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

${\begin{array}{lll}\scriptstyle {[(3).(8).*1\cdot 11]}&\scriptstyle {\vdash :\sim p\supset q.\supset .\sim q\supset \sim (\sim p)}&\scriptstyle {~(9)}\\\scriptstyle {\left[*2\cdot 05{\frac {\sim p\supset q,\sim q\supset \sim (\sim p),\sim q\supset p}{p,\quad ~q,\quad ~r}}\right]}&\scriptstyle {\vdash ::\sim q\supset \sim (\sim p).\supset .\sim q\supset p:}&\\&\scriptstyle {\quad \supset :.\sim p\supset q.\supset .\sim q\supset \sim (\sim p):\supset :\sim p\supset q.\supset .\sim q\supset p}&\scriptstyle {(10)}\\\scriptstyle {[(6).(10).*1\cdot 11]}&\scriptstyle {\vdash :.\sim p\supset q.\supset .\sim q\supset \sim (\sim p):\supset :}&\\&\scriptstyle {\quad \sim p\supset q.\supset .\sim q\supset p}&\scriptstyle {(11)}\\\scriptstyle {[(9).(11).*1\cdot 11]}&\scriptstyle {\vdash :\sim p\supset q.\supset .\sim q\supset p}&\end{array}}$

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

"$\scriptstyle {[{\text{Syll}}]~\vdash .(a).(b).(c).\supset \vdash .(d)}$,"

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 sorites 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

"$\scriptstyle {\vdash .p_{1}.\supset }$

[etc.]

$\scriptstyle {\vdash .p_{2}}$,"

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.

*2·16.⁠$\scriptstyle {\vdash :p\supset q.\supset .\sim q\supset \sim p}$

Dem.

${\begin{array}{lll}\scriptstyle {[*2\cdot 12]\quad }&\scriptstyle {\vdash .q\supset \sim (\sim q).\supset }&\\\scriptstyle {[*2\cdot 05]}&\scriptstyle {\vdash :p\supset q.\supset .p\supset \sim (\sim q)}&\scriptstyle {(1)}\\\scriptstyle {\left[*2\cdot 03{\frac {\sim q}{q}}\right]}&\scriptstyle {\vdash :p\supset \sim (\sim q).\supset .\sim q\supset \sim p}&\scriptstyle {(2)}\\\scriptstyle {[{\text{Syll}}]}&\scriptstyle {\vdash .(1).(2).\supset \vdash :p\supset q.\supset .\sim q\supset \sim p}&\end{array}}$