Introduction to Logic and to the Methodology of Deductive Sciences

• VII •

LAWS OF ORDER FOR NUMBERS

43. Primitive terms of the theory under construction; axioms concerning fundamental relations among numbers

With a certain amount of knowledge of the fields of logic and methodology at our disposal, we shall now undertake to lay the foundations of a particular and, incidentally, very elementary mathematical theory. This will be a good opportunity for us to assimilate better our previously acquired knowledge, and even to expand it to some extent.

The theory with which we shall concern ourselves constitutes a fragment of the arithmetic of real numbers. It contains fundamental theorems concerning the basic relations less than and greater than among numbers, as well as the basic operations on numbers, namely of addition and subtraction. It presupposes nothing but logic.

The primitive terms which we shall adopt in this theory are the following:

real number,
is less than,
is greater than,
sum.

Instead of “real number” we shall, as before, simply say “number”. Also, it is slightly more convenient to consider, instead of the term “number”, the expression “the set of all numbers” as a primitive term, which, for brevity, we will replace by the symbol “N” thus, in order to express that x is a number, we write:

We may, on the other hand, stipulate that the universe of discourse of our theory consists of real numbers only and that variables such as “x”, “y”, … stand exclusively for the names of numbers; in this case, the term “real number” would be altogether dispensable in the formulations of statements of our theory, and the symbol “N” might, when needed, be replaced by “∨” (cf. Section 23).

The expressions “is less than” and “is greater than” are to be treated as if they were entities consisting of a single word each; they will be replaced by the briefer symbols “<” and “>”, respectively. Instead of “is not less than” and “is not greater than” we shall employ the usual symbols “” and “”. Further, instead of “the sum of the numbers (the summands) x and y” or “the result of adding x and y” we shall use the customary notation:

x + y.

Thus, the symbol “N” designates a certain set, the symbols “<” and “>” certain two-termed relations, and finally the symbol “+” a certain binary operation.

Among the axioms of the theory under consideration two groups may be distinguished. The axioms of the first group express fundamental properties of the relations less than and greater than, whereas those of the second are primarily concerned with addition. For the time being we shall consider the first group only; it consists, altogether, of five statements:

AXIOM 1. For any numbers x and y (i.e., for arbitrary elements of the set N) we have: x = y or x < y or x > y.

AXIOM 2. If x < y, then .

AXIOM 3. If x > y, then .

AXIOM 4. If x < y and y < z, then x < z.

AXIOM 5. If x > y and y > z, then x > z.

The axioms listed here, just as any arithmetical theorem of a universal character stating that arbitrary numbers x, y, … have such and such a property, should really begin with the words “for any numbers x, y, …” or “for any elements x, y, … of the set N” or, simply, “for any x, y, …” (if we agree that the variables “x”, “y”, … here denote numbers only). But since we want to conform to the usage discussed in Section 3, we often omit such a phrase and merely add it in our mind; this holds not only for the axioms but also for the theorems and definitions which will occur in the course of our considerations. Axiom 2, for instance, is meant to be read as follows:

For any x and y (or for any elements x and y of the set N), if x < y then .

We shall refer to Axiom 1 as the WEAK LAW OF TRICHOTOMY (with the strong law of trichotomy we shall become acquainted later). Axioms 2–5 express the fact that the relations less than and greater than are asymmetrical and transitive (cf. Section 29); accordingly they are called the LAWS OF ASYMMETRY and LAWS OF TRANSITIVITY for the relations less than and greater than. The axioms of the first group and the theorems following from them are called the LAWS OF ORDER FOR NUMBERS.

The relations < and >, together with the logical relation of identity =, will be here referred to as the FUNDAMENTAL RELATIONS AMONG NUMBERS.

44. Laws of irreflexivity for the fundamental relations; indirect proofs

Our next task consists in the derivation of a number of theorems from the axioms adopted by us. Since we do not aim at a systematic presentation, in this and the following chapter only those theorems will be stated which may serve to help illustrate certain concepts and facts of the fields of logic and methodology.

THEOREM 1. No number is smaller than itself: .

