Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/134

*2·47.⁠$\scriptstyle {\vdash :\sim (p\lor q).\supset .\sim p\lor q\qquad \left[*2\cdot 45.*2\cdot 2{\frac {\sim p}{p}}.{\text{Syll}}\right]}$

*2·48.⁠$\scriptstyle {\vdash :\sim (p\lor q).\supset .p\lor \sim q\qquad \left[*2\cdot 46.*1\cdot 3{\frac {\sim q}{q}}.{\text{Syll}}\right]}$

*2·49.⁠$\scriptstyle {\vdash :\sim (p\lor q).\supset .\sim p\lor \sim q\qquad \left[*2\cdot 45.*2\cdot 2{\frac {\sim p,\sim q}{p,~~q}}.{\text{Syll}}\right]}$

*2·5.⁠$\scriptstyle {\vdash :\sim (p\supset q).\supset .\sim p\supset q\qquad \left[*2\cdot 47{\frac {\sim p}{p}}\right]}$

*2·51.⁠$\scriptstyle {\vdash :\sim (p\supset q).\supset .p\supset \sim q\qquad \left[*2\cdot 48{\frac {\sim p}{p}}\right]}$

*2·52.⁠$\scriptstyle {\vdash :\sim (p\supset q).\supset .p\supset \sim q\qquad \left[*2\cdot 49{\frac {\sim p}{p}}\right]}$

*2·521.⁠$\scriptstyle {\vdash :\sim (p\supset q).\supset .q\supset p\qquad [*2\cdot 52\cdot 17]}$

*2·53.⁠$\scriptstyle {\vdash :p\lor q.\supset .\sim p\supset q}$

Dem.

$\scriptstyle {\vdash .*2\cdot 12\cdot 38.\supset \vdash :p\lor q.\supset .\sim (\sim p)\lor q:\supset \vdash .{\text{Prop}}}$

*2·54.⁠$\scriptstyle {\vdash :\sim p\supset q.\supset .p\lor q\qquad [*2\cdot 14\cdot 38]}$

*2·55.⁠$\scriptstyle {\vdash :.\sim p.\supset :p\lor q.\supset .q\quad [*2\cdot 53.{\text{Comm}}]}$

*2·56.⁠$\scriptstyle {\vdash :.\sim q.\supset :p\lor q.\supset .p\quad \left[*2\cdot 55{\frac {q,p}{p,q}}.{\text{Perm}}\right]}$

*2·6.⁠$\scriptstyle {\vdash :.\sim p\supset q.\supset :p\supset q.\supset .q}$

Dem.

${\begin{aligned}&{\begin{array}{lll}\scriptstyle {[*2\cdot 38]}&\scriptstyle {\vdash :.\sim p\supset q.\supset :\sim p\lor q.\supset .q\lor q}&\scriptstyle {(1)}\\\scriptstyle {[{\text{Taut}}.{\text{Syll}}]}&\scriptstyle {\vdash :.\sim p\lor q.\supset .q\lor q:\supset :\sim p\lor q.\supset .q\qquad }&\scriptstyle {(2)}\end{array}}\\&\scriptstyle {\vdash .(1).(2).{\text{Syll}}.\supset \vdash :.\sim p\supset q.\supset :\sim p\lor q.\supset .q:.\supset \vdash .{\text{Prop}}}\end{aligned}}$

*2·61.⁠$\scriptstyle {\vdash :.p\supset q.\supset :\sim p\supset q.\supset .q\quad [*2\cdot 6.{\text{Comm}}]}$

*2·61.⁠$\scriptstyle {\vdash :.p\supset q.\supset :\sim p\supset q.\supset .q\qquad [*2\cdot 6.{\text{Comm}}]}$

*2·62.⁠$\scriptstyle {\vdash :.p\lor q.\supset :p\supset q.\supset .q\qquad [*2\cdot 53\cdot 6.{\text{Syll}}]}$

*2·621.⁠$\scriptstyle {\vdash :.p\supset q.\supset :p\lor q.\supset .q\qquad [*2\cdot 62.{\text{Comm}}]}$

*2·63.⁠$\scriptstyle {\vdash :.p\lor q.\supset :\sim p\lor q.\supset .q\qquad [*2\cdot 62]}$

*2·64.⁠$\scriptstyle {\vdash :.p\lor q.\supset :p\lor \sim q.\supset .p\qquad \left[*2\cdot 63{\frac {q,p}{p,q}}.{\text{Perm}}\right]}$

*2·65.⁠$\scriptstyle {\vdash :.p\supset q.\supset :p\supset \sim q.\supset .\sim p\quad \left[*2\cdot 64{\frac {\sim p}{p}}\right]}$

*2·67.⁠$\scriptstyle {\vdash :.p\lor q.\supset .q:\supset .p\supset q}$

Dem.

${\begin{aligned}&{\begin{array}{lll}\scriptstyle {[*2\cdot 54.{\text{Syll}}]~}&\scriptstyle {\vdash :.p\lor q.\supset .q:\supset :\sim p\supset q.\supset .q\qquad }&\scriptstyle {(1)}\\\scriptstyle {[*2\cdot 24.{\text{Syll}}]}&\scriptstyle {\vdash :.\sim p\supset q.\supset .q:\supset .p\supset q}&\scriptstyle {(2)}\end{array}}\\&\scriptstyle {\vdash .(1).(2).{\text{Syll}}.\supset \vdash .{\text{Prop}}}\end{aligned}}$