Book:Alfred Tarski/Introduction to Logic and to the Methodology of Deductive Sciences/Second Edition

Subject Matter

 * Set Theory
 * Logic
 * Mathematical Logic

Contents

 * Harvard University September 1940
 * University of California, Berkeley, August 1945
 * University of California, Berkeley, August 1945




 * ELEMENTS OF LOGIC. DEDUCTIVE METHOD
 * ELEMENTS OF LOGIC. DEDUCTIVE METHOD


 * I.
 * 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.
 * 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.
 * 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.
 * 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.
 * 27. Relations, their domains and counter-domains; relations and sequential 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


 * VI.
 * 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


 * APPLICATIONS OF LOGIC AND METHODOLOGY IN CONSTRUCTING MATHEMATICAL THEORIES
 * APPLICATIONS OF LOGIC AND METHODOLOGY IN CONSTRUCTING MATHEMATICAL THEORIES


 * VII.
 * 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.
 * 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.
 * 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.
 * 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