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THE POSET OF ALL LOGICS II: LEIBNIZ CLASSES AND HIERARCHY

Part of: Varieties Model theory Algebraic logic

Published online by Cambridge University Press:  10 June 2021

R. JANSANA  and
T. MORASCHINI
R. JANSANA
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BARCELONA CARRER DE MONTALEGRE 6 08001 BARCELONA, SPAIN E-mail: jansana@ub.edu
T. MORASCHINI*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BARCELONA CARRER DE MONTALEGRE 6 08001 BARCELONA, SPAIN E-mail: jansana@ub.edu
*
E-mail: tommaso.moraschini@ub.edu
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Abstract

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.

Keywords

Abstract algebraic logicLeibniz hierarchyLeibniz conditionMaltsev conditioninterpretabilityposet of all logics

MSC classification

Primary: 03G27: Abstract algebraic logic
Secondary: 03C05: Equational classes, universal algebra 08B05: Equational logic, Mal′cev (Mal′tsev) conditions
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 1 , March 2023 , pp. 324 - 362
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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