Introduction to Logic

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