An Algebraic Approach to Non-Classical Logics

Name: An Algebraic Approach to Non-Classical Logics
Brand: Elsevier
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Lev D. Beklemishev(Author)

Elsevier (Publisher)

Published on 14. May 2014

421 pages

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978-0-08-095488-2 (ISBN)

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Front Cover
An Algebraic Approach to Non-Classical Logics
Copyright Page
Contents
PART ONE: IMPLICATIVE ALGEBRAS AND LATTICES
Chapter I. Preliminary set-theoretical, topological and algebraic notions
Introduction
1. Sets, mappings
2. Topological spaces
3. Ordered sets and quasi-ordered sets
4. Abstract algebras
5. Exercises
Chapter II. Implicative algebras
Introduction
1. Definition and elementary properties
2. Positive implication algebras
3. Implicative filters in positive implication algebras
4. Representation theorem for positive implication algebras
5. Implication algebras
6. Implicative filters in implication algebras
7. Representation theorem for implication algebras
8. Exercises
Chapter III. Distributive lattices and quasi-Boolean algebras
Introduction
1. Lattices
2. Distributive lattices
3. Quasi-Boolean algebras
4. Exercises
Chapter IV. Relatively pseudo-complemented lattices, contrapositionally complemented lattices, semi-complemented lattices and pseudo-Boolean algebras
Introduction
1. Relatively pseudo-complemented lattices
2. Filters in relatively pseudo-complemented lattices
3. Representation theorem for relatively pseudo-complemented lattices
4. Contrapositionally complemented lattices
5. Semi-complemented lattices
6. Pseudo-Boolean algebras
7. Exercises
Chapter V. Quasi-pseudo-Boolean algebras
Introduction
1. Definition and elementary properties
2. Equational definability of quasi-pseudo-Boolean algebras
3. Examples of quasi-pseudo-Boolean algebras
4. Filters in quasi-pseudo-Boolean algebras
5. Representation theorem for quasi-pseudo-Boolean algebras
6. Exercises
Chapter VI. Boolean algebras and topological Boolean algebras
Introduction
1. Definition and elementary properties of Boolean algebras
2. Subalgebras of Boolean algebras
3. Filters and implicative filters in Boolean algebras
4. Representation theorem for Boolean algebras
5. Topological Boolean algebras
6. I-filters in topological Boolean algebras
7. Representation theorem for topological Boolean algebras
8. Strongly compact topological spaces
9. A lemma on imbedding for topological Boolean algebras
10. Connections between topological Boolean algebras, pseudo-Boolean algebras, relatively pseudo-complemented lattices, contrapositionally complemented lattices and semi-complemented lattices
11. Lemmas on imbeddings for pseudo-Boolean algebras, relatively pseudo-complemented lattices, contrapositionally complemented lattices and semi-complemented lattices.
12. Exercises
Chapter VII. Post algebras
Introduction
1. Definition and elementary properties
2. Examples of Post algebras
3. Filters and D-filters in Post algebras
4. Post homomorphisms
5. Post fields of sets
6. Representation theorem for Post algebras
7. Exercises
PART TWO NON-CLASSICAL LOGICS
Chapter VIII. Implicative extensional propositional calculi
Introduction
1. Formalized languages of zero order
2. The algebra of formulas
3. Interpretation of formulas as mappings
4. Consequence operations in formalized languages of zero order
5. The class S of standard systems of implicative extensional propositional calculi
6. g-algebras
7. Completeness theorem
8. Logically equivalent systems
9. L-theories of zero order
10. Standard systems of implicative extensional propositional calculi with semi-negation.
11. Theorems on logically equivalent systems in S
12. Deductive filters
13. The connection between L-theories and deductive filters
14. Exercises
Chapter IX. Positive implicative logic and classical implicative logic
Introduction.
1. Propositional calculus I pl of positive implicative logic
2. gpl-algebras
3. Positive implicative logic Lpl
4. Lpl-theories of zero order
5. The connection between Lpl-theories and implicative filters
6. Propositional calculus Ixl of classical implicative logic
7. gxl-algebras
8. Classical implicative logic Lxl
9. Lxl-theories of zero order
10. The connection between Lxl-theories of zero order and implicative filters
11. Exercises
Chapter X. Positive logic
Introduction
1. Propositional calculus gx of positive logic
2. Ip-algebras
3. Positive logic Lp
4. On disjunctions derivable in the propositional calculi of positive logic
5. Lp-theories of zero order
6. The connection between Lp-theories and filters
7. Exercises
Chapter XI. Minimal logic, positive logic with semi-negation and intuitionistic logic
Introduction
1. Propositional calculus Ip of minimal logic
2. Minimal logic Lµ
3. Lµ-theories of zero order and their connection with filters
4. Propositional calculus Iv of positive logic with semi-negation
5. Positive logic with semi-negation Lv
6. Lv-thecries of zero order and their connection with filters
7. Propositional calculus Ix of intuitionistic logic
8. Intuitionistic logic Lx
9. Lx-theories of zero order and their connection with filters
10. Prime Lx-theories
11. Exercises
Chapter XII. Constructive logic with strong negation
Introduction
1. Propositional calculus IN of constructive logic with strong negation
2. IN-algebras
3. Constructive logic with strong negation LN
4. Connections between constructive logic with strong negation and intuitionistic logic
5. A topological characterization of formulas derivable in propositional calculi of constructive logic with strong negation
6. LN-theories of zero order
7. The connection between LN-theories of zero order and special filters of the first kind
8. Prime LN-theories
9. Exercises
Chapter XIll. Classical logic and modal logic
Introduction
1. Propositional calculus Ix of classical logic
2. Classical logic Lx
3. Lx-theories of zero order and their connection with filters
4. Propositional calculus I? of modal logic
5. Modal logic L?
6. L?-theories of zero order and their connection with I-filters
7. I-prime L?-theories
8. Exercises
Chapter XIV. Many-valued logics
Introduction
1. Propositional calculus Im of m-valued logic
2. Im-algebras
3. m-valued logic Lm
4. Lm-theories of zero order and their connection with D-filters
5. Exercises
SUPPLEMENT
First order predicate calculi of non-classical logics
Introduction
1. Formalized languages of first order
2. First order predicate calculi of a logic L
3. Elementary L-theories
4. The algebra of terms
5. Realizations of terms
6. Implicative algebras with generalized joins and meets
7. The algebra of an elementary L-theory
8. Realizations of first order formalized languages associated with a logic L
9. Canonical realizations for elementary L-theories
10. L-models
11. The completeness theorem for the first order predicate calculi of a logic L
12. The existence of L-models for consistent elementary L-theories
13. Exercises
Bibliography
List of symbols
Author index
Subject index

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