Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/140

*3·33.$\scriptstyle {\vdash :p\supset q.q\supset r.\supset .p\supset r\quad [{\text{Syll}}.{\text{Imp}}]}$

*3·34.$\scriptstyle {\vdash :q\supset r.p\supset q.\supset .p\supset r\quad {\text{Syll}}.{\text{Imp}}]}$

These two propositions will hereafter be referred to as "Syll"; they are usually more convenient than either *2·05 or *2·06.

*3·35.$\scriptstyle {\vdash :p.p\supset q.\supset .q\quad [*2\cdot 27.{\text{Imp}}]}$

*3·37.$\scriptstyle {\vdash :.p.q.\supset .r:\supset :p.\sim r.\supset .\sim q}$

Dem.

${\begin{array}{llr}\scriptstyle {\vdash .{\text{Transp}}.}&\scriptstyle {\supset \vdash :q\supset r.\supset .\sim r\supset \sim q:}\\\scriptstyle {[{\text{Syll}}]}&\scriptstyle {\supset \vdash :.p.\supset .q\supset r:\supset :p.\supset .\sim r\supset \sim q\qquad }&\scriptstyle {(1)}\\\scriptstyle {\vdash .{\text{Exp}}.}&\scriptstyle {\supset \vdash :.p.q.\supset .r:\supset :p.\supset .q\supset r}&\scriptstyle {(2)}\\\scriptstyle {\vdash .{\text{Imp}}.}&\scriptstyle {\supset \vdash :.p.\supset .\sim r\supset \sim q:\supset :p.\sim r.\supset .\sim q}&\scriptstyle {(3)}\\\scriptstyle {\vdash .(2).(1).}&\scriptstyle {(3).{\text{Syll}}.\supset \vdash .{\text{Prop}}}\end{array}}$

This is another form of transposition.

*3·4.$\scriptstyle {\vdash :p.q.\supset .p\supset q\qquad [*2\cdot 51.{\text{Transp}}.(*1\cdot 01.*3\cdot 01)]}$

*3·41.$\scriptstyle {\vdash :.p\supset r.\supset :p.q.\supset .r\quad [*3\cdot 26.{\text{Syll}}]}$

*3·42.$\scriptstyle {\vdash :.q\supset r.\supset :p.q.\supset .r\quad [*3\cdot 27.{\text{Syll}}]}$

*3·43.$\scriptstyle {\vdash :.p\supset q.p\supset r.\supset :p.\supset .q.r}$

Dem.

${\begin{array}{llr}\scriptstyle {\vdash .*3\cdot 2.\supset \vdash :.q.\supset :r.\supset }&\scriptstyle {.q.r}&\scriptstyle {(1)}\\\scriptstyle {\vdash .(1).{\text{Syll}}.\supset \vdash ::p\supset q.}&\scriptstyle {\supset :.p.\supset :r.\supset .q.r:.}\\\scriptstyle {[*2\cdot 77]}&\scriptstyle {\supset :.p\supset r.\supset :p.\supset .q.r\qquad }&\scriptstyle {(2)}\\\scriptstyle {\vdash .(2).{\text{Imp}}.\supset \vdash .{\text{Prop}}}\end{array}}$

*3·44.$\scriptstyle {\vdash :.q\supset p.r\supset p.\supset :q\lor r.\supset .p}$

This principle is analogous to *3·43. The analogy between *3·43 and *3·44 is of a sort which generally subsists between formulae concerning products and formulae concerning sums.

Dem.

${\begin{array}{llr}\scriptstyle {\vdash .{\text{Syll}}.\supset \vdash :.\sim q\supset r.r\supset p}&\scriptstyle {.\supset :\sim q\supset p:}\\\scriptstyle {[*2\cdot 6]}&\scriptstyle {~\supset :q\supset p.\supset .p}&\scriptstyle {(1)}\\\scriptstyle {\vdash .(1).{\text{Exp}}.\supset \vdash ::\sim q\supset r.}&\scriptstyle {\supset :.r\supset p.\supset :q\supset p.\supset .p:.}\\\scriptstyle {[{\text{Comm}}.{\text{Imp}}]}&\scriptstyle {\supset :.q\supset p.r\supset p.\supset .p\qquad }&\scriptstyle {(2)}\\\scriptstyle {\vdash .(2).{\text{Comm}}.\supset \vdash :.q\supset p.}&\scriptstyle {r\supset p.\supset :\sim q\supset r.\supset .p:.}\\\scriptstyle {[*2\cdot 53.{\text{Syll}}]\quad \supset \vdash .{\text{Prop}}}\end{array}}$