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Universal Algebraic Logic Dedicated to the Unity of Science

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 rela...

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Bibliographic Details
Main Authors: Andréka, Hajnal, Gyenis, Zalán (Author), Németi, István (Author), Sain, Ildikó (Author)
Format: eBook
Language:English
Published: Cham Birkhäuser 2022, 2022
Edition:1st ed. 2022
Series:Studies in Universal Logic
Subjects:
Mathematical Logic
Algebra
Mathematical Logic And Foundations
Online Access:
Collection: Springer eBooks 2005- - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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
  • 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

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