PROOF. Suppose our theorem were false. Then there would be a number x satisfying the formula:

Now Axiom 2 refers to arbitrary numbers x and y (which need not be distinct), so that it remains valid if in place of “y” we write the variable “x”; we then obtain:

But from (1) and (2) it follows immediately that

this consequence, however, forms an obvious contradiction to formula (1). We must, therefore, reject the original assumption and accept the theorem as proved.

We shall now show how to transform this argument into a complete proof, using for clarity the logical symbolism (cf. Sections 13 and 15). To this end we resort to the so-called LAW OF REDUCTIO AD ABSURDUM of sentential calculus:

(I) [p → (~ p)] → (~ p)¹

We further use Axiom 2 in the following symbolic form:

(II) (x < y) → [~ (y < x)]

Our proof is based exclusively upon the sentences (I) and (II). First we apply the rule of substitution to (I), replacing “p” in it throughout by “(x < x)”:

(III) {(x < x) → [~ (x < x)]} → [~ (x < x)]

We next apply the rule of substitution to (II), replacing “y” by “x”:

(IV) (x < x) → [~ (x < x)]

Finally we observe that the sentence (IV) is the hypothesis of the conditional sentence (III), so that the rule of detachment may be applied. We are thus led to the formula:

(V) ~ (x < x)

which is the symbolic form of the theorem to be proved.

The proof of Theorem 1 represents an example of what is called an INDIRECT PROOF, also known as a PROOF BY REDUCTIO AD ABSURDUM. Proofs of this kind may quite generally be characterized as follows: in order to prove a theorem, we assume the theorem to be false, and derive from that certain consequences which compel us to reject the original assumption. Indirect proofs are very common in mathematics. They do not all fall under the schema of the proof of Theorem 1, however; on the contrary, the latter represents a comparatively rare form of indirect proof, and we shall meet with more typical examples of indirect proofs further below.

The axiom system adopted by us is perfectly symmetrical with respect to the two symbols “<” and “>”. To every theorem concerning the relation less than, we therefore automatically obtain the corresponding theorem concerning the relation greater than, the proofs being entirely analogous, so that the proof of the second theorem may be omitted altogether. In particular, corresponding to Theorem 1 we have:

THEOREM 2. No number is greater than itself: .

While the relation of identity =, as we know from logic, is reflexive, Theorems 1 and 2 show that the other two fundamental relations among numbers, < and >, are irreflexive; these theorems are therefore called the LAWS OF IRREFLEXIVITY (for the relations less than and greater than).

45. Further theorems on the fundamental relations

We shall next prove the following theorem:

THEOREM 3. x > y if, and only if, y < x.

PROOF. It has to be shown that the formulas:

x > y and y < x

are equivalent, that is to say, that the first implies the second, and vice versa (cf. Section 10).

Suppose, first, that

By Axiom 1 we must have at least one of the three cases:

If we had x = y, we could, by virtue of the fundamental law of the theory of identity, i.e. LEIBNIZ’S law (cf. Section 17), replace the variable “x” by “y” in formula (1); the resulting formula:

y < y

constitutes an obvious contradiction to Theorem 1. Hence we have:

But we also have:

since, by Axiom 2, the formulas:

x < y and y < x

cannot hold simultaneously. On account of (2), (3) and (4), we find that the third case must apply:

We thus have shown that the formula (5) is implied by the formula (1); conversely, the implication in the opposite direction can be established by an analogous procedure. The two formulas are, therefore, indeed equivalent, q.e.d.²

Using the terminology of the calculus of relations (cf. Section 28), we may say that, according to Theorem 3, each of the relations < and > is the converse of the other.

THEOREM 4. If x ≠ y, then x < y or y < x.

PROOF. Since

x ≠ y,

we have, by Axiom 1:

x < y or x > y;

the second of these formulas implies, by Theorem 3:

y < x.

Hence we have:

x < y or y < x, q.e.d.

Analogously we can prove

THEOREM 5. If x ≠ y, then x > y or y > x.

By Theorems 4 and 5 the relations < and > are connected; accordingly these theorems are known as the LAWS OF CONNEXITY (for the relations less than and greater than). Axioms 2–5, together with Theorems 4 and 5, show that the set of numbers N is ordered by either of the relations < and >.

THEOREM 6. Any numbers x and y satisfy one, and only one, of the three formulas: x = y, x < y and x > y.

