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On the Use of Variables
Constants and variables
Expressions containing variables—sentential and designatory functions
Construction of sentences in which variables occur—universal and existential sentences
Universal and existential quantifiers free and bound variables
The importance of variables in mathematics
Exercises
On the Sentential Calculus
Logical constants the old logic and the new logic
Sentential calculus the negation of a sentence, the conjunction and the disjunction of sentences
Implications or conditional sentences implications in the material meaning
The use of implications in mathematics
Equivalence of sentences
The formulation of definitions and its rules
Laws of sentential calculus
The symbolism of sentential calculus compound sentential functions and truth tables
An application of laws of sentential calculus in inference
Rules of inference, complete proofs
Exercises
On the Theory of Identity
Logical concepts outside sentential calculus the concept of identity
Fundamental laws of the theory of identity
Identity of objects and identity of their designations the use of quotation marks
Equality in arithmetic and in geometry, and its relationship to logical identity
Numerical quantifiers
Exercises
On the Theory of Classes
Classes and their elements
Classes and sentential functions with one free variable
The universal class and the null class
The fundamental relations among classes
Operations on classes
Equinumerous classes, the cardinal number of a class, finite and infinite classes arithmetic as a part of logic
Exercises
On the Theory of Relations
Relations, their domains and counter-domains relations and sentential functions with two free variables
The algebra of relations
Several kinds of relations
Relations which are reflexive, symmetric, and transitive
Ordering relations examples of other relations
Many-one relations or functions
One-one relations or bijective functions, and one-to-one correspondences
Many-place relations functions of several variables and operations
The importance of logic for other sciences
Exercises
On the Deductive Method
Fundamental constituents of a deductive theory—primitive and defined terms, axioms and theorems
Models and interpretations of a deductive theory
The law of deduction formal character of deductive sciences
Selection of axioms and primitive terms their independence
Formalization of definitions and proofs, formalized deductive theories
Consistency and completeness of a deductive theory the decision problem
The widened conception of the methodology of deductive sciences
Exercises
Construction of a Mathematical Theory: Laws of Order for Numbers
The primitive terms of the theory to be constructed the axioms concerning the fundamental relations among numbers
The laws of irreflexivity for the fundamental relations indirect proofs
Further theorems on the fundamental relations
Other relations among numbers
Exercises
Construction of a Mathematical Theory: Laws of Addition and Subtraction
The axioms concerning addition general properties of operations, the concept of a group and the concept of an Abelian group
Commutative and associative laws for larger numbers of summands
The laws of monotonicity for addition and their converses
Closed systems of sentences
A few consequences of the laws of monotonicity
The definition of subtraction inverse operations
Definitions whose definiendum contains the identity sign
Theorems on subtraction
Exercises
Methodological Considerations on the Constructed Theory
Elimination of superfluous axioms from the original axiom system
Independence of the axioms of the reduced system
Elimination of a superfluous primitive term and the subsequent simplification of the axiom system the concept of an ordered Abelian group
Further simplification of the axiom system possible transformations of the system of primitive terms
The problem of consistency of the constructed theory
The problem of completeness of the constructed theory
Exercises
Extension of the Constructed Theory: Foundations of Arithmetic of Real Numbers
The first axiom system for the arithmetic of real numbers
Closer characterization of the first axiom system its methodological advantages and didactic disadvantages
The second axiom system for the arithmetic of real numbers
Closer characterization of the second axiom system the concept of a field and that of an ordered field
Equipollence of the two axiom systems methodological disadvantages and didactic advantages of the second system
Exercises
Index