Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/29

 $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ as arguments. It is also called the logical product of $$\scriptstyle{p}$$ and $\scriptstyle{q}$. Accordingly $$\scriptstyle{p \cdot q}$$ means that both $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ are true. It is easily seen that this function can be defined in terms of the two preceding functions. For when $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ are both true it must be false that either $$\scriptstyle{\sim p}$$ or $$\scriptstyle{\sim q}$$ is true. Hence in this book $$\scriptstyle{p \cdot q}$$ is merely a shortened form of symbolism for If any further idea attaches to the proposition "both $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ are true," it is not required here.

The Implicative Function is a propositional function with two arguments $$\scriptstyle{p}$$ and $\scriptstyle{q}$, and is the proposition that either not-$\scriptstyle{p}$ or $$\scriptstyle{q}$$ is true, that is, it is the proposition $$\scriptstyle{\sim p \or q}$$. Thus if $$\scriptstyle{p}$$ is true $$\scriptstyle{\sim p}$$ is false, and accordingly the only alternative left by the proposition $$\scriptstyle{\sim p \or q}$$ is that $$\scriptstyle{q}$$ is true. In other words if $$\scriptstyle{p}$$ and $$\scriptstyle{\sim p \or q}$$ are both true, then $$\scriptstyle{q}$$ is true. In this sense the proposition $$\scriptstyle{\sim p \or q}$$ will be quoted as stating that $$\scriptstyle{p}$$ implies $\scriptstyle{q}$. The idea contained in this propositional function is so important that it requires a symbolism which with direct simplicity represents the proposition as connecting $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ without the intervention of $$\scriptstyle{\sim p}$$. But "implies" as used here expresses nothing else than the connection between $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ also expressed by the disjunction "not-$\scriptstyle{p}$ or $\scriptstyle{q}$." The symbol employed for "$\scriptstyle{p}$ implies $\scriptstyle{q}$," i.e. for "$\scriptstyle{\sim p \or q}$," is "$\scriptstyle{p \supset q}$." This symbol may also be read "if $\scriptstyle{p}$, then $\scriptstyle{q}$." The association of implication with the use of an apparent variable produces an extension called "formal implication." This is explained later: it is an idea derivative from "implication" as here defined. When it is necessary explicitly to discriminate "implication" from "formal implication," it is called "material implication." Thus "material implication" is simply "implication" as here defined. The process of inference, which in common usage is often confused with implication, is explained immediately.

These four functions of propositions are the fundamental constant (i.e. definite) propositional functions with propositions as arguments, and all other constant propositional functions with propositions as arguments, so far as they are required in the present work, are formed out of them by successive steps. No variable propositional functions of this kind occur in this work.

Equivalence. The simplest example of the formation of a more complex function of propositions by the use of these four fundamental forms is furnished by "equivalence." Two propositions $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ are said to be "equivalent" when $$\scriptstyle{p}$$ implies $$\scriptstyle{q}$$ and $$\scriptstyle{q}$$ implies $\scriptstyle{p}$. This relation between $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ is denoted by "$$\scriptstyle{p \equiv q}$$. Thus "$$\scriptstyle{p \equiv q}$$" stands for "$$\scriptstyle{(p \supset q) \cdot (q \supset p)}$$." It is easily seen that two propositions are equivalent when, and only when, they are both true or are both false. Equivalence rises in the scale of importance when we come to "formal implication" and thus to "formal equivalence." It must not be supposed that two propositions which are equivalent are in