This book connects logic with algebra. It covers universal logic and the question of what logic is as well as universal algebraic logic and duality theories between the world of logics and the world of algebra. It also discusses algebraic logic proper including algebras of relations of various ranks, Tarski's cylindric algebras, relation algebras, Halmos' polyadic algebras and other kinds of algebras of logic.
This book connects logic with algebra. It covers universal logic and the question of what logic is as well as universal algebraic logic and duality theories between the world of logics and the world of algebra. It also discusses algebraic logic proper including algebras of relations of various ranks, Tarski's cylindric algebras, relation algebras, Halmos' polyadic algebras and other kinds of algebras of logic.
1. Introduction.-
2. Notation, elementary concepts: 2.1 Sets, classes,
tuples, simple operations on sets.- 2.2 Binary relations, equivalence
relations, functions.-
2.3 Orderings, ordinals, cardinals.- 2.4 Sequences.- 2.5 Direct product of
families of sets.- 2.6 Relations of higher ranks.- 2.7 First-order logic
(FOL).-
3. Basics from universal algebra: 3.1 Examples for algebras.- 3.2 Building
new algebras from old ones (operations on algebras): 3.2.1 Subalgebra.-
3.2.2 Homomorphic image.- 3.2.3 Cartesian product, direct decomposition.-
3.2.4 Subdirect decomposition.- 3.2.5 Ultraproduct, reduced product.-
3.3 Variety characterization, quasi-variety characterization.- 3.4
Discriminator varieties.- 3.5 Boolean algebras.- 3.6 Boolean algebras with
operators.-
4. General framework for studying logics: 4.1 Defining the
framework.- 4.2 Concrete logics in the new framework.- 4.2.1 Distinguished
logics.- 4.2.2 The finite variable fragment of FOL. Finite model theory.-
4.2.3 Summary.- 4.2.4 Solutions for some exercises of subsections 4.2.1 and
4.2.2.-
5. Bridge between the world of logics and the world of algebras: 5.1
Fine-tuning the framework.- 5.2 Algebraic characterizations of completeness
and compactness properties.-
6. Generalizations.-
7. Further equivalence
results (definability, interpolation, omitting types)
8. New kinds of logics
9. The guarded fragment and its finite variable hierarchy.-
10. Distinguished
algebras of logic: 10.1 Cylindric algebras.- 10.2 Relation algebras,
substitution cylindric algebras, polyadic algebras.- 10.3 Relativization. The
non-square approach to ``taming logics''.- 10.4 Brief return to guarded
fragment and related positive results.-
11. Applications of the algebraic
results to logic (via our bridge).-
12. Connections with set theory and
foundational thinking.-
13. Connections with abstract model theory.-
14.
Extending the bridge to including logic of spacetime, connections with
relativity.-
15. On the problems in the books Henkin-Monk-Tarski: Cylindric
Algebras Parts I,II and Henkin-Monk-Tarski-Andreka-Nemeti.- Some further
problems.