PROOF. It follows from Axiom 1 that at least one of the formulas stated must be satisfied. In order to prove that the formulas:

x = y and x > y

exclude each other, we proceed as in the proof of Theorem 3: we replace in the second of these formulas “x” by “y” and arrive at a contradiction to Theorem 1. Similarly it can be shown that the formulas:

x = y and x > y

exclude each other. And finally, the two formulas:

x < y and x > y

cannot hold simultaneously, because, by Theorem 3, we would then have:

x < y and y < x,

in contradiction to Axiom 2. Hence, any numbers x and y satisfy one and no more of the three formulas in question, q.e.d.

Theorem 6 we will call the STRONG LAW OF TRICHOTOMY, or simply the LAW OF TRICHOTOMY; according to this law, one and only one of the three fundamental relations holds between any two given numbers. Using the phrase “either … or …” in the meaning proposed in Section 7, we can formulate Theorem 6 in a more concise manner:

For any numbers x and y we have either x = y or x < y
or x > y.

46. Other relations among numbers

Apart from the fundamental relations, three other relations play an important part in arithmetic. One of these is the logical relation of diversity ≠ which we know already; the other are the relations and which will be discussed now.

The meaning of the symbol “” is explained by the following definition:

DEFINITION 1. We say that if, and only if, x = y or x < y.

The formula:

is to be read: “x is less than or equal to y” or “x is at most equal to y”.

Although the content of the definition as stated appears to be clear, experience shows that in practical applications it sometimes becomes the source of certain misunderstandings. Some people who believe they understand the meaning of the symbol “” perfectly well protest nevertheless against its application to definite numbers. They do not only reject a formula like:

as obviously false—and this rightly so—, but they also consider as meaningless or even false such formulas as:

for they maintain that there is no sense in saying that or that since it is known that 0 = 0 and 0 < 1. In other words, it is not possible to exhibit a single pair of numbers which, in their opinion, satisfies the formula:

This view is palpably mistaken. Just because 0 < 1 holds, it follows that the sentence:

0 = 1 or 0 < 1

is true, for the disjunction of two sentences is certainly true provided one of them is true (cf. Section 7); but according to Definition 1 this disjunction is equivalent to the formula:

For a quite analogous reason the formula:

is also true.

The source of these misunderstandings, presumably, lies in certain habits of everyday life (*to which we have already called attention at the end of Section 7*). In ordinary language it is customary to assert the disjunction of two sentences only if we know that one of the sentences is true without knowing which. It does not occur to us to say that 0 = 1 or 0 < 1, though this is undoubtedly true, since we can say something that is simpler and at the same time logically stronger, namely, that 0 < 1. In mathematical considerations, however, it is not always advantageous to state everything that we know in its strongest possible form. For example, we sometimes assert of a quadrangle merely that it is a parallelogram, although we know it to be a square, and this because we may want to apply a general theorem concerning arbitrary parallelograms. For similar reasons it may occur that it is known of a number x (for instance, of the number 0) that it is less than 1, and yet it may merely be asserted that , that is, that either x = 1 or x < 1.

We will now state two theorems concerning the relation .

THEOREM 7. if, and only if, .

PROOF. This theorem is an immediate consequence of Theorem 6, i.e. the law of trichotomy. In fact, if

and hence, by Definition 1,

it is impossible for the formula:

x > y

to hold. Conversely, if

we must have (2) and hence, again by Definition 1, formula (1) must hold. The formulas (1) and (3) are thus equivalent, q.e.d.

In the terminology of Section 28, Theorem 7 states that the relation is the negation of the relation >.

On account of its structure, Theorem 7 might be looked upon as the definition of the symbol “”; it would be a different one from that adopted here but equivalent to it. The statement of this theorem may also contribute to dispel any last doubts about the usage of the symbol “”; for nobody will hesitate any longer to recognize as true such formulas as:

in view of the fact that they are equivalent to the formulas:

If we wished, we could avoid the use of the symbol “” completely, by always employing “” instead.

THEOREM 8. x < y if, and only if, and x ≠ y.

