Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/120

From Wikisource
Jump to navigation Jump to search
This page has been proofread, but needs to be validated.
98
MATHEMATICAL LOGIC
[PART I

will mean "it is false that either is false or is false," which is equivalent to " and are both true"; and so on. For the present, and must be elementary propositions.

The above are all the primitive ideas required in the theory of deduction. Other primitive ideas will be introduced in Section B.

Definition of Implication. When a proposition follows from a proposition , so that if is true, must also be true, we say that implies . The idea of implication, in the form in which we require it, can be defined. The meaning to be given to implication in what follows may at first sight appear somewhat artificial; but although there are other legitimate meanings, the one here adopted is very much more convenient for our purposes than any of its rivals. The essential property that we require of implication is this: "What is implied by a true proposition is true." It is in virtue of this property that implication yields proofs. But this property by no means determines whether anything, and if so what, is implied by a false proposition. What it does determine is that, if implies , then it cannot be the case that is true and is false, i.e. it must be the case that either is false or is true. The most convenient interpretation of implication is to say, conversely, that if either is false or is true, then " implies " is to be true. Hence " implies " is to be defined to mean: "Either is false or is true." Hence we put:

*1·01.

Here the letters "" stand for "definition." They and the sign of equality together are to be regarded as forming one symbol, standing for "is defined to mean[1]." Whatever comes to the left of the sign of equality is defined to mean the same as what comes to the right of it. Definition is not among the primitive ideas, because definitions are concerned solely with the symbolism, not with what is symbolised; they are introduced for practical convenience, and are theoretically unnecessary.

In virtue of the above definition, when "" holds, then either is false or is true; hence if is true, must be true. Thus the above definition preserves the essential characteristic of implication; it gives, in fact, the most general meaning compatible with the preservation of this characteristic.

Primitive Propositions.

*1·1. Anything implied by a true elementary proposition is true. [2].

The above principle will be extended in *9 to propositions which are not elementary. It is not the same as "if is true, then if implies , is

  1. The sign of equality not followed by the letters "" will have a different meaning, to be defined later.
  2. The letters "" stand for "primitive proposition," as with Peano.