This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. A thought-provoking introduction to the fundamentals and the perfect adjunct to courses in logic and the foundations of mathematics. Exercises appear throughout.
Includes bibliographical references (p. 227-230) and index.
PREFACE
FROM THE PREFACE TO THE ORIGINAL EDITION
FIRST PART ELEMENTS OF LOGIC. DEDUCTIVE METHOD
I. ON THE USE OF VARIABLES
1. Constants and variables
2. Expressions containing variables-sentential and designatory functions
3. Formation of sentences by means of variables-universal and existential sentences
4. Universal and existential quantifiers; free and bound variables
5. The importance of variables in mathematics
Exercises
II. ON THE SENTENTIAL CALCULUS
6. Logical constants; the old logic and the new logic
7. "Sentential calculus; negation of a sentence, conjunction and disjunction of sentences"
8. Implication or conditional sentence; implication in material meaning
9. The use of implication in mathematics
10. Equivalence of sentences
11. The formulation of definitions and its rules
12. Laws of sentential calculus
13. Symbolism of sentential calculus; truth functions and truth tables
14. Application of laws of sentential calculus in inference
15. "Rules of inference, complete proofs"
Exercises
III. ON THE THEORY OF IDENTITY
16. Logical concepts outside sentential calculus; concept of identity
17. Fundamental laws of the theory of identity
18. Identity of things and identity of their designations; use of quotation marks
19. "Equality in arithmetic and geometry, and its relation to logical identity"
20. Numerical quantifiers
Exercises
IV. ON THE THEORY OF CLASSES
21. Classes and their elements
22. Classes and sentential functions with one free variable
23. Universal class and null class
24. Fundamental relations among classes
25. Operations on classes
26. "Equinumerous classes, cardinal number of a class, finite and infinite classes; arithmetic as a part of logic"
Exercises
V. ON THE THEORY OF RELATIONS
27. "Relations, their domains and counter-domains; relations and sentential functions with two free variables"
28. Calculus of relations
29. Some properties of relations
30 "Relations which are reflexive, symmetrical and transitive"
31. Ordering relations; examples of other relations
32. One-many relations or functions
33. "One-one relations or biunique functions, and one-to-one correspondences"
34. Many-termed relations; functions of several variables and operations
35. The importance of logic for other sciences
Exercises
VI. ON THE DEDUCTIVE METHOD
36. "Fundamental constituents of a deductive theory-primitive and defined terms, axioms and theorems"
37. Model and interpretation of a deductive theory
38. Law of deduction; formal character of deductive sciences
39. Selection of axioms and primitive terms; their independence
40. "Formalization of definitions and proofs, formalized deductive theories"
41. Consistency and completeness of a deductive theory; decision problem
42. The widened conception of the methodology of deductive sciences
Exercises
SECOND PART APPLICATIONS OF LOGIC AND METHODOLOGY IN CONSTRUCTING MATHEMATICAL THEORIES
VII. CONSTRUCTION OF A MATHEMATICAL THEORY: LAWS OF ORDER FOR NUMBERS
43. Primitive terms of the theory under construction; axioms concerning fundamental relations among numbers
44. Laws of irreflexivity for the fundamental relations; indirect proofs
45. Further theorems on the fundamental relations
46. Other relations among numbers
Exercises
VIII. CONSTRUCTION OF A MATHEMATICAL THEORY: LAWS OF ADDITION AND SUBTRACTION
47. "Axioms concerning addition; general properties of operations, concepts of a group and of an Abelian group"
48. Commutative and associative laws for a larger number of summands
49. Laws of monotony for addition and their converses
50. Closed systems of sentences
51. Consequences of the laws of monotony
52. Definition of subtraction; inverse operations
53. Definitions whose definiendum contains the identity sign
54. Theorems on subtraction
Exercises
IX. METHODOLOGICAL CONSIDERATIONS ON THE CONSTRUCTED THEORY
55. Elimination of superfluous axioms in the original axiom system
56. Independence of the axioms of the simplified system
57. Elimination of superfluous primitive terms and subsequent simplification of the axiom system; concept of an ordered Abelian group
58. Further simplification of the axiom system; possible transformations of the system of primitive terms
59. Problem of the consistency of the constructed theory
60. Problem of the completeness of the constructed theory
Exercises
X. EXTENSION OF THE CONSTRUCTED THEORY. FOUNDATIONS OF ARITHMETIC OF REAL NUMBERS
61. First axiom system for the arithmetic of real numbers
62. Closer characterization of the first axiom system; its methodological advantages and didactical disadvantages
63. Second axiom system for the arithmetic of real numbers
64. Closer characterization of the second axiom system; concepts of a field and of an ordered field
65. Equipollence of the two axiom systems; methodological disadvantages and didactical advantages of the second system
Exercises
SUGGESTED READINGS
INDEX