# Algebraic Methods of Mathematical Logic

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Algebraic Methods of Mathematical Logic focuses on the algebraic methods of mathematical logic, including Boolean algebra, mathematical language, and arithmetization. The book first offers information on the dialectic of the relation between mathematical and metamathematical aspects; metamathematico-mathematical parallelism and its natural limits; practical applications of methods of mathematical logic; and principal mathematical tools of mathematical logic. The text then elaborates on the language of mathematics and its symbolization and recursive construction of the relation of consequence. Discussions focus on recursive construction of the relation of consequence, fundamental descriptively-semantic rules, mathematical logic and mathematical language as a material system of signs, and the substance and purpose of symbolization of mathematical language. The publication examines expressive possibilities of symbolization; intuitive and mathematical notions of an idealized axiomatic mathematical theory; and the algebraic theory of elementary predicate logic. Topics include the notion of Boolean algebra based on joins, meets, and complementation, logical frame of a language and mathematical theory, and arithmetization and algebraization. The manuscript is a valuable reference for mathematicians and researchers interested in the algebraic methods of mathematical logic.

## Table of Contents

Preface

Contents

1. 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 Sense

2. 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 Generalizations

3. 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 Theorem

4. 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 Equality

5. 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 Quantification

6. 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 Representation

7. 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 Algebras

8. 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 Logic

Bibliography

Index

## Product details

- No. of pages: 210
- Language: English
- Copyright: © Academic Press 1967
- Published: January 1, 1967
- Imprint: Academic Press
- eBook ISBN: 9781483270524

## About the Author

### Ladislav Rieger

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