Algebraic Methods of Mathematical Logic

PrefaceContents1. Introduction 1.1 Directions for the Use of the Introduction 1.2 A General Characterization of Mathematical Logic 1.3 Formalization, Mathematization, Interpretation 1.4 The Dialectic of the Relation Between Mathematical and Metamathematical Aspects 1.5 Metamathematico-Mathematical Parallelism and its Natural Limits 1.6 Practical Applications of Methods of Mathematical Logic 1.7 Principal Mathematical Tools of Mathematical Logic 1.8 Constructivism in Metamathematics 1.9 Philosophy and Mathematical Logic 1.10 Methodological Tasks and Achievements of Mathematical Logic within Mathematics 1.11 Semantics and Pragmatics 1.12 Mathematical Logic and Logic in the Broad Sense2. The Language of Mathematics and its Symbolization 2.1 Mathematical Logic and Mathematical Language as a Material System of Signs 2.2 The Technique of Symbolization of the Language of Mathematics 2.3 The Substance and Purpose of Symbolization of Mathematical Language 2.4 Logical Syntax and Logical Semantics 2.5 The Idealized Symbolical Mathematical Theory (without Individual Constants) and its Generalizations3. Recursive Construction of the Relation of Consequence 3.1 Fundamental Descriptively-Syntactic Rules 3.2 Fundamental Descriptively-Semantic Rules; Definition of Truth of a Sentence 3.3 Recursive Construction of the Relation of Consequence 3.4 Theorems on the Relation of Consequence; Duality; the Deduction Theorem4. Expressive Possibilities of the Present Symbolization 4.1 Symbolic Expression of Operations and Functions 4.2 Possibility of Elementary Symbolization of Classical Mathematics 4.3 Individual Constants and their Elimination 4.4 The Syntactic Approach to Individual Constants 4.5 Formalization of Mathematical Theories without Primitive Equality5. Intuitive and Mathematical Notions of an Idealized Axiomatic Mathematical Theory 5.1 Critical Annotations; Arithmetization and Algebraization 5.2 Logical Frame of a Language and Mathematical Theory 5.3 The Algorithmic Condition for a Finite Sequence of Signs to be an Expression; Uniqueness of Decomposition of Expressions; The Replacement Theorem; Scope of Quantification6. The Algebraic Theory of Elementary Predicate Logic 6.1 The Notion of Boolean Algebra Based on the Order Relation 6.2 The Notion of Boolean Algebra Based on Joins, Meets and Complementation 6.3 Basic Algebraic Tools of Mathematical Logic; Boolean Subalgebras; Homomorphisms; Ideals and Prime Ideals; Set Representation7. Foundations of the Algebraic Theory of Logical Syntax 7.1 Free Boolean Algebras, Construction and Representation 7.2 The Algebraic Aspect of Propositional Calculi 7.3 Algebraic Properties of Basic Lindenbaum Algebras 7.3 Basic Properties of Joins and Meets in Boolean Algebras 7.5 Congruence of Sentential Expressions, Renaming of Bound Individual Indeterminates 7.6 Joins and Meets in Lindenbaum Algebras Given by Logical Quantifiers; Freedom of Basic Lindenbaum Algebras 7.7 Indexed Boolean Algebras 7.8 Homomorphisms and Ideals in Indexed Boolean Algebras 7.9 Substitutive i-Ideals and Formalized Theories 7.10 Abstract Characterization of Lindenbaum Algebras8. Algebraic Laws of Semantics of First-Order Predicate Logic 8.1 The Relation of Satisfaction; Truth-Valuations; Models of Formalized Theories 8.2 Algebraic Characterization of Basic Semantical Notions 8.3 Theorem on i-Prime Over-Ideals for Substitutively Indexed Algebras 8.4 Algebraic Formulation of Completeness of Predicate LogicBibliographyIndex