Introduction to Logic and to the Methodology of Deductive Sciences

Introduction to Logic and to the Methodology of Deductive Sciences

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

Product details

  • Paperback | 239 pages
  • 138 x 215 x 13mm | 287g
  • Dover Publications Inc.
  • New York, United States
  • English
  • 048628462X
  • 9780486284620
  • 117,774

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 INDEXshow more

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