Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/149

*4·73. $\scriptstyle {\vdash :.q.\supset :p.\equiv .p.q\quad [{\text{Simp}}.*4\cdot 71]}$

This proposition is very useful, since it shows that a true factor may be omitted from a product without altering its truth or falsehood, just as a true hypothesis may be omitted from an implication.

*4·74. $\scriptstyle {\vdash :.\sim p.\supset :q.\equiv .p\lor q}$$\scriptstyle {[*2\cdot 21.*4\cdot 72]}$

*4·76. $\scriptstyle {\vdash :.p\supset q.p\supset r.\equiv :p.\supset .q.r}$$\scriptstyle {\left[*4\cdot 41{\frac {\sim p}{p}}.(*1\cdot 01)\right]}$

*4·77. $\scriptstyle {\vdash :.q\supset p.r\supset p.\equiv :q\lor r.\supset .p}$$\scriptstyle {[*3\cdot 44.{\text{Add}}.*2\cdot 2]}$

*4·78. $\scriptstyle {\vdash :.p\supset q.\lor .p\supset r:\equiv :p.\supset .q\lor r}$

Dem.

${\begin{array}{ll}\scriptstyle {\vdash .*4\cdot 2.(*1\cdot 01).\supset \vdash :.p\supset q.\lor .p\supset r:}&\scriptstyle {\equiv :\sim p\lor q.\lor .\sim p\lor r:}\\\scriptstyle {[*4\cdot 33]}&\scriptstyle {\equiv .\sim p.\lor .q\lor \sim p\lor r:}\\\scriptstyle {[*4\cdot 31\cdot 37]}&\scriptstyle {\equiv :\sim p.\lor .\sim p\lor q\lor r:}\\\scriptstyle {[*4\cdot 33]}&\scriptstyle {\equiv :\sim p\lor \sim p.\lor .q\lor r:}\\\scriptstyle {[*4\cdot 25\cdot 37]}&\scriptstyle {\equiv :\sim p.\lor .q\lor r:}\\\scriptstyle {[*4\cdot 2.(*1\cdot 01)]}&\scriptstyle {\equiv :p.\supset .q\lor r:.\supset \vdash .{\text{Prop}}}\end{array}}$

*4·79. $\scriptstyle {\vdash :.q\supset p.\lor .r\supset p:\equiv :q.r.\supset .p}$

Dem.

${\begin{array}{ll}\scriptstyle {\vdash .*4\cdot 1\cdot 39.\supset \vdash :.q\supset p.\lor .r\supset p:}&\scriptstyle {\equiv :\sim p\supset \sim q.\lor .\sim p\supset \sim r:}\\\scriptstyle {[*4\cdot 78]}&\scriptstyle {\equiv :\sim p.\supset .\sim q\lor \sim r:}\\\scriptstyle {[*2\cdot 15]}&\scriptstyle {\equiv :\sim (\sim q\lor \sim r).\supset .p:}\\\scriptstyle {[*4\cdot 2.(*3\cdot 01)]}&\scriptstyle {\equiv :q.r.\supset .p:.\supset \vdash .{\text{Prop}}}\end{array}}$

Note. The analogues, for classes, of *4·78·79 are false. Take, e.g. *4·78, and put $\scriptstyle {p=}$English people, $\scriptstyle {q=}$men, $\scriptstyle {r=}$women. Then $\scriptstyle {p}$ is contained in $\scriptstyle {q}$ or $\scriptstyle {r}$, but is not contained in $\scriptstyle {q}$ and is not contained in $\scriptstyle {r}$.

*4·8. $\scriptstyle {\vdash :p\supset \sim p.\equiv .\sim p}$$\scriptstyle {[*2\cdot 01.{\text{Simp}}]}$

*4·81. $\scriptstyle {\vdash :\sim p\supset p.\equiv .p}$$\scriptstyle {[*2\cdot 18.{\text{Simp}}]}$

*4·82. $\scriptstyle {\vdash :p\supset q.p\supset \sim q.\equiv .\sim p}$$\scriptstyle {[*2\cdot 65.{\text{Imp}}.*2\cdot 21.{\text{Comp}}]}$

*4·83. $\scriptstyle {\vdash :p\supset q.\sim p\supset q.\equiv .q}$$\scriptstyle {[*2\cdot 61.{\text{Imp}}.{\text{Simp}}.{\text{Comp}}]}$

Note. *4·82·83 may also be obtained from *4·43, of which they are virtually other forms.

*4·84. $\scriptstyle {\vdash :.p\equiv q.\supset :p\supset r.\equiv .q\supset r}$$\scriptstyle {[*2\cdot 06.*3\cdot 47]}$

*4·85. $\scriptstyle {\vdash :.p\equiv q.\supset :r\supset p.\equiv .r\supset q}$$\scriptstyle {[*2\cdot 05.*3\cdot 47]}$

*4·86. $\scriptstyle {\vdash :.p\equiv q.\supset :p\equiv r.\equiv .q\equiv r}$$\scriptstyle {[*4\cdot 21\cdot 22]}$

*4·87. $\scriptstyle {\vdash :.p.q.\supset .r:\equiv :p.\supset .q\supset r:\equiv :q.\supset .p\supset r:\equiv :q.p.\supset .r}$

$\scriptstyle {[{\text{Exp}}.{\text{Comm}}.{\text{Imp}}]}$

*4·87 embodies in one proposition the principles of exportation and importation and the commutative principle.