Abstract Algebraic Logic - An Introductory Textbook

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Josep Maria Font
Title: Abstract Algebraic Logic - An Introductory Textbook
Series: Studies in Logic
Volume: 60
Publisher: College Publications
City: London
Year: 2016

Abstract algebraic logic is the more general and abstract side of algebraic logic, the branch of mathematics that studies the connections between logics and their algebra-based semantics. This emerging subfield of mathematical logic is increasingly becoming an indispensable tool to approach the algebraic study of any (mainly sentential) logic in a systematic way.

This book takes a bottom-up approach and guides readers, by means of successive steps of generalization and abstraction, to meet more and more complicated algebra-based semantics. An entire chapter is devoted to Blok and Pigozzi's theory of algebraizable logics, proving the main theorems and incorporating later developments by other scholars. After a chapter on the basics of the classical theory of matrices, one chapter is devoted to an in-depth exposition of the semantics of generalized matrices. Two more avanced chapters provide introductions to the two hierachies that organize the logical landscape according to the criteria of abstract algebraic logic, the Leibniz hierarchy and the Frege hierarchy. Dozens of examples of particular logics are presented and classified, among which, needless to say, many-valued logics. The properties of particular logics are coveniently summarized in an Appendix.

The book is addressed to mathematicians and logicians with little or no previous exposure to algebraic logic. Some acquaintance with non-classical logics is desirable. The book is written with students (or beginners in the field) in mind, and combines a textbook style in its main sections, including more than 400 carefully graded exercises, with a survey style in the discussion of newer research directions. The book includes some historical notes, numerous bibliographic references, and a set of six comprehensive indices, one of them being an index of logics.

Table of Contents

A letter to the reader

Introduction and Reading Guide

1 Mathematical and logical preliminaries

2 The first steps in the algebraic study of a logic

3 The semantics of algebras

4 The semantics of matrices

5 The semantics of generalized matrices

6 Introduction to the Leibniz hierarchy

7 Introduction to the Frege hierarchy

Appendix Summary of properties of particular logics