This book gives a comprehensive introduction to Universal Algebraic Logic. The three main themes are (i) universal logic and the question of what logic is, (ii) duality theories between the world of logics and the world of algebra, and (iii) Tarskian algebraic logic proper including algebras of relations of various ranks, cylindric algebras, relation algebras, polyadic algebras and other kinds of algebras of logic. One of the strengths of our approach is that it is directly applicable to a wide range of logics including not only propositional logics but also e.g. classical first order logic and other quantifier logics. Following the Tarskian tradition, besides the connections between logic and algebra, related logical connections with geometry and eventually spacetime geometry leading up to relativity are also part of the perspective of the book. Besides Tarskian algebraizations of logics, category theoretical perspectives are also touched upon.
This book, apart from being a monograph containing state of the art results in algebraic logic, can be used as the basis for a number of different courses intended for both novices and more experienced students of logic, mathematics, or philosophy. For instance, the first two chapters can be used in their own right as a crash course in Universal Algebra.
Author(s): Hajnal Andréka, Zalán Gyenis, István Németi, Ildikó Sain
Series: Studies in Universal Logic
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 336
City: Cham
Preface
Acknowledgement
Contents
Chapter 1 Notation, Elementary Concepts
1.1 Sets, classes, tuples, simple operations on sets
1.2 Binary relations, equivalence relations, functions
1.3 Orderings, ordinals, cardinals
1.4 Sequences
1.5 Direct product of families of sets
1.6 Relations of higher ranks
1.7 Closure systems
1.8 First{order logic (FOL)
Chapter 2 Basics from Universal Algebra
2.1 Examples for algebras
2.2 Building new algebras from old ones (operations on algebras)
2.2.1 Subalgebra
2.2.2 Homomorphic image
2.2.3 A distinguished example: Lattices
2.2.4 Congruence relation
2.2.5 Cartesian product, direct decomposition
2.2.6 Subdirect decomposition
2.2.7 Ultraproduct, reduced product
2.3 Categories
2.4 Variety characterization, quasi{variety characterization
2.5 Free algebras
2.6 Boolean Algebras
2.7 Discriminator varieties
2.8 Boolean Ordered Algebras (Boas), Boolean Algebras with Operators (BAOs)
Chapter 3 General framework and algebraization
3.1 De ning the framework for studying logics
3.2 Concrete logics in the new framework
3.3 Algebraization
3.3.1 Having connectives, formula algebra
3.3.2 Compositionality, tautological formula algebra
3.3.3 Algebraic counterparts of a logic
3.3.4 Substitution properties
3.3.5 Filter property
3.3.6 General Logics
3.4 Connections with Abstract Algebraic Logic, Abstract Model Theory and Institutions
Chapter 4 Bridge between logic and algebra
4.1 Algebraic characterization of compactness properties
4.2 Algebraic characterizations of completeness properties
4.2.1 Hilbert-type inference systems
4.2.2 Completeness and soundness
4.3 Algebraic characterization of definability properties
4.3.1 Syntactical Beth definability property
4.3.2 Beth definability property
4.3.3 Local Beth definability property
4.3.4 Weak Beth definability property
4.4 Algebraic characterization of interpolation properties
4.4.1 Interpolation properties
4.4.2 Amalgamation and interpolation properties
4.5 Decidability
4.6 Gödel's incompleteness property
Chapter 5 Applying the machinery: Examples
5.1 Classical propositional logic LC
5.2 Arrow logic LREL
5.3 Finite-variable fragments of first-order logic, with substituted atomic formulas, L'n
5.4 n-variable fragment Ln of first-order logic, for n ≤ ω
5.5 First-order logic with nonstandard semantics, Lan
5.6 Variable-dependent rst-order logic, Lnvd
5.7 First-order logic, ranked version, LFOLranked
5.8 First-order logic, rank-free (or type-less) version, LFOLrf
Chapter 6 Generalizations and new kinds of logics
6.1 Generalizations
6.2 New kinds of logics
Chapter 7 Appendix: Algebras of relations
7.1 Algebras of binary relations
7.2 Algebras of finitary relations
7.3 All finitary relations together
Bibliography
Index
Index of symbols
