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

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SECTION A]

EQUIVALENCE AND FORMAL RULES

121

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

$\scriptstyle {p\equiv q.\supset .f(p)\equiv f(q)}$ .

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:

*4·1. $\scriptstyle {\vdash :p\supset q.\equiv .\sim q\supset \sim p}$

*4·11. $\scriptstyle {\vdash :p\equiv q.\equiv .\sim p\equiv \sim q}$

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

*4·13. $\scriptstyle {\vdash .p\equiv \sim (\sim p)}$

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

*4·2. $\scriptstyle {\vdash .p\equiv p}$

*4·21. $\scriptstyle {\vdash :p\equiv q.\equiv .q\equiv p}$

*4·22. $\scriptstyle {\vdash :p\equiv q.q\equiv r.\supset .p\equiv r}$

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

*4·24. $\scriptstyle {\vdash :p.\equiv .p\lor p}$

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.