Writings of Charles S. Peirce: A Chronological Edition, Volume 8. Charles S. Peirce

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      (6). If y z, while it is false that x z, then it is false that x y.

      (7). If it is false that x z, it is either false that x y or that y z.

      By Rules I and III,

      (8). It is either false that v x or false that x u.

      By Rule II,

      (9). Either v x or x u.

      [Note B. An entire calculus of logic might be made with the sign alone. Some of the most important formulae would be, as follows:—

      x x.

      If x (y z), then y (x z).

      (v x) x.

      But such a calculus would be useless on account of its complexity.]

      [Note C. The equation x = y means, of course, that x and y are at the same point of the scale. That is to say, the definition of x = y is contained in the following propositions:

      A. If x = y, x y.

      B. If x = y, y x.

      C. If x y and y x, then x = y.

      From these propositions, it follows that logical equality is subject to the following rules:—

      i. x = x.

      ii. If x = y, then y = x.

      iii. If x = y and y = z, then x = z.

      iv. Either x = y or y = z or z = x.

      The proof of these from A, B, C, by means of Rules I, II, III, is left to the student.]

      §3. LOGICAL ADDITION AND MULTIPLICATION

      [Note. The sign as explained above, is, we may trust, free from every trace of ambiguity. But while it does not hesitate between two meanings, it does carry two meanings at one and the same time. The expression x y means that either x is false or y true; but it also means that x is at least as low as y upon a scale. In short, x y not only states something, but states it under a particular aspect; and though it is anything but a poetical or rhetorical expression, it conveys its purport by means of an arithmetical simile. Now, elegance requires that this simile, once adopted, should be adhered to; and elegance, as we shall find, is every whit as important a consideration in the art of reasoning as it is in the more sensuous modes to which the name of Art is commonly appropriated. Following out this analogy, then, we proceed to inquire what are to be the logical significations of addition, subtraction, multiplication, and division.]

      Any two numbers whatever (say 5 and 2) might be chosen for u and v, the representatives of the false and the true; though there is some convenience in making v the larger. Then, the principle of contradiction is satisfied by these being different numbers; for a number, x, cannot at once be equal to 5 and to 2, and therefore the proposition represented by x cannot be at once true and false. But in order to satisfy the principle of excluded middle, that every proposition is either true or false, every letter, x, signifying a proposition must, considered as a number, be supposed subject to a quadratic equation whose roots are u and v. In short, we must have

      (x − u)(v − x) = 0.

      Since the product forming the left hand member of this equation vanishes, one of the factors must vanish. So that either x − u = 0 and x = u, or v − x = 0 and x = v. Another way of expressing the principle of excluded middle would be:

      It will be found, however, that occasion seldom arises for taking explicit account of the principle of excluded middle.

      The propositions

      Either x is false or y is true,

      and

      Either y is false or z is true,

      are expressed by the equations

      (x − u)(v − y) = 0

      (y − u)(v − z) = 0.

      For, as before, to say that the product forming the first member of each equation vanishes, is equivalent to saying that one or other factor vanishes.

      Let us now eliminate y from the above two equations. For this purpose, we multiply the first by (v − z) and the second by (x − u).

      We, thus, get

      (x − u)(v − y)(v − z) = 0

      (x − u)(y − u)(v − z) = 0.

      We now add these two equations and get

      (x − u)(v − u)(v − z) = 0.

      But the factor v − u does not vanish. We, therefore, divide by it, and so find

      (x − u)(v − z) = 0.

      The signification of this is,

      Either x is false or z is true;