Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/126

*2·01.$\scriptstyle {\vdash :p\supset \sim p.\supset .\sim p}$

This proposition states that, if $\scriptstyle {p}$ implies its own falsehood, then $\scriptstyle {p}$ is false. It is called the "principle of the reductio ad absurdum," and will be referred to as "Abs."^[1]. The proof is as follows (where "Dem." is short for demonstration"):

Dem. (1)

${\begin{array}{ll}\scriptstyle {\left[{\text{Taut}}{\frac {\sim p}{p}}\right]\quad }&\scriptstyle {\vdash :\sim p\lor \sim p.\supset .\sim p}\\\scriptstyle {[(1).(*1\cdot 01)]\quad }&\scriptstyle {\vdash :p\supset \sim p.\supset .\sim p}\end{array}}$

*2·02.$\scriptstyle {\vdash :q.\supset .p\supset q}$

Dem. (1)

${\begin{array}{ll}\scriptstyle {\left[{\text{Add}}{\frac {\sim p}{p}}\right]\quad }&\scriptstyle {\vdash :q.\supset .\sim p\lor q}\\\scriptstyle {[(1).(*1\cdot 01)]\quad }&\scriptstyle {\vdash :q.\supset .p\supset q}\end{array}}$

*2·03.$\scriptstyle {\vdash :p\supset \sim q.\supset .q\supset \sim p}$

Dem. (1)

${\begin{array}{ll}\scriptstyle {\left[{\text{Perm}}{\frac {\sim p,\sim q}{p,~~q}}\right]\quad }&\scriptstyle {\vdash :\sim p\lor \sim q.\supset .\sim q\lor \sim p}\\\scriptstyle {[(1).(*1\cdot 01)]\quad }&\scriptstyle {\vdash :p\supset \sim q.\supset .q\supset \sim p}\end{array}}$

*2·04.$\scriptstyle {\vdash :.p.\supset .q\supset r:\supset :q.\supset .p\supset r}$

Dem. (1)

${\begin{array}{ll}\scriptstyle {\left[{\text{Assoc}}{\frac {\sim p,\sim q}{p,~~q}}\right]\quad }&\scriptstyle {\vdash :.\sim p\lor (\sim q\lor r).\supset .\sim q\lor (\sim p\lor r)}\\\scriptstyle {[(1).(*1\cdot 01)]\quad }&\scriptstyle {\vdash :.p.\supset .q\supset r:\supset :q.\supset .p\supset r}\end{array}}$

*2·05.$\scriptstyle {\vdash :.q\supset r.\supset :p\supset q.\supset .p\supset r}$

Dem. (1)

${\begin{array}{ll}\scriptstyle {\left[{\text{Sum}}{\frac {\sim p}{p}}\right]\quad }&\scriptstyle {\vdash :.q\supset r.\supset :\sim p\lor q.\supset .\sim p\lor r}\\\scriptstyle {[(1).(*1\cdot 01)]\quad }&\scriptstyle {\vdash :.q\supset r.\supset :p\supset q.\supset .p\supset r}\end{array}}$

*2·06.$\scriptstyle {\vdash :.p\supset q.\supset :q\supset r.\supset .p\supset r}$

Dem. (1)
(2)

${\begin{array}{lll}\scriptstyle {\left[{\text{Comm}}{\frac {q\supset r,p\supset q,p\supset r}{p,~~q,~~r}}\right]\quad }&\scriptstyle {\vdash ::}&\scriptstyle {q\supset r.\supset :p\supset q.\supset .p\supset r:.}\\&&\scriptstyle {\supset :.p\supset q.\supset :q\supset r.\supset .p\supset r}\\\scriptstyle {[*2\cdot 05]\quad }&\scriptstyle {\vdash :.}&\scriptstyle {q\supset r.\supset :p\supset q.\supset .p\supset r}\\\scriptstyle {[(1).(2).*1\cdot 11]\quad }&\scriptstyle {\vdash :.}&\scriptstyle {p\supset q.\supset :q\supset r.\supset .p\supset r}\end{array}}$

1. There is an interesting historical article on this principle by Vailati, "A proposito d'un passo del Teeteto e di una dimostrazione di Euclide," Rivista di Filosofia e scienze affine, 1904.