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1 Local Warehouse Mathematics- Foundations and Logic

This title in other editions

Introduction To Logic & To the Methodology 2ND Edition

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Introduction To Logic & To the Methodology 2ND Edition Cover

 

Synopses & Reviews

Publisher Comments:

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.

Book News Annotation:

**** Reprint of the Olaf Helmer translation originally published by Oxford U. Press in 1946 (& cited in BCL3).
Annotation c. Book News, Inc., Portland, OR (booknews.com)

Synopsis:

Classic undergraduate text sharpens students’ grasp of the deductive method, including variables, sentential calculus, theory of identity, more. Exercises.

Description:

Includes bibliographical references (p. 227-230) and index.

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

Product Details

ISBN:
9780486284620
Translator:
Helmer, Olaf
Author:
Helmer, Olaf
Author:
Tarski, Alfred
Publisher:
Dover Publications
Location:
New York :
Subject:
General
Subject:
Philosophy
Subject:
General science
Subject:
Mathematics
Subject:
Logic
Subject:
Arithmetic
Subject:
Arithmetic -- Foundations.
Subject:
General Science
Subject:
Mathematics -- Philosophy.
Subject:
Mathematics-Logic and Philosophy
Copyright:
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
no. 56
Publication Date:
19950331
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
272
Dimensions:
8.5 x 5.38 in 0.65 lb

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Introduction To Logic & To the Methodology 2ND Edition Used Trade Paper
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Product details 272 pages Dover Publications - English 9780486284620 Reviews:
"Synopsis" by ,
Classic undergraduate text sharpens students’ grasp of the deductive method, including variables, sentential calculus, theory of identity, more. Exercises.

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