Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/130

Note. The proposition to be proved will be called "$\scriptstyle {\text{Prop}}$," and when a proof ends, like that of *2·16, by an implication between asserted propositions, of which the consequent is the proposition to be proved, we shall write "$\scriptstyle {\vdash .{\text{etc.}}\supset \vdash .{\text{Prop}}}$". Thus "$\scriptstyle {\supset \vdash .{\text{Prop}}}$" ends a proof, and more or less corresponds to "q.e.d."

*2·17.⁠$\scriptstyle {\vdash :\sim q\supset \sim p.\supset .p\supset q}$

Dem.

${\begin{array}{lll}\scriptstyle {\left[*2\cdot 03{\frac {\sim q,p}{p,~q}}\right]\quad }&\scriptstyle {\vdash :\sim q\supset \sim p.\supset .p\supset \sim (\sim q)\qquad \qquad }&\scriptstyle {(1)}\\\scriptstyle {[*2\cdot 14]\quad }&\scriptstyle {\vdash :\sim (\sim q)\supset q:\supset }&\\\scriptstyle {[*2\cdot 05]\quad }&\scriptstyle {\vdash :p\supset \sim (\sim q).\supset .p\supset q\qquad \qquad }&\scriptstyle {(2)}\\\scriptstyle {[{\text{Syll}}]\quad }&\scriptstyle {\vdash .(1).(2).\supset \vdash .{\text{Prop}}}\end{array}}$

• 2·15, *2·16 and *2·17 are forms of the principle of transposition, and will be all referred to as "Transp."

*2·18.⁠$\scriptstyle {\vdash :\sim p\supset p.\supset .p}$

Dem.

${\begin{array}{lll}\scriptstyle {[*2\cdot 12]\quad }&\scriptstyle {\vdash .p\supset \sim (\sim p).\supset }\\\scriptstyle {[*2\cdot 05]\quad }&\scriptstyle {\vdash .\sim p\supset p.\supset .\sim p\supset \sim (\sim p)\qquad \qquad }&\scriptstyle {(1)}\\\scriptstyle {\left[*2\cdot 01{\frac {\sim p}{p}}\right]\quad }&\scriptstyle {\vdash :\sim p\supset \sim (\sim p).\supset .\sim (\sim p)\qquad \qquad }&\scriptstyle {(2)}\\\scriptstyle {[{\text{Syll}}]\quad }&\scriptstyle {\vdash .(1).(2).\supset \vdash :\sim p\supset p.\supset .\sim (\sim p)\qquad \qquad }&\scriptstyle {(3)}\\\scriptstyle {[*2\cdot 14]\quad }&\scriptstyle {\vdash .\sim (\sim p)\supset p\qquad \qquad }&\scriptstyle {(4)}\\\scriptstyle {[{\text{Syll}}]\quad }&\scriptstyle {\vdash .(3).(4).\supset \vdash .{\text{Prop}}}\end{array}}$

This is the complement of the principle of the reductio ad absurdum. It states that a proposition which follows from the hypothesis of its own falsehood is true.

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

Dem.

${\begin{array}{ll}\scriptstyle {\vdash .{\text{Add}}.\supset \vdash :p.\supset .q\lor p\qquad \qquad }&\scriptstyle {(1)}\\\scriptstyle {[{\text{Perm}}]\vdash :q\lor p.\supset .p\lor q\qquad \qquad }&\scriptstyle {(2)}\\\scriptstyle {[{\text{Syll}}]~~\vdash .(1).(2).\supset \vdash .{\text{Prop}}}\end{array}}$

*2·21.⁠$\scriptstyle {\vdash :\sim p.\supset .p\supset q\quad \left[*2\cdot 2{\frac {\sim p}{p}}\right]}$

The above two propositions are very frequently used.

*2·24.⁠$\scriptstyle {\vdash :p.\supset .\sim p\supset q\quad [*2\cdot 21.{\text{Comm}}]}$