Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/132

that it is $\scriptstyle {b}$, not "$\scriptstyle {a\supset b}$," that implies $\scriptstyle {c}$. But we put two dots after $\scriptstyle {d}$, to show that now the whole proposition "$\scriptstyle {a\supset d}$" is concerned. If "$\scriptstyle {a\supset d}$" is not the proposition to be proved, but is to be used subsequently in the proof, we put

${\begin{array}{ll}\scriptstyle {\vdash :a.}&\scriptstyle {\supset .b.}\\\scriptstyle {[{\text{etc.}}]}&\scriptstyle {\supset .c.}\\\scriptstyle {[{\text{etc.}}]}&\scriptstyle {\supset .d\qquad \qquad \qquad \qquad \qquad {\text{(1),}}}\end{array}}$

and then "$\scriptstyle {(1)}$" means "$\scriptstyle {a\supset d}$." The proof of *2·31 is as follows: Dem.

${\begin{array}{ll}\scriptstyle {[*2\cdot 3]\vdash :p\lor (q\lor r).}&\scriptstyle {\supset .p\lor (r\lor q).}\\\scriptstyle {\left[{\text{Assoc}}{\frac {r,q}{q,r}}\right]\quad }&\scriptstyle {\supset .r\lor (p\lor q).}\\\scriptstyle {[{\text{Perm}}{\frac {r,p\lor q}{p,~~q}}]\quad }&\scriptstyle {\supset .(p\lor q)\lor r:\supset \vdash .{\text{Prop}}}\end{array}}$

*2·32.⁠$\scriptstyle {\vdash :(p\lor q)\lor r.\supset .p\lor (q\lor r)}$

Dem.

${\begin{array}{ll}\scriptstyle {\left[{\text{Perm}}{\frac {p\lor q,r}{p,~q}}\right]\vdash :(p\lor q)\lor r.}&\scriptstyle {\supset .r\lor (p\lor q)}\\\scriptstyle {\left[{\text{Assoc}}{\frac {r,p,q}{p,q,r}}\right]}&\scriptstyle {\supset .p\lor (r\lor q)}\\\scriptstyle {[*2\cdot 3]}&\scriptstyle {\supset .p\lor (q\lor r):\supset \vdash .{\text{Prop}}}\end{array}}$

*2·33.⁠$\scriptstyle {p\lor q\lor r.=.(p\lor q)\lor r\quad {\text{Df}}}$

This definition serves only for the avoidance of brackets.

*2·36.⁠$\scriptstyle {\vdash :.q\supset r.\supset :p\lor q.\supset .r\lor p}$

Dem.

${\begin{array}{lll}\scriptstyle {[{\text{Perm}}]}&\scriptstyle {\vdash :p\lor r.\supset .r\lor p:}\\\scriptstyle {\left[{\text{Syll}}{\frac {p\lor q,p\lor r,r\lor p}{p,~~q,~~r}}\right]\supset }&\scriptstyle {\vdash :.p\lor q.\supset .p\lor r:\supset :p\lor q.\supset .r\lor p\quad }&\scriptstyle {(1)}\\\scriptstyle {[{\text{Sum}}]}&\scriptstyle {\vdash :.q\supset r.\supset :p\lor q.\supset .p\lor r}&\scriptstyle {(2)}\\\scriptstyle {\vdash .(1).(2).Syll.\supset \vdash .{\text{Prop}}}\end{array}}$

*2·37.⁠$\scriptstyle {\vdash :.q\supset r.\supset :q\lor p.\supset .p\lor r}$

$\scriptstyle {[{\text{Syll}}.{\text{Perm}}.{\text{Sum}}]}$

*2·38.⁠$\scriptstyle {\vdash :.q\supset r.\supset :q\lor p.\supset .r\lor p}$

$\scriptstyle {[{\text{Syll}}.{\text{Perm}}.{\text{Sum}}]}$

The proofs of *2·37·38 are exactly analogous to that of *2·36. (We use "*2·37·38" as an abbreviation for "*2·37 and *2·38." Such abbreviations will be used throughout.)