Introduction to Logic

Häftad, Engelska, 2003

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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.

Produktinformation

Utgivningsdatum2003-03-28
Mått140 x 215 x 15 mm
Vikt280 g
FormatHäftad
SpråkEngelska
SerieDover Books on MaTHEMA 1.4tics
Antal sidor272
FörlagDover Publications Inc.
ISBN9780486284620

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PREFACEFROM THE PREFACE TO THE ORIGINAL EDITIONFIRST PART ELEMENTS OF LOGIC. DEDUCTIVE METHODI. ON THE USE OF VARIABLES1. Constants and variables2. Expressions containing variables-sentential and designatory functions3. Formation of sentences by means of variables-universal and existential sentences4. Universal and existential quantifiers; free and bound variables5. The importance of variables in mathematicsExercisesII. ON THE SENTENTIAL CALCULUS6. Logical constants; the old logic and the new logic7. "Sentential calculus; negation of a sentence, conjunction and disjunction of sentences"8. Implication or conditional sentence; implication in material meaning9. The use of implication in mathematics10. Equivalence of sentences11. The formulation of definitions and its rules12. Laws of sentential calculus 13. Symbolism of sentential calculus; truth functions and truth tables14. Application of laws of sentential calculus in inference15. "Rules of inference, complete proofs"ExercisesIII. ON THE THEORY OF IDENTITY16. Logical concepts outside sentential calculus; concept of identity17. Fundamental laws of the theory of identity18. Identity of things and identity of their designations; use of quotation marks19. "Equality in arithmetic and geometry, and its relation to logical identity"20. Numerical quantifiersExercisesIV. ON THE THEORY OF CLASSES21. Classes and their elements22. Classes and sentential functions with one free variable23. Universal class and null class24. Fundamental relations among classes25. Operations on classes26. "Equinumerous classes, cardinal number of a class, finite and infinite classes; arithmetic as a part of logic"ExercisesV. ON THE THEORY OF RELATIONS27. "Relations, their domains and counter-domains; relations and sentential functions with two free variables"28. Calculus of relations29. Some properties of relations30 "Relations which are reflexive, symmetrical and transitive"31. Ordering relations; examples of other relations32. One-many relations or functions33. "One-one relations or biunique functions, and one-to-one correspondences"34. Many-termed relations; functions of several variables and operations35. The importance of logic for other sciencesExercisesVI. ON THE DEDUCTIVE METHOD36. "Fundamental constituents of a deductive theory-primitive and defined terms, axioms and theorems"37. Model and interpretation of a deductive theory38. Law of deduction; formal character of deductive sciences39. Selection of axioms and primitive terms; their independence40. "Formalization of definitions and proofs, formalized deductive theories"41. Consistency and completeness of a deductive theory; decision problem42. The widened conception of the methodology of deductive sciencesExercisesSECOND PART APPLICATIONS OF LOGIC AND METHODOLOGY IN CONSTRUCTING MATHEMATICAL THEORIESVII. CONSTRUCTION OF A MATHEMATICAL THEORY: LAWS OF ORDER FOR NUMBERS43. Primitive terms of the theory under construction; axioms concerning fundamental relations among numbers44. Laws of irreflexivity for the fundamental relations; indirect proofs45. Further theorems on the fundamental relations46. Other relations among numbersExercisesVIII. CONSTRUCTION OF A MATHEMATICAL THEORY: LAWS OF ADDITION AND SUBTRACTION47. "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 summands49. Laws of monotony for addition and their converses50. Closed systems of sentences51. Consequences of the laws of monotony52. Definition of subtraction; inverse operations53. Definitions whose definiendum contains the identity sign54. Theorems on subtractionExercisesIX. METHODOLOGICAL CONSIDERATIONS ON THE CONSTRUCTED THEORY55. Elimination of superfluous axioms in the original axiom system56. Independence of the axioms of the simplified system57. Elimination of superfluous primitive terms and subsequent simplification of the axiom system; concept of an ordered Abelian group58. Further simplification of the axiom system; possible transformations of the system of primitive terms59. Problem of the consistency of the constructed theory60. Problem of the completeness of the constructed theoryExercisesX. EXTENSION OF THE CONSTRUCTED THEORY. FOUNDATIONS OF ARITHMETIC OF REAL NUMBERS61. First axiom system for the arithmetic of real numbers62. Closer characterization of the first axiom system; its methodological advantages and didactical disadvantages63. Second axiom system for the arithmetic of real numbers64. Closer characterization of the second axiom system; concepts of a field and of an ordered field65. Equipollence of the two axiom systems; methodological disadvantages and didactical advantages of the second systemExercisesSUGGESTED READINGSINDEX