Consequence Relations: An Introduction to the Lindenbaum-Tarski Method (Oxford Logic Guides)

Foreword
Preface
Acknowledgments
Contents
1. Introduction
	1.1 Overview of the key concepts
	1.2 Overview of the contents
2. Preliminaries
	2.1 Preliminaries from set theory
	2.2 Preliminaries from topology
	2.3 Preliminaries from algebra
		2.3.1 Subalgebras, homomorphisms, direct products and subdirect products
		2.3.2 Class operators
		2.3.3 Term algebra, varieties and quasi-varieties
		2.3.4 Hilbert algebras
		2.3.5 Distributive lattices
		2.3.6 Brouwerian semilattices
		2.3.7 Relatively pseudo-complemented lattices
		2.3.8 Boolean algebras
		2.3.9 Heyting algebras
	2.4 Preliminaries from model theory
	2.5 Preliminaries from computability theory
		2.5.1 Algorithmic functions and relations on ℕ
		2.5.2 Word sets and functions
		2.5.3 Effectively denumerable sets
3. Sentential Formal Languages and Their Interpretation
	3.1 Sentential formal languages
	3.2 Semantics of sentential formal language
		3.2.1 The logic of a two-valued matrix
		3.2.2 The Łukasiewicz logic of a three-valued matrix
		3.2.3 The Łukasiewicz modal logic of a three-valued matrix
		3.2.4 The Gödel n-valued logics
		3.2.5 The Dummett denumerable matrix
		3.2.6 Two infinite generalizations of Łukasiewicz’s Ł₃
	3.3 Historical notes
4. Logical Consequence
	4.1 Consequence relations
	4.2 Consequence operators
	4.3 Realizations of abstract logic
		4.3.1 Defining new abstract logics from given ones by substitution
		4.3.2 Consequence operator via a closure system
		4.3.3 Defining abstract logic semantically: a general approach
		4.3.4 Consequence relation via logical matrices
		4.3.5 Consequence relation via inference rules
	4.4 Abstract logics defined by modus rules
		4.4.1 General characterization
		4.4.2 Abstract logics ????*
		4.4.3 The class of ????-models
		4.4.4 Modus rules vs non-finitary rules
	4.5 Extensions of abstract logics
	4.6 Historical notes
5. Matrix Consequence
	5.1 Single-matrix consequence
	5.2 Finitary matrix consequence
	5.3 The conception of separating tools
	5.4 Historical notes
6. Unital Abstract Logics
	6.1 Unital algebraic expansions
	6.2 Unital abstract logics
		6.2.1 Definition and some properties
		6.2.2 Some subclasses of unital logics
	6.3 Lindenbaum-Tarski algebras
		6.3.1 Definition and properties
		6.3.2 Implicational unital logics
		6.3.3 Examples of Lindenbaum-Tarski algebras
		6.3.4 Rieger–Nishimura algebra
	6.4 Examples of application of Lindenbaum-Tarski algebras
		6.4.1 Glivenko’s theorem
		6.4.2 Completeness
		6.4.3 Admissible and derivable structural inference rules
		6.4.4 Classes of algebras corresponding and fully corresponding to a calculus
	6.5 An alternative approach to the concept of Lindenbaum-Tarski algebra
	6.6 Historical notes
7. Equational Consequence
	7.1 Language for equalities and its semantics
	7.2 Equational consequence
	7.3 ℰˆ as a deductive system
	7.4 Lindenbaum-Tarski matrices for logic ℰ
	7.5 Mal’cev matrices
	7.6 Equational consequence based on implicational logic
		7.6.1 Abstract logic ℰ_B
		7.6.2 Abstract logic ℰ_H
	7.7 Philosophical and historical notes
8. Equational L-Consequence
	8.1 Equational L-consequence
	8.2 ℰˆL as a deductive system
	8.3 Lindenbaum-Tarski matrices for logic ℰ_L
	8.4 Interconnection between ℰ and ℰ_L
	8.5 ℰ_L-models
	8.6 Examples of logics ℰ_L
	8.7 L-equational consequence based on implicational logic
	8.8 Historical notes
9. ????-Consequence
	9.1 ????-Languages
	9.2 Semantics of ℒ_????
	9.3 ????-Consequence
		9.3.1 ????-consequence defined by ????-matrices
		9.3.2 ????-consequence defined via inference rules
	9.4 Lindenbaum ????-matrices
	9.5 Lindenbaum-Tarski ????-structures
	9.6 Three ????-matrix consequences
	9.7 Historical notes
10. Decidability
	10.1 Abstract logics defined by finite atlases
		10.1.1 Indistinguishable formulas
		10.1.2 Matrix equivalences
		10.1.3 Restricted Lindenbaum matrices and atlases
	10.2 Enumerating procedure
	10.3 Solution to Problem (m.1)
	10.4 Solution to Problem (m.2)
	10.5 Solution to Problem (m.3)
	10.6 Historical notes
References
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Index
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