Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/129

SECTION A]

Note on the proof of *2·15. In the above proof, it will be seen that (3), (4), (6) are respectively of the forms $\scriptstyle{p_1\supset p_2}$, $\scriptstyle{p_2\supset p_3}$, $\scriptstyle{p_3\supset p_4}$, where $$\scriptstyle{p_1\supset p_4}$$ is the proposition to be proved. From $\scriptstyle{p_1\supset p_2}$, $\scriptstyle{p_2\supset p_3}$, $$\scriptstyle{p_3\supset p_4}$$ the proposition $$\scriptstyle{p_1\supset p_4}$$ results by repeated applications of *2·05 or *2·06 (both of which are called "Syll."). It is tedious and unnecessary to repeat this process every time it is used; it will therefore be abbreviated into where $$\scriptstyle{(a)}$$ is of the form $\scriptstyle{p_1\supset p_2}$, $$\scriptstyle{(b)}$$ of the form $\scriptstyle{p_2\supset p_3}$, $$\scriptstyle{(c)}$$ of the form $\scriptstyle{p_3\supset p_4}$, and $$\scriptstyle{(d)}$$ of the form $\scriptstyle{p_1\supset p_4}$. The same abbreviation will be applied to a soritesseries [sic] of any length.

Also where we have "$\scriptstyle{\vdash.p_1}$" and "$\scriptstyle{\vdash.p_1\supset p_2}$," and $$\scriptstyle{p_2}$$ is the proposition to be proved, it is convenient to write simply where "etc." will be a reference to the previous propositions in virtue of which the implication "$\scriptstyle{p_1\supset p_2}$" holds. This form embodies the use of *1·11 or *1·1, and makes many proofs at once shorter and easier to follow. It is used in the first two lines of the following proof.

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

Dem.