Introduction to Logic and to the Methodology of Deductive Sciences

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
  • New York, United States
  • English
  • 048628462X
  • 9780486284620
  • 131,985

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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