Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/130

108 Note. The proposition to be proved will be called "$\scriptstyle{\text{Prop}}$,"|undefined and when a proof ends, like that of *2·16, by an implication between asserted propositions, of which the consequent is the proposition to be proved, we shall write "$\scriptstyle{\vdash.\text{etc.}\supset\vdash.\text{Prop}}$".|undefined Thus "$\scriptstyle{\supset\vdash.\text{Prop}}$"|undefined ends a proof, and more or less corresponds to

$$\scriptstyle{\vdash:\sim q\supset\sim p.\supset.p\supset q}$$

Dem.


 * 2·15, *2·16 and *2·17 are forms of the principle of transposition, and will be all referred to as "Transp."

$$\scriptstyle{\vdash:\sim p\supset p.\supset.p}$$

Dem.

This is the complement of the principle of the reductio ad absurdum. It states that a proposition which follows from the hypothesis of its own falsehood is true.

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

Dem.

$$\scriptstyle{\vdash:\sim p.\supset.p\supset q\quad\left[*2\cdot2\frac{\sim p}{p}\right]}$$

The above two propositions are very frequently used.

$$\scriptstyle{\vdash:p.\supset.\sim p\supset q\quad[*2\cdot21.\text{Comm}]}$$