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

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## Alfred Tarski:

## Alfred Tarski: *Introduction to Logic and to the Methodology of Deductive Sciences (2nd Edition)*

Published $\text {1946}$, **Oxford University Press** (translated by Olaf Helmer)

### Subject Matter

### Contents

- Preface
- Harvard University September 1940
- University of California, Berkeley, August 1945

- 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 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. 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

## Source work progress

- 1946: Alfred Tarski:
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