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 beinga 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.
Table des matières
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 Godel’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.
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