Synopsis "Universal Algebraic Logic: Dedicated to the Unity of Science"
Preface.- Acknowledgement.- 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).- 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 Boas and BAOs.- 3 General framework and algebraization.- 3.1 Defining 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.- 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 G odel's incompleteness property.- 5 Applying the machinery: Examples.- 5.1 Classical propositional logic LC.- 5.2 Arrow logic L_{REL}.- 5.3 Finite-variable fragments of first-order logic, with substituted atomic formulas, L'_n.- 5.4 n-variable fragment L_n of rst-order logic, for n le omega.- 5.5 First-order logic with nonstandard semantics, L {a}_{n}.- 5.6 Variable-dependent first-order logic, L {vd}_{n}.- 5.7 First-order logic, ranked version, L {ranked}_{FOL}.- 5.8 First-order logic, rank-free (or type-less) version, L {rf}_{FOL}.- 6 Generalizations and new kinds of logics.- 6.1 Generalizations.- 6.2 New kinds of logics.- 7 Appendix: Algebras of relations.- 7.1 Algebras of binary relations.- 7.2 Algebras of unitary relations.- 7.3 All unitary relations together.- Bibliography.- Index.- Index of symbols.