PROOF. If

then, by Definition 1,

while, by the law of trichotomy, the formula:

x = y

cannot hold. Conversely, if formula (2) holds, then by Definition 1 we obtain:

but if, at the same time, we have:

x ≠ y

we have to accept the first part of the disjunction (3), that is, formula (1). The implication therefore holds in both directions, q.e.d.

A number of other theorems concerning the relation we shall pass over; among them, there are, in particular, theorems to the effect that this relation is reflexive and transitive. The proofs of none of these theorems afford any difficulties.

The definition of the symbol “” is entirely analogous to Definition 1; and from the theorems concerning the relation we automatically obtain corresponding theorems concerning the relation by merely replacing the symbols “”, “<” and “>” throughout by the symbols “”, “>” and “<”.

Formulas of the form:

x = y

in which the places of “x” and “y” may be taken by constants, variables or compound expressions denoting numbers are usually called EQUATIONS. Similar formulas of the form:

x < y or x > y

are called INEQUALITIES (IN THE NARROWER SENSE); among the INEQUALITIES IN THE WIDER SENSE we have, in addition, formulas of the form:

The expressions occurring on the left and right sides of the symbols “=”, “<”, and so on, in these formulas are referred to as the LEFT AND RIGHT SIDES OR THE EQUATION OR OF THE INEQUALITY.

Exercises

1. Consider two relations among men: that of being of a smaller stature, and that of being of a larger stature. What condition has to be satisfied by an arbitrary set of people, so that it together with those two relations forms a model of the first group of axioms (of. Section 37)?

2. Let the formula:

express the fact that the numbers x and y satisfy one of the following conditions: (i) the number x has a smaller absolute value than the number y, or (ii) if the absolute values of x and y are the same, x is negative and y is positive. Further, let the formula:

have the same meaning as the formula:

Show, on the basis of arithmetic, that the set of all numbers and the relations and just defined constitute a model of the first group of axioms.

Give other examples of interpretations of these axioms within arithmetic and geometry.

3. From Theorem 1 derive the following theorem:

if x < y, then x ≠ y.

Conversely, derive Theorem 1 from the theorem just stated, without making use of any other arithmetical statements. Are these two inferences indirect and do they fall under the schema of the proof of Theorem 1 of Section 44?

4. Generalize the proof of Theorem 1 of Section 44, and thereby establish the following general law of the theory of relations (cf. remarks made in Section 37):

every relation R which is asymmetrical in the class K is also irreflexive in that class.

5. Show that, if Theorem 1 is adopted as a new axiom, the old Axiom 2 can be derived as a theorem from this axiom together with Axiom 4.

As a generalization of this argument, prove the following general law of the theory of relations:

every relation R which is irreflexive and transitive in the class K is also asymmetrical in that class.

*6. At the end of Section 44 we tried to explain why the proof of Theorem 2 may be omitted. These remarks represent an application of certain general considerations of Chapter VI. Explain this in detail, and, in particular, specify the considerations to which this refers.

7. Derive the following theorems from the first group of axioms:

(a) x = y if, and only if, and ;

(b) if x < y, then x < z or z < y.

8. Derive the following theorems from Axiom 4 and Definition 1:

(a) if x < y and , then x < z;

(b) if and y < z, then x < z;

9. Show that the relations and are reflexive, transitive and connected. Are these relations symmetrical or asymmetrical?

10. Show that, between any two numbers, exactly three of the following six relations hold: =, <, >, ≠, and .

11. Both the converse and the negation of any of the relations listed in the preceding exercise are again among these six relations. Show in detail that this is the case.

*12. Between which of the relations given in Exercise 10 does the relation of inclusion hold? What will be the sum, the product and the relative product of any pair among these relations?

Hint: Recall the terms explained in Section 28. Do not omit to consider pairs consisting of two equal relations, and remember that the relative product may depend upon the order of the factors (cf. Exercise 5 of Chapter V). Altogether 36 pairs of relations should be examined.

¹ This law, together with a related one of the same name:

[(~p) → p] → p,

has been used in many intricate and historically important arguments in logic and mathematics. The Italian logician and mathematician G. VAILATI (1863–1909) devoted a special monograph to its history.

² The letters “q.e.d.” are the customary abbreviation of the expression “quod erat demonstrandum”, meaning “which was to be proved”.