We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus and by a set of conditions on its closure operator.
Concerning the relation between the two logics, we prove that the truth-preserving logic is the extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.
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- Ordering protoalgebraic logics | Journal of Logic and Computation | Oxford Academic.
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Implicational (semilinear) logics III: completeness properties
Abstract Algebraic Logic is a relatively new subfield of Mathematical Logic. It is a natural evolution of Algebraic Logic, a branch of Mathematical Logic studying logical systems by giving them an algebra-based semantics. It can be traced back to George Boole and his study of classical propositional logic by means of a two-element algebra that became its canonical semantics. Other non-classical logics enjoy a strong connection with algebras as well intuitionistic logic and Heyting algebras, substructural logic and residuated lattices, etc. Abstract Algebraic Logic AAL was born as the natural next step to be taken in this evolution: the abstract study of logical systems through their interplay with algebraic semantics.
One of the starting points of AAL is the book by Helena Rasiowa  where she studies logics possessing a reasonable implication connective. A crucial technical notion used is this process is the Leibniz operator, which maps any theory of a logic to the congruence relation of the formulae which are provably equivalent in the presence of such theory.
Protoalgebraic Logics | Janusz Czelakowski | Springer
Logics were classified by means of properties of Leibniz operator, which gave rise to the two-dimensional Leibniz hierarchy. This classification, starting with the biggest class of protoalgebraic logics, became the core theory of AAL due to its robustness, the characterizations of its classes, and their usefulness for obtaining bridge theorems, i. The aim of this course is to present a self-contained introduction to AAL. For didactic reasons we present the full Leibniz hierarchy at the end of the tutorial only and, for most of the time, we simplify our account to yet another generalization of Rasiowa approach: the class of weakly implicative logics [2,3].
Presently distributed by the Association of Symbolic Logic. Lecture notes: A and B. Abstract algebraic logic AAL deals with the algebraic study of logics in a way that views both terms "algebraic" and "logic" under a general perspective.
Algebraic Propositional Logic
It studies different abstract processes of assigning a class of algebras to a given logic, and the general properties of these processes, their limitations and their different variants or enhancements, instead of focusing on the resulting class of algebras for a particular logic. The clarification of the different ways in which a logic can be algebraized has thrown light on the behaviour of several logical systems that are not amenable to the standard settings, but it has also clarified the deep roots of the algebraic properties of classical logic.
In its recent developments AAL has proven useful for the study of non-monotonic and substructural logics, and of other Gentzen-style systems formulated with hypersequents or many-sided sequents. It does also provide definite criteria to clarify in which sense a logic is said to correspond to a given class of algebras, and it has stimulated a reflection on the issue of what a logic is. This tutorial is conceived as a reading guide of the available scholarly material. AAL is a mathematical theory, and by now it is deep and technically involved, with several ramifications and its own terminology and notation.
A 3-session tutorial can only show the essentials, that is, the "what" and the "why", but not much of the "how".
Related Protoalgebraic Logics: Volume 10 (Trends in Logic)
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