Logic and Implication: An Introduction to the General Algebraic Study of Non-Classical Logics

Springer Verlag (2021)
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Abstract

This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions. Based on their recent work, the authors develop a powerful uniform framework for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics. The book may be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics.

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Chapters

Weakly Implicative Logics

This chapter is the real beginning of our story and hence it is devoted to presenting its main object of study: the class of weakly implicative logics.We start by introducing basic syntactical notions and giving a purely syntactical definition of logics as mathematical objects . After testing the de... see more

Introduction

This chapter gives the main motivations of the book and some historical perspective.

First-Order Predicate Logics

The last chapter gives a short introduction to the study of first-order predicate logics built over weakly implicative logics. We follow a semantics-first approach in which we start from semantically defined predicate logics and then propose suitable Hilbert-style axiomatizations and prove correspon... see more

On Lattice and Residuated Connectives

Clearly, not only implication, but also other logical connectives are crucial for the theory and the applications of particular logics. Hence, this chapter is devoted to the study of two groups of important connectives: lattice and residuated connectives.We start by exploring the logical and algebra... see more

Semilinear Logics

This chapter focuses on the other family that motivated the general study of logics with implication: semilinear logics, defined as logics complete with respect to linearly ordered reduced matrices.We start by formulating and proving useful characterizations of semilinear logics in terms of linear f... see more

Generalized Disjunctions

The general kind of disjunctions enjoying the proof by cases property obtained for substructural logics in the previous chapter motivates the abstract study of generalized disjunctions that we present in the fifth chapter.First, we introduce several forms of proof by cases property and corresponding... see more

Completeness Properties

This chapter presents the foundations of the theory of logical matrices with a special focus on the question of which classes of matrices provide a complete semantics for a given logic. We identify three kinds of completeness based on how we restrict the cardinality of the sets of premises: we disti... see more

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Citations of this work

No cause for collapse.Dustin Gooßens & Andrew Tedder - 2023 - Asian Journal of Philosophy 2 (2):1-19.
On Implicative and Positive Implicative GE Algebras.Andrzej Walendziak - 2020 - Bulletin of the Section of Logic:19 pp..

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