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

98 $$\scriptstyle{\sim(\sim p\or\sim q)}$$ will mean "it is false that either $$\scriptstyle{p}$$ is false or $$\scriptstyle{q}$$ is false," which is equivalent to "$\scriptstyle{p}$ and $$\scriptstyle{q}$$ are both true"; and so on. For the present, $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ 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 $$\scriptstyle{q}$$ follows from a proposition $\scriptstyle{p}$, so that if $$\scriptstyle{p}$$ is true, $$\scriptstyle{q}$$ must also be true, we say that $$\scriptstyle{p}$$ implies $\scriptstyle{q}$. 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 $$\scriptstyle{p}$$ implies $\scriptstyle{q}$, then it cannot be the case that $$\scriptstyle{p}$$ is true and $$\scriptstyle{q}$$ is false, i.e. it must be the case that either $$\scriptstyle{p}$$ is false or $$\scriptstyle{q}$$ is true. The most convenient interpretation of implication is to say, conversely, that if either $$\scriptstyle{p}$$ is false or $$\scriptstyle{q}$$ is true, then "$\scriptstyle{p}$ implies $\scriptstyle{q}$" is to be true. Hence "$\scriptstyle{p}$ implies $\scriptstyle{q}$" is to be defined to mean: "Either $$\scriptstyle{p}$$ is false or $$\scriptstyle{q}$$ is true." Hence we put:

$$\scriptstyle{p\supset q.=.\sim p\or q\quad\text{Df.}}$$

Here the letters "$\scriptstyle{\text{Df}}$"|undefined stand for "definition." They and the sign of equality together are to be regarded as forming one symbol, standing for "is defined to mean ." 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 "$\scriptstyle{p\supset q}$" holds, then either $$\scriptstyle{p}$$ is false or $$\scriptstyle{q}$$ is true; hence if $$\scriptstyle{p}$$ is true, $$\scriptstyle{q}$$ 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.

Anything implied by a true elementary proposition is true. $\scriptstyle{\text{Pp}}$ .|undefined

The above principle will be extended in *9 to propositions which are not elementary. It is not the same as "if $$\scriptstyle{p}$$ is true, then if $$\scriptstyle{p}$$ implies $\scriptstyle{q}$, $$\scriptstyle{q}$$ is