This well-organized book was designed to introduce students to a way of thinking that encourages precision and accuracy. As the text for a course in modern logic, it familiarizes readers with a complete theory of logical inference and its specific applications to mathematics and the empirical sciences.
Part I deals with formal principles of inference and definition, including a detailed attempt to relate the formal theory of inference to the standard informal proofs common throughout mathematics. An in-depth exploration of elementary intuitive set theory constitutes Part II, with separate chapters on sets, relations, and functions. The final section deals with the set-theoretical foundations of the axiomatic method and contains, in both the discussion and exercises, numerous examples of axiomatically formulated theories. Topics range from the theory of groups and the algebra of the real numbers to elementary probability theory, classical particle mechanics, and the theory of measurement of sensation intensities.
Ideally suited for undergraduate courses, this text requires no background in mathematics or philosophy.
Author(s): Patrick Suppes
Publisher: Dover Publications
Year: 1999
Language: English
Commentary: Front and back covers.
Pages: 336
PREFACE
TABLE OF CONTENTS
INTRODUCTION
PART I - PRINCIPLES OF INFERENCE AND DEFINITION
1. THE SENTENTIAL CONNECTIVES
1.1 Negation and Conjunction
1.2 Disjunction
1.3 Implication: Conditional Sentences
1.4 Equivalence: Biconditional Sentences
1.5 Grouping and Parentheses
1.6 Truth Tables and Tautologies
1.7 Tautological Implication and Equivalence
2. SENTENTIAL THEORY OF INFERENCE
2.1 Two Major Criteria of Inference and Sentential Interpretations
2.2 The Three Sentential Rules of Derivation
2.3 Some Useful Tautological Implications
2.4 Consistency of Premises and Indirect Proofs
3. SYMBOLIZING EVERYDAY LANGUAGE
3.1 Grammar and Logic
3.2 Terms
3.3 Predicates
3.4 Quantifiers
3.5 Bound and Free Variables
3.6 A Final Example
4. GENERAL THEORY OF INFERENCE
4.1 Inference Involving Only Universal Quantifiers
4.2 Interpretations and Validity
4.3 Restricted Inferences with Existential Quantifiers
4.4 Interchange of Quantifiers
4.5 General Inferences
4.6 Summary of Rules of Inference
5. FURTHER RULES OF INFERENCE
5.1 Logic of Identity
5.2 Theorems of Logic
5.3 Derived Rules of Inference
6. POSTSCRIPT ON USE AND MENTION
6.1 Names and Things Named
6.2 Problems of Sentential Variables
6.3 Juxtaposition of Names
7. TRANSITION FROM FORMAL TO INFORMAL PROOFS
7.1 General Considerations
7.2 Basic Number Axioms
7.3 Comparative Examples of Formal Derivations and Informal Proofs
7.4 Examples of Fallacious Informal Proofs
7.5 Further Examples of Informal Proofs
8. THEORY OF DEFINITION
8.1 Traditional Ideas
8.2 Criteria for Proper Definitions
8.3 Rules for Proper Definitions
8.4 Definitions Which are Identities
8.5 The Problem of Division by Zero
8.6 Conditional Definitions
8.7 Five Approaches to Division by Zero
8.8 Padoa's Principle and Independence of Primitive Symbols
PART II - ELEMENTARY INTUITIVE SET THEORY
9. SETS
9.1 Introduction
9.2 Membership
9.3 Inclusion
9.4 The Empty Set
9.5 Operations on Sets
9.6 Domains of Individuals
9.7 Translating Everyday Language
9.8 Venn Diagrams
9.9 Elementary Principles About Operations on Sets
10. RELATIONS
10.1 Ordered Couples
10.2 Definition of Relations
10.3 Properties of Binary Relations
10.4 Equivalence Relations
10.5 Ordering Relations
10.6 Operations on Relations
11. FUNCTIONS
11.1 Definition
11.2 Operations on Functions
11.3 Church's Lambda Notation
12. SET-THEORETICAL FOUNDATIONS OF THE AXIOMATIC METHOD
12.1 Introduction
12.2 Set-Theoretical Predicates and Axiomatizations of Theories
12.3 Isomorphism of Models for a Theory
12.4 Example: Probability
12.5 Example: Mechanics
INDEX
