Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/143

SECTION A] of these ideas alone. We shall give the name of a truth-function to a function $$\scriptstyle{f(p)}$$ whose argument is a proposition, and whose truth-value depends only upon the truth-value of its argument. All the functions of propositions with which we shall be specially concerned will be truth-functions, i.e. we shall have The reason of this is, that the functions of propositions with which we deal are all built up by means of the primitive ideas of *1. But it is not a universal characteristic of functions of propositions to be truth-functions. For example, "$\scriptstyle{A}$ believes $\scriptstyle{p}$" may be true for one true value of $$\scriptstyle{p}$$ and false for another.

The principal propositions of this number are the following:

$$\scriptstyle{\vdash:p\supset q.\equiv.\sim q\supset\sim p}$$

$$\scriptstyle{\vdash:p\equiv q.\equiv.\sim p\equiv\sim q}$$

These are both forms of the "principle of transposition."

$$\scriptstyle{\vdash.p\equiv\sim(\sim p)}$$

This is the principle of double negation, i.e. a proposition is equivalent to the falsehood of its negation.

$$\scriptstyle{\vdash.p\equiv p}$$

$$\scriptstyle{\vdash:p\equiv q.\equiv.q\equiv p}$$

$$\scriptstyle{\vdash:p\equiv q.q\equiv r.\supset.p\equiv r}$$

These propositions assert that equivalence is reflexive, symmetrical and transitive.

$$\scriptstyle{\vdash:p.\equiv.p\or p}$$

I.e. $$\scriptstyle{p}$$ is equivalent to "$\scriptstyle{p}$ and $\scriptstyle{p}$" and to "$\scriptstyle{p}$ or $\scriptstyle{p}$," which are two forms of the law of tautology, and are the source of the principal differences between the algebra of symbolic logic and ordinary algebra.

$$\scriptstyle{\vdash:p.q.\equiv.q.p}$$

This is the commutative law for the product of propositions.

$$\scriptstyle{\vdash:p\or q.\equiv.q\or p}$$

This is the commutative law for the sum of propositions.

The associative laws for multiplication and addition of propositions, namely

$$\scriptstyle{\vdash:(p.q).r.\equiv.p.(q.r)}$$

$$\scriptstyle{\vdash:(p\or q)\or r.\equiv.p\or(q\or r)}$$

The distributive law in the two forms