# Introduction to Logic and to the Methodology of Deductive Sciences : And to the Methodology of Deductive Sciences

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## Description

Reprint of the Olaf Helmer translation originally published by Oxford U. Press in 1946 (& cited in BCL3 ). Annotation copyright Book News, Inc. Portland, Or.

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## Product details

- Paperback | 239 pages
- 138 x 215 x 13mm | 287g
- 01 Dec 1995
- Dover Publications Inc.
- New York, United States
- English
- 048628462X
- 9780486284620
- 131,985

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## Table of contents

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

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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

show more