Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/139

Note that, in the above proof, "(1)" stands for the proposition

"$\scriptstyle {\sim (q.p).\supset .\sim (p.q)}$,"

as was explained in the proof of *2·31.

*3·24.⁠$\scriptstyle {\vdash .\sim (p.\sim p)}$

Dem.

${\begin{array}{ll}\scriptstyle {\left[*2\cdot 11{\frac {\sim p}{p}}\right]\quad }&\scriptstyle {\vdash .\sim p\lor \sim (\sim p).\supset }\\\scriptstyle {\left[*3\cdot 14{\frac {\sim p}{q}}\right]}&\scriptstyle {\vdash .\sim (p.\sim p)}\end{array}}$

The above is the law of contradiction.

*3·26.⁠$\scriptstyle {\vdash :p.q.\supset .p}$

Dem.

${\begin{array}{llr}\scriptstyle {\left[*2\cdot 02{\frac {q,p}{p,q}}\right]}&\scriptstyle {~\vdash :p.\supset .q\supset p\qquad }&\scriptstyle {(1)}\\\scriptstyle {[(1).(*1\cdot 01)]}&\scriptstyle {~\vdash :\sim p.\lor .\sim q\lor p:}\\\scriptstyle {[*2\cdot 31]}&\scriptstyle {\supset \vdash :\sim p\lor \sim q.\lor .p:}\\\scriptstyle {\left[*2\cdot 53{\frac {\sim p\lor \sim q,p}{p,\quad q}}\right]}&\scriptstyle {\supset \vdash :\sim (\sim p\lor \sim q).\supset .p\qquad }&\scriptstyle {(2)}\\\scriptstyle {[(2).(*3\cdot 01)]}&\scriptstyle {~\vdash :p.q.\supset .p}\end{array}}$

*3·27.⁠$\scriptstyle {\vdash :p.q.\supset .q}$

Dem.

${\begin{array}{ll}\scriptstyle {[*3\cdot 22]\quad \vdash :p.q.}&\scriptstyle {\supset .q.p.}\\\scriptstyle {\left[*3\cdot 26{\frac {q,p}{p,q}}\right]}&\scriptstyle {\supset .q:\supset \vdash .{\text{Prop}}}\end{array}}$

• 3·26·27 will both be called the "principle of simplification," like *2·02, from which they are deduced. They will be referred to as "Simp."

*3·3.⁠$\scriptstyle {\vdash :.p.q.\supset .r:\supset :p.\supset .q\supset r}$

Dem.

${\begin{array}{ll}\scriptstyle {[{\text{Id}}.(*3\cdot 01)]\quad \vdash :.p.q.\supset .r:}&\scriptstyle {\supset :\sim (\sim p\lor \sim q).\supset .r:}\\\scriptstyle {[{\text{Transp}}]}&\scriptstyle {\supset :\sim r.\supset .\sim p\lor \sim q:}\\\scriptstyle {[{\text{Id}}.(*1\cdot 01)]}&\scriptstyle {\supset :\sim r.\supset .p\supset \sim q:}\\\scriptstyle {[{\text{Comm}}]}&\scriptstyle {\supset :p.\supset .\sim r\supset \sim q:}\\\scriptstyle {[{\text{Transp}}.{\text{Syll}}]}&\scriptstyle {\supset :p.\supset .q\supset r:.\supset \vdash .{\text{Prop}}}\end{array}}$

*3·31.⁠$\scriptstyle {\vdash :.p.\supset .q\supset r:\supset :p.q.\supset .r}$

Dem.

${\begin{array}{ll}\scriptstyle {[{\text{Id}}.(*1\cdot 01)]\quad \vdash :.p.\supset .q\supset r:}&\scriptstyle {\supset :\sim p.\lor .\sim q\lor r:}\\\scriptstyle {[*2\cdot 31]}&\scriptstyle {\supset :\sim p\lor \sim q.\lor .r:}\\\scriptstyle {\left[*2\cdot 53{\frac {\sim p\lor \sim q,r}{p,\quad q}}\right]}&\scriptstyle {\supset :\sim (\sim p\lor \sim q).\supset .r:}\\\scriptstyle {[{\text{Id}}.(*3\cdot 01)]}&\scriptstyle {\supset :p.q.\supset .r:.\supset \vdash .{\text{Prop}}}\end{array}}$