Introduction to Logic and to the Methodology of the Deductive Sciences
1st Edition
Published on 24. March 1994
254 pages
978-0-19-802139-1 (ISBN)
Description
Now in its fourth edition, this classic work clearly and concisely introduces the subject of logic and its applications. The first part of the book explains the basic concepts and principles which make up the elements of logic. The author demonstrates that these ideas are found in all branches of mathematics, and that logical laws are constantly applied in mathematical reasoning. The second part of the book shows the applications of logic in mathematical theory building with concrete examples that draw upon the concepts and principles presented in the first section. Numerous exercises and an introduction to the theory of real numbers are also presented. Students, teachers and general readers interested in logic and mathematics will find this book to be an invaluable introduction to the subject.
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Content
- Intro
- Contents
- From Author's Prefaces to Previous Editions
- Editor's Preface
- A Short Biographical Sketch of Alfred Tarski
- 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 Construction of sentences in which variables occur-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
- the negation of a sentence, the conjunction and the disjunction of sentences
- 8 Implications or conditional sentences
- implications in the material meaning
- 9 The use of implications in mathematics
- 10 Equivalence of sentences
- 11 The formulation of definitions and its rules
- 12 Laws of sentential calculus
- 13 The symbolism of sentential calculus
- compound sentential functions and truth tables
- 14 An 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
- the concept of identity
- 17 Fundamental laws of the theory of identity
- 18 Identity of objects and identity of their designations
- the use of quotation marks
- 19 Equality in arithmetic and in geometry, and its relationship 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 The universal class and the null class
- 24 The fundamental relations among classes
- 25 Operations on classes
- 26 Equinumerous classes, the 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 The algebra of relations
- 29 Several kinds of relations
- 30 Relations which are reflexive, symmetric, and transitive
- 31 Ordering relations
- examples of other relations
- 32 Many-one relations or functions
- 33 One-one relations or bijective functions, and one-to-one correspondences
- 34 Many-place 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 Models and interpretations of a deductive theory
- 38 The 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
- the 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 The primitive terms of the theory to be constructed
- the axioms concerning the fundamental relations among numbers
- 44 The 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 The axioms concerning addition
- general properties of operations, the concept of a group and the concept of an Abelian group
- 48 Commutative and associative laws for larger numbers of summands
- 49 The laws of monotonicity for addition and their converses
- 50 Closed systems of sentences
- 51 A few consequences of the laws of monotonicity
- 52 The 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 from the original axiom system
- 56 Independence of the axioms of the reduced system
- 57 Elimination of a superfluous primitive term and the subsequent simplification of the axiom system
- the concept of an ordered Abelian group
- 58 Further simplification of the axiom system
- possible transformations of the system of primitive terms
- 59 The problem of consistency of the constructed theory
- 60 The problem of completeness of the constructed theory
- Exercises
- X: Extension of the Constructed Theory: Foundations of Arithmetic of Real Numbers
- 61 The first axiom system for the arithmetic of real numbers
- 62 Closer characterization of the first axiom system
- its methodological advantages and didactic disadvantages
- 63 The second axiom system for the arithmetic of real numbers
- 64 Closer characterization of the second axiom system
- the concept of a field and that of an ordered field
- 65 Equipollence of the two axiom systems
- methodological disadvantages and didactic advantages of the second system
- Exercises
- Index
- A
- B
- C
- D
- E
- F
- G
- H
- I
- K
- L
- M
- N
- O
- P
- Q
- R
- S
- T
- U
- V
- W
- Z
