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

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СКАЧАТЬ then, may be summed up in the following propositions, which I mark L, M, N, for convenience of future reference.

      L. If x = y, then either x is true or y is false.¹

      M. If x = y, then either x is false or y is true.

      N. If x and y are either both true or false, then x = y.

      From this definition of the sign of equality, it follows that in this algebra it is subject to precisely the same rules as in ordinary algebra.² These rules are as follows:

      Rule 1. x = x.

      Rule 2. If x = y, then y = x.

      Rule 3. If x = y, and y = z, then x = z.

      I proceed to give formal proofs of these rules; for though they are evidently true, it may not be quite evident that their truth follows necessarily, or how it does so, from the propositions L, M, N. At any rate the proofs will be valuable as examples of demonstration carried to the last pitch of formalism.

      Rule 1. Any proposition, x, is either true or false. Call this statement E. In N, write x in place both of x and of y. From N, so stated, together with E, we conclude x = x.

      Rule 2. Suppose x = y, which statement we may refer to as P. Then, all we have to prove is that y = x. From L and P, it follows that either x is false or y is true. Call these alternatives A and A′ respectively. We examine first the alternative A. By M and P, either x is true or y is false. Call this statement (having two alternatives) B. But no proposition, x, is both true and false. Call this statement C. From B and C, we conclude that y is false. Thus, the first alternative, A, is that x is false and y is false. Next, we examine the other alternative, A′. From M and P, we conclude B, as before. But no proposition, y, is both true and false. Call this statement C′. From B and C′, we conclude that x is true. Then the second alternative is that y is true and x is true. Thus, there are but two alternatives, either that x and y are both true or that they are both false. Call this compound statement D. In the statement of N, substitute x for y and y for x. Then, from N so stated, together with D, we conclude that y = x, which is all we had to prove.

      Rule 3. Any proposition, y, is either true or false. Call these two alternatives A and A′. We first examine the alternative A. No proposition, y, is both true and false. Call this statement C. By M, A, and C, if x = y, then x is true. Call this conditional proposition B. In the statement of L substitute y for x and z for y. Then, from L, so stated, A, and C, we conclude that if y = z then z is true. Call this conditional proposition D. In the statement of N, substitute z for y. Then, from N, so stated, from A, D, it follows that if x = y and y = z, then x = z. Next, I examine the other alternative A′. From L, A′, and C, it follows that if x = y, then x is false. Call this statement B′. From M, stated as before, A′, and C, we conclude that if y = z, then z is false. Call this statement D′. Then from N, stated as before, A′, B′, and D′, we conclude that if x = y and y = z, then x = z. This being the case under both alternatives, we conclude it unconditionally.

      Addition and multiplication, and their cognate words and algebraical signs, are used in such sense that x + y means that either x or y is true (without excluding the possibility of both being so), while xy means that both x and y are true. More explicitly, the meanings of the sum and product are summed up in the following propositions, which are lettered A, B, C, X, Y, Z, for convenience of reference.

A. Either x is false or x + y is true.	X. Either xy is false or x is true.
B. Either y is false or x + y is true.	Y. Either xy is false or y is true.
C. Either x + y is false or y is true.	Z. Either x or y is false or xy is true.

      From these definitions it follows that in this algebra, all the ordinary rules of addition and multiplication hold good, together with some other rules besides. The rules common to logical and arithmetical algebra are the following.

      Rule 4. The associative principle of addition.

      (x + y) + z = x + (y + z).

      Rule 5. The associative principle of multiplication. (xy)z = x(yz).

      Rule 6. The commutative principle of addition. x + y = y + x.

      Rule 7. The commutative principle of multiplication. xy = yx.

      Rule 8. The distributive principle of multiplication with reference to addition. x(y + z) = xy + xz.

      The rules peculiar to logical algebra may be stated as follows:

      Rule 9. x + x = xx.

      Rule 10. If x + y = xz, then either x + y = x or xz = x.

      EXERCISE 1. Prove the above rules, from propositions L, M, N, A, B, C, X, Y, Z.

      The above rules are made to conform as much as possible to those of ordinary algebra, and suppose that we are dealing with equations. But as a general rule, we shall not have any equations, but having written down a statement, the problem before us will be to ascertain what follows from it. In that case, it will be better to work by the following system of rules, which for the sake of distinction, I shall term principles.

      Principle I. The commutative principle. The order of factors and additive terms is indifferent, that is, x + y = y + x and xy = yx.

      Principle II. The principle of erasing parentheses. We always have a right to erase a parenthesis in any asserted proposition. This includes the associative principle, and also permits us to infer x + yz from (x + y)z.

      Principle III. From any part of an asserted proposition, we have the right to erase any factor; and to any part we have a right to logically add anything we like. Thus, from xy we can infer x + z.

      Principle IV. We have a right to repeat any factor, and to drop any additive term that is equal to another such term. Thus, from x we can infer xx, and from x + x we can infer x.

      EXERCISE 2. Prove the above four principles from propositions L, M, and N, together with rules 4 to 10.

      EXERCISE 3.

      1. By means of the ten rules alone, prove that addition is distributive with respect to multiplication; that is, that

      x + yz = (x + y)(x + z).

      2. By means of the four principles alone, show that from x(y + z) we can infer xy + xz.

      3. СКАЧАТЬ