Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/146

*4·34.$\scriptstyle {p.q.r.=.(p.q).r\quad {\text{Df}}}$

*4·36.$\scriptstyle {\vdash :.p\equiv q.\supset :p.r.\equiv .q.r}$$\scriptstyle {[{\text{Fact}}.*3\cdot 47]}$

*4·37.$\scriptstyle {\vdash :.p\equiv q.\supset :p\lor r.\equiv .q\lor r}$$\scriptstyle {[{\text{Sum}}.*3\cdot 47]}$

*4·38.$\scriptstyle {\vdash :.p\equiv r.q\equiv s.\supset :p.q.\equiv .r.s}$$\scriptstyle {[*3\cdot 47.*4\cdot 32.*3\cdot 22]}$

*4·39.$\scriptstyle {\vdash :.p\equiv r.q\equiv s.\supset :p\lor q.\equiv .r\lor s}$$\scriptstyle {[*3\cdot 48.*4\cdot 32.*3\cdot 22]}$

• 4·4.$\scriptstyle {\vdash :.p.q\lor r.\equiv :p.q.\lor .p.r}$

This is the first form of the distributive law.

Dem.

${\begin{array}{llr}\scriptstyle {\vdash .*3\cdot 2.}&\scriptstyle {\supset \vdash ::p.\supset :q.\supset .p.q:.p.\supset :r.\supset .p.r::}\\\scriptstyle {[{\text{Comp}}]}&\scriptstyle {\supset \vdash ::p.\supset :.q.\supset .p.q:r.\supset .p.r:.}\\\scriptstyle {[*3\cdot 48]}&\scriptstyle {\qquad \supset :.q\lor r.\supset :p.q.\lor .p.r}&\scriptstyle {\qquad (1)}\\\scriptstyle {\vdash .(1).{\text{Imp}}.}&\scriptstyle {\supset \vdash :.p.q\lor r.\supset :p.q.\lor .p.r}&\scriptstyle {(2)}\\\scriptstyle {\vdash .*3\cdot 26.}&\scriptstyle {\supset \vdash :.p.q.\supset .p:p.r.\supset .p:.}\\\scriptstyle {[*3\cdot 44]}&\scriptstyle {\supset \vdash :.p.q.\lor .p.r:\supset .p}&\scriptstyle {(3)}\\\scriptstyle {\vdash .*3\cdot 27.}&\scriptstyle {\supset \vdash :.p.q.\supset .q:p.r.\supset .r:.}\\\scriptstyle {[*3\cdot 48]}&\scriptstyle {\supset \vdash :.p.q.\lor .p.r:\supset .q\lor r}&\scriptstyle {(4)}\\\scriptstyle {\vdash .(3).(4).{\text{Comp}}.}&\scriptstyle {\supset \vdash :.p.q.\lor .p.r:\supset .p.q\lor r}&\scriptstyle {(5)}\\\scriptstyle {\vdash .(2).(5).}&\scriptstyle {\supset \vdash .{\text{Prop}}}\end{array}}$

• 4·41.$\scriptstyle {\vdash :.p.\lor .q.r:\equiv .p\lor q.p\lor r}$

This is the second form of the distributive law—a form to which there is nothing analogous in ordinary algebra. By the conventions as to dots, "$\scriptstyle {p.\lor .q.r}$" means "$\scriptstyle {p\lor (q.r)}$."

Dem.

${\begin{array}{lllr}\scriptstyle {\vdash .*3\cdot 26.{\text{Sum}}.}&\scriptstyle {\supset \vdash :.p.\lor .q.r:}&\scriptstyle {\supset .p\lor q}&\scriptstyle {\qquad (1)}\\\scriptstyle {\vdash .*3\cdot 27.{\text{Sum}}.}&\scriptstyle {\supset \vdash :.p.\lor .q.r:}&\scriptstyle {\supset .p\lor r}&\scriptstyle {(2)}\\\scriptstyle {\vdash .(1).(2).{\text{Comp}}.}&\scriptstyle {\supset \vdash :.p\lor .q.r:}&\scriptstyle {\supset .p\lor q.p\lor r}&\scriptstyle {(3)}\\\scriptstyle {\vdash .*2\cdot 53.*3\cdot 47.}&\scriptstyle {\supset \vdash :.p\lor q.p\lor r.}&\scriptstyle {\supset :\sim p\supset q.\sim p\supset r:}\\\scriptstyle {[{\text{Comp}}]}&&\scriptstyle {\supset :\sim p.\supset .q.r:}\\\scriptstyle {[*2\cdot 54]}&&\scriptstyle {\supset :p.\lor .q.r}&\scriptstyle {(4)}\\\scriptstyle {\vdash .(3).(4).}&\scriptstyle {\supset \vdash .{\text{Prop}}}\end{array}}$

*4·42.$\scriptstyle {\vdash :.p.\equiv :p.q.\lor .p.\sim q}$

Dem.

${\begin{array}{llr}\scriptstyle {\vdash .*3\cdot 21.}&\scriptstyle {\supset \vdash :.q\lor \sim q.\supset :p.\supset .p.q\lor \sim q:.}\\\scriptstyle {[*2\cdot 11]}&\scriptstyle {\supset \vdash :p.\supset .p.q\lor \sim q}&\scriptstyle {\qquad (1)}\\\scriptstyle {\vdash .*3\cdot 26.}&\scriptstyle {\supset \vdash :p.q\lor \sim q.\supset .p}&\scriptstyle {(2)}\\\scriptstyle {\vdash .(1).(2).}&\scriptstyle {\supset \vdash :.p.\equiv :p.q\lor \sim q:}\\\scriptstyle {[*4\cdot 4]}&\scriptstyle {\qquad \equiv :p.q.\lor .p.\sim q:.\supset \vdash .{\text{Prop}}}\end{array}}$