Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/138

*3·45.⁠$\scriptstyle {\vdash :.p\supset q.\supset :p.r.\supset .q.r}$

I.e. both sides of an implication may be multiplied by a common factor. This is called by Peano the "principle of the factor." It will be referred to as "Fact."

*3·47.⁠$\scriptstyle {\vdash :.p\supset r.q\supset s.\supset :p.q.\supset .r.s}$

I.e. if $\scriptstyle {p}$ implies $\scriptstyle {q}$ and $\scriptstyle {r}$ implies $\scriptstyle {s}$, then $\scriptstyle {p}$ and $\scriptstyle {q}$ jointly imply $\scriptstyle {r}$ and $\scriptstyle {s}$ jointly. The law of contradiction, "$\scriptstyle {\vdash .\sim (p.\sim p)}$," is proved in this number (*3·24); but in spite of its fame we have found few occasions for its use.

---

*3·01.⁠$\scriptstyle {p.q.=.\sim (\sim p\lor \sim q)\quad {\text{Df}}}$

*3·02.⁠$\scriptstyle {p\supset q\supset r.=.p\supset q.q\supset r\quad {\text{Df}}}$

*3·03.⁠Given two asserted elementary propositional functions "$\scriptstyle {\vdash .\phi p}$" and "$\scriptstyle {\vdash .\psi p}$" whose arguments are elementary propositions, we have $\scriptstyle {\vdash .\phi p.\psi p}$.

Dem.

${\begin{array}{lr}\scriptstyle {\vdash .*1\cdot 7\cdot 72.*2\cdot 11.\supset \vdash :\sim \phi p\lor \sim \psi p.\lor \sim (\sim \phi p\lor \sim \psi p)\quad }&\scriptstyle {(1)}\\\scriptstyle {\vdash .(1).*2\cdot 32.(*1\cdot 01).\supset \vdash :.\phi p.\supset :\psi p.\supset .\sim (\sim \phi p\lor \sim \psi p)}&\scriptstyle {(2)}\\\scriptstyle {\vdash .(2).(*3\cdot 03).\supset \vdash :.\phi p.\supset :\psi p.\supset .\phi p.\psi p}&\scriptstyle {(3)}\\\scriptstyle {\vdash .(3).*1\cdot 11.\supset \vdash .{\text{Prop}}}\end{array}}$

*3·1.⁠$\scriptstyle {\vdash :p.q.\supset .\sim (\sim p\lor \sim q)\quad [{\text{Id}}.(*3\cdot 01)]}$

*3·11.⁠$\scriptstyle {\vdash :\sim (\sim p\lor \sim q).\supset .p.q\quad [{\text{Id}}.(*3\cdot 01)]}$

*3·12.⁠$\scriptstyle {\vdash :\sim p.\lor .\sim q.\lor .p.q\quad \left[*2\cdot 11{\frac {\sim p\lor \sim q}{p}}\right]}$

*3·13.⁠$\scriptstyle {\vdash :\sim (p.q).\supset .\sim p\lor \sim q\quad [*3\cdot 11.{\text{Transp}}]}$

*3·14.⁠$\scriptstyle {\vdash :\sim p\lor \sim q.\supset .\sim (p.q)\quad [*3\cdot 1.{\text{Transp}}]}$

*3·2.⁠$\scriptstyle {\vdash :.p.\supset :q.\supset .p.q\qquad [*3\cdot 12]}$

*3·21.⁠$\scriptstyle {\vdash :.q.\supset :p.\supset .p.q\qquad [*3\cdot 2.{\text{Comm}}]}$

*3·22.⁠$\scriptstyle {\vdash :p.q.\supset .q.p}$

This is one form of the commutative law for logical multiplication, A more complete form is given in *4·3.

Dem.

${\begin{array}{ll}\scriptstyle {\left[*3\cdot 13{\frac {q,p}{p,q}}\right]\quad \vdash :\sim (q.p).}&\scriptstyle {\supset .\sim q\lor \sim p.}\\\scriptstyle {[{\text{Perm}}]}&\scriptstyle {\supset .\sim p\lor \sim q.}\\\scriptstyle {[*3\cdot 14]}&\scriptstyle {\supset .\sim (p.q)\qquad \qquad (1)}\\\scriptstyle {\vdash .(1).{\text{Transp}}.\supset \vdash .{\text{Prop}}}\end{array}}$