Introduction to Logic

and to the Methodology of Deductive Sciences

Alfred Tarski

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486318899
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

First published in Polish in 1936, this classic work was originally written as a popular scientific book — one that would present to the educated lay reader a clear picture of certain powerful trends of thought in modern logic. According to the author, these trends sought to create a unified conceptual apparatus as a common basis for the whole of human knowledge.
Because these new developments in logical thought tended to perfect and sharpen the deductive method, an indispensable tool in many fields for deriving conclusions from accepted assumptions, the author decided to widen the scope of the work. In subsequent editions he revised the book to make it also a text on which to base an elementary college course in logic and the methodology of deductive sciences. It is this revised edition that is reprinted here.
Part One deals with elements of logic and the deductive method, including the use of variables, sentential calculus, theory of identity, theory of classes, theory of relations and the deductive method. The Second Part covers applications of logic and methodology in constructing mathematical theories, including laws of order for numbers, laws of addition and subtraction, methodological considerations on the constructed theory, foundations of arithmetic of real numbers, and more. The author has provided numerous exercises to help students assimilate the material, which not only provides a stimulating and thought-provoking introduction to the fundamentals of logical thought, but is the perfect adjunct to courses in logic and the foundation of mathematics.

FIRST PART

•

ELEMENTS OF LOGIC DEDUCTIVE METHOD

• I •

ON THE USE OF VARIABLES

1. Constants and variables

Every scientific theory is a system of sentences which are accepted as true and which may be called LAWS or ASSERTED STATEMENTS or, for short, simply STATEMENTS. In mathematics, these statements follow one another in a definite order according to certain principles which will be discussed in detail in Chapter VI, and they are, as a rule, accompanied by considerations intended to establish their validity. Considerations of this kind are referred to as PROOFS, and the statements established by them are called THEOREMS.

Among the terms and symbols occurring in mathematical theorems and proofs we distinguish CONSTANTS and VARIABLES.

In arithmetic, for instance, we encounter such constants as “number”, “zero” (“0”), “one” (“1“), “sum” (“+”), and many others.¹ Each of these terms has a well-determined meaning which remains unchanged throughout the course of the considerations.

As variables we employ, as a rule, single letters, e.g. in arithmetic the small letters of the English alphabet: “a”, “b”, “c”, …, “x”, “y”, “z”. As opposed to the constants, the variables do not possess any meaning by themselves, Thus, the question:

does zero have such and such a property?

e.g.:

is zero an integer?

can be answered in the affirmative or in the negative; the answer may be true or false, but at any rate it is meaningful. A question concerning x, on the other hand, for example the question:

is x an integer?

cannot be answered meaningfully.

In some textbooks of elementary mathematics, particularly the less recent ones, one does occasionally come across formulations which convey the impression that it is possible to attribute an independent meaning to variables. Thus it is said that the symbols “x”, “y”, … also denote certain numbers or quantities, not “constant numbers” however (which are denoted by constants like “0”, “1”, … ), but the so-called “variable numbers” or rather “variable quantities”. Statements of this kind have their source in a gross misunderstanding. The “variable number” x could not possibly have any specified property, for instance, it could be neither positive nor negative nor equal to zero; or rather, the properties of such a number would change from case to case, that is to say, the number would sometimes be positive, sometimes negative, and sometimes equal to zero. But entities of such a kind we do not find in our world at all; their existence would contradict the fundamental laws of thought. The classification of the symbols into constants and variables, therefore, does not have any analogue in the form of a similar classification of the numbers.

2. Expressions containing variables—sentential and designatory functions

In view of the fact that variables do not have a meaning by themselves, such phrases as:

x is an integer

are not sentences, although they have the grammatical form of sentences; they do not express a definite assertion and can be neither confirmed nor refuted. From the expression:

x is an integer

we only obtain a sentence when we replace “x” in it by a constant denoting a definite number; thus, for instance, if “x” is replaced by the symbol “1”, the result is a true sentence, whereas a false sentence arises on replacing “x” by “

”. An expression of this kind, which contains variables and, on replacement of these variables by constants, becomes a sentence, is called a SENTENTIAL FUNCTION. But mathematicians, by the way, are not very fond of this expression, because they use the term “function” with a different meaning. More often the word “CONDITION” is employed in this sense; and sentential functions and sentences which are composed entirely of mathematical symbols (and not of words of everyday language), such as:

x + y = 5,

are usually referred to by mathematicians as FORMULAS. In place of “sentential function” we shall sometimes simply say “sentence”—but only in cases where there is no danger of any mis-understanding.

The role of the variables in a sentential function has sometimes been compared very adequately with that of the blanks left in a questionnaire; just as the questionnaire acquires a definite content only after the blanks have been filled in, a sentential function becomes a sentence only after constants have been inserted in place of the variables. The result of the replacement of the variables in a sentential function by constants—equal constants taking the place of equal variables—may lead to a true sentence; in that case, the things denoted by those constants are said to SATISFY the given sentential function. For example, the numbers 1, 2 and 2

satisfy the sentential function:

x < 3,

but the numbers 3, 4 and 4

do not.

Besides the sentential functions there are some further expressions containing variables that merit our attention, namely, the so-called DESIGNATORY or DESCRIPTIVE FUNCTIONS. They are expressions which, on replacement of the variables by constants, turn into designations (“descriptions”) of things. For example, the expression:

2x + 1

is a designatory function, because we obtain the designation of a certain number (e.g., the number 5), if in it we replace the variable “x” by an arbitrary numerical constant, that is, by a constant denoting a number (e.g., “2”).

Among the designatory functions occurring in arithmetic, we have, in particular, all the so-called algebraic expressions which are composed of variables, numerical constants and symbols of the four fundamental arithmetical operations, such as:

Algebraic equations, on the other hand, that is to say, formulas consisting of two algebraic expressions connected by the symbol “=”, are sentential functions. As far as equations are concerned, a special terminology has become customary in mathematics; thus, the variables occurring in an equation are referred to as the unknowns, and the numbers satisfying the equation are called the roots of the equation. E.g., in the equation:

x² + 6 = 5x

the variable “x” is the unknown, while the numbers 2 and 3 are roots of the equation.

Of the variables “x”, “y”, … employed in arithmetic it is said that they STAND FOR DESIGNATIONS OF NUMBERS or that numbers are VALUES of these variables. Thereby approximately the following is meant: a sentential function containing the symbols “x”, “y”, …...