An Introduction to Mathematical Logic

Preface
Contents
Chapter 1 Mathematics and Reality
	1-1 Abstract or applied
	1-2 Definitions and undefined expressions
	1-3 Unproved statements
	1-4 Models and interpretations
	1-5 Where do little axioms come from?
	1-6 Mathematical systems
	1-7 The plan of this book
Chapter 2 Sentential Variables, Operators, and Formulas
	2-1 Introduction
	2-2 Sentential variables and sentential formulas
	2-3 Negation
	2-4 Conjunction and disjunction
	2-5 The conditional
	2-6 The biconditional
	2-7 Brackets and scope conventions. Sentential operators
Chapter 3 Truth Tables. The Sentential Calculus
	3-1 Basic truth tables
	3-2 Truth tables for compound sentences
	3-3 Tautologies
	3-4 A short test for tautology
	3-5 Choice of formulas
	3-6 Useful tautological formulas
	3-7 Reducing the number of basic operators
Chapter 4 The Deeper Structure of Statements
	4-1 Quantifiers
	4-2 The significance of quantifiers
	4-3 Purifying quantifiers
	4-4 Atomic statements
	4-5 Terms
	4-6 Constants aiu1 variables
	4-7 Bound variables and free variables
	4-8 Predicate notation
	4-9 Special pairs of statement forms
Chapter 5 The Demonstration
	5-1 Introduction
	5-2 The turnstile
	5-3 Rules of assumption and tautology
	5-4 Rules of instance and example
	5-5 Rules of choice and generalization. Specified variables
	5-6 Rules of reflexivity of equality, equality substitution, and biconditional substitution
	5-7 Rule of detachment
	5-8 Presenting a demonstration
	5-9 Three sample demonstrations
	5-10 Equivalence relations
Chapter 6 Two Major Theorems
	6-1 Introduction
	6-2 The lemma theorem
	6-3 The general deduction theorem
	6-4 Three corollaries
Chapter 7 Supplementary Inference Rules
	7-1 The inference rule symbol
	7-2 The pattern of proof for new inference rules
	7-3 Inference rule forms of the rules of instance, example, choice, generalization, equality substitution, and biconditional substitution
	7-4 Iterated use of the rules of instance and example
	7-5 The tautological inference family
	7-6 Generalized versions of certain rules
Chapter 8 Universal Theorems
	8-1 Introduction
	8-2 Some particularly useful universal theorems
	8-3 Equality as an equivalence relation
	8-4 Formal proof
	8-5 Applications in group theory
Chapter 9 Techniques of Negation
	9-1 Motivation
	9-2 The basic relations and their applications
Chapter 10 Terms and Definitions
	10-1 Variables as terms
	10-2 Other terms
	10-3 Terms by postulate
	10-4 Terms by substitution
	10-5 Terms by definition
	10-6 Constants
	10-7 Defining new statement forms
	10-8 Definitions and the demonstration
	10-9 More formal versions of four basic rules
	10-10 A theorem on equality
Chapter 11 The System {/|, ∈}
	11-1 Introduction
	11-2 Undefined expressions
	11-3 Criteria for statements
	11-4 Axioms and definitions
	11-5 Some simple basic theorems
	11-6 Membership and equality
	11-7 Subset theorems and tautologies
	11-8 Illustrative proofs
Chapter 12 The Boolean Approach: the System {/|, ∩, ∪}
	12-1 Introduction
	12-2 Axioms and definitions
	12-3 A list of theorems
	12-4 Moves toward informality
	12-5 The duality principle for {/|, ∩, ∪}
Answers to Odd-Numbered Exercises
Index