Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/36

assertion with the $\scriptstyle {x}$ in the other. This will be the case—so our axiom states—if both assertions present $\scriptstyle {x}$ as the argument to some one function, that is to say, if $\scriptstyle {\phi x}$ is a constituent in both assertions (whatever propositional function $\scriptstyle {\phi }$ may be), or, more generally, if $\scriptstyle {\phi (x,y,z,\ldots )}$ is a constituent in one assertion, and $\scriptstyle {\phi (x,u,v,\ldots )}$ is a constituent in the other. This axiom introduces notions which have not yet been explained; for a fuller account, see the remarks accompanying *3·03 (which is the statement of this axiom) in the body of the work, as well as the explanation of propositional functions and ambiguous assertion to be given shortly.

Some simple propositions. In addition to the primitive propositions we have already mentioned, the following are among the most important of the elementary properties of propositions appearing among the deductions.

The law of excluded middle:

$\scriptstyle {\vdash .p\lor \sim p}$.

This is *2·11 below. We shall indicate in brackets the numbers given to the following propositions in the body of the work. The law of contradiction (*3·24):

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

The law of double negation (*4·13):

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

The principle of transposition, i.e. "if $\scriptstyle {p}$ implies $\scriptstyle {q}$, then not-$\scriptstyle {q}$ implies not-$\scriptstyle {p}$," and vice versa: this principle has various forms, namely

${\begin{array}{ll}\scriptstyle {(*4\cdot 1)}&\scriptstyle {\vdash :p\supset q.\equiv .\sim q\supset \sim p,}\\\scriptstyle {(*4\cdot 11)}&\scriptstyle {\vdash :p\equiv q.\equiv .\sim p\equiv \sim q,}\\\scriptstyle {(*4\cdot 14)}&\scriptstyle {\vdash :.p.q.\supset .r:\equiv :p.\sim r.\supset .\sim q,}\end{array}}$

as well as others which are variants of these. The law of tautology, in the two forms:

${\begin{aligned}\\\scriptstyle {(*4\cdot 24)\ \ }&\scriptstyle {\vdash :p.\equiv .p.p,}\\\scriptstyle {(*4\cdot 25)\ \ }&\scriptstyle {\vdash :p.\equiv .p\lor p,}\\\end{aligned}}$

i.e. "$\scriptstyle {p}$ is true" is equivalent to "$\scriptstyle {p}$ is true and $\scriptstyle {p}$ is true," as well as to "$\scriptstyle {p}$ is true or $\scriptstyle {p}$ is true." From a formal point of view, it is through the law of tautology and its consequences that the algebra of logic is chiefly distinguished from ordinary algebra. The law of absorption:

$\scriptstyle {(*4\cdot 71)\ \ \vdash :.p\supset q.\equiv :p.\equiv .p.q}$,

i.e. "$\scriptstyle {p}$ implies $\scriptstyle {q}$" is equivalent to "$\scriptstyle {p}$ is equivalent to $\scriptstyle {p.q}$." This is called the law of absorption because it shows that the factor $\scriptstyle {q}$ in the product is