Rules of the Copula
The sign is subject to three algebraical rules,2 as follows:—
RULE I. If x y and y z, then x z. This is called the principle of the transitiveness of the copula.
RULE II. Either x y or y z.
RULE III. There are two propositions, u and v, such that v u is false.
To these is to be added the following:
RULE OF INTERPRETATION. If y is true, v y; and if y is false, y u.
Rule I can be proved from propositions, A, B, C. For by C, if x y either x is false or y is true. In the statement of A, substitute z for y. It then reads that if x is false, x z. Hence, if x y, either x z or y is true. Call this proposition P. In the statement of C, substitute y for x, and z for y. It then reads that if y z, either y is false or z is true. Combining this with P, we see that if x y and y z, either x z or z is true. Call this proposition Q. In the statement of B, substitute z for y. It then reads that if z is true, x z. Combining this with Q, we conclude that if x y and y z, x z. Q.E.D.
Rule II can be proved from propositions A and B alone. For in the statement of A, substitute y for x and z for y. It then reads that if y is false, y z. But by B, if y is true, x y. Hence, either x y or y z. Q.E.D.
Rule III can be proved from proposition C alone. For in the statement of C, substitute v for x and u for y; and it reads that if v u, either v is false or u is true. If, therefore, v is any true proposition, and u any false one, it is not true that v u. Thus, it is possible so to take u and v that v u shall be false. Q.E.D.
The rule of interpretation evidently follows from propositions A and B.
That these four rules fully represent propositions A, B, C, can be shown by deducing the latter from the former. It is left to the student to construct these proofs.
Let us consider the three algebraical rules by themselves, independently of the rule of interpretation. Rule I shows that x y expresses a relation between x and y analogous to that of numbers on a scale, the number x being at least as low on the scale as y. For if x is at least as low as y, and y at least as low as z, then x is at least as low as z. Rule II shows that this scale has not more than two places upon it. For if one number, y, could be lower on the scale than a second, x, and at the same time higher than a third, z, neither x y nor y z would be true. Rule III shows that the scale has at least two places. For if it had but one, any one number would be as low as any other and we should have x y for all values of x and y. Finally the rule of interpretation shows that the higher point on the scale represents the truth and the lower falsity. [Note A. In the ordinary logic, the fact that there are not more than two varieties of propositions in respect to truth is expressed by the so-called Principle of Excluded Middle, which is that every proposition is either true or false, (or A is either B or not-B); while the fact that there are at least two different varieties is expressed by the so-called Principle of Contradiction, which is that nothing is both true and false, (or A is not not-A).]
I now proceed to deduce a few useful formulae from Rules I, II, III. In Rule II, substitute x for z, and we have
(1). Either x y or y x.
In (1), substitute x for y, and we have
(2). x x.
By Rules II and III,
(3). x v.
(4). u x.
By Rule I,
(5). If x y, while it is false that x z, then it is false that y
