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

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

Dem.

$$\scriptstyle{\vdash:(p\or q)\or r.\supset.p\or(q\or r)}$$

Dem.

$$\scriptstyle{p\or q\or r.=.(p\or q)\or r\quad\text{Df}}$$

This definition serves only for the avoidance of brackets.

$$\scriptstyle{\vdash:.q\supset r.\supset:p\or q.\supset.r\or p}$$

Dem.

$$\scriptstyle{\vdash:.q\supset r.\supset:q\or p.\supset.p\or r}$$

$$\scriptstyle{\vdash:.q\supset r.\supset:q\or p.\supset.r\or p}$$

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.)