Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/127

SECTION A] In the last line of this proof, "$\scriptstyle{(1).(2).*1\cdot11}$" means that we are inferring in accordance with *1·11, having before us a proposition, namely $\scriptstyle{p\supset q.\supset:q\supset r.\supset.p\supset r}$, which, by (1), is implied by $\scriptstyle{q\supset r.\supset:p\supset q.\supset.p\supset r}$, which, by (2), is true. In general, in such cases, we shall omit the reference to *1·11.

The above two propositions will both be referred to as the "principle of the syllogism" (shortened to "Syll."), because, as will appear later, the syllogism in Barbara is derived from them.

$$\scriptstyle{\vdash:p.\supset.p\or p\quad\left[*1\cdot3\frac{p}{q}\right]}$$

Here we put nothing beyond "$\scriptstyle{*1\cdot3\frac{p}{q}}$,"|undefined because the proposition to be proved is what *1·3 becomes when $$\scriptstyle{p}$$ is written in place of $\scriptstyle{q}$.

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

Dem.

$$\scriptstyle{\vdash.\sim p\or p\quad[\text{Id.}(*1\cdot01)]}$$

$$\scriptstyle{\vdash.p\or\sim p}$$

Dem.

This is the law of excluded middle.

$$\scriptstyle{\vdash.p\supset\sim(\sim p)}$$

Dem.