This text was developed for a course at the upper-sophomore or junior level within the mathematics curriculum. It is intended as a course in logic useful to the student of mathematics rather than as a beginning course for a prospective specialist in philosophy. I have no doubt that the sophisticated reader will find sins of omission and commission in this presentation-no text is entirely satisfactory to-anyone, its author included. Most of the sins, I hope, will have resulted from deliberate choices of what I considered the lesser evil in the light of the aims of the text. One notable example is the omission of any discussion of language levels or of the distinction between use and mention of a symbol. In some places quotation marks are used to emphasize that a symbol is mentioned, but consistency in that usage was abandoned after the first draft, because it frequently turned out to have a distracting influence instead of a clarifying one.
With the exception of a few dispensable examples from the elementary calculus, a background in high school mathematics is all that is actually drawn upon. With a reasonably capable class, the whole text can be covered in a standard three semester-hour course. The plan of the book is described in § 1-7 (to thwart the student who habitually skips prefaces), and the reader might do well to turn to it at this point.
Author(s): gerson robison
Edition: 1
Publisher: Prentice-Hall, Englewood Cliffs, NJ
Year: 1969
Language: English
Pages: 212
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
