and inspired some investigations in algebraic logic [5,6,23,24], the problem of understanding whether these hierarchies could be unified remained open. This should probably be attributed to the fact that, until recently, a framework in which a positive solution could be formulated was missing. In this paper we show that such a framework is provided by the poset of all logics Log.
Presented by Jacek Malinowski; Received July 24, 2019
Studia Logica https://doi.org/10.1007/s11225-020-09916-z
c Springer Nature B.V. 2020
R. Jansana, T. Moraschini
To this end, it is convenient to introduce some new concept. A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite axioms and rules. Moreover, finitely presentable Leibniz classes are the classes of logics that can be faithfully identified with filters of Log generated by (equivalence classes of) logics that are both finitely presentable and finitely equivalential. Equivalently, they can be characterized in terms of closure properties as the classes of finitely equivalential logics closed under the formation of term-equivalent logics, compatible expansions, and binary non-indexed products that, moreover, satisfy the following requirement: each of their members compatibly extends some of their fi
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