Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/133

The use of a general principle of deduction, such as either form of "$\scriptstyle {Syll}$," in a proof, is different from the use of the particular premisses to which the principle of deduction is applied. The principle of deduction gives the general rule according to which the inference is made, but is not itself a premiss in the inference. If we treated it as a premiss, we should need either it or some other general rule to enable us to infer the desired conclusion, and thus we should gradually acquire an increasing accumulation of premisses without ever being able to make any inference. Thus when a general rule is adduced in drawing an inference, as when we write "$\scriptstyle {[{\text{Syll}}]\vdash .(1).(2).\supset \vdash .{\text{Prop}}}$," the mention of "$\scriptstyle {\text{Syll}}$" is only required in order to remind the reader how the inference is drawn.

The rule of inference may, however, also occur as one of the ordinary premisses, that is to say, in the case of "$\scriptstyle {\text{Syll}}$" for example, the proposition "$\scriptstyle {p\supset q.\supset :q\supset r.\supset .p\supset r}$" may be one of those to which our rules of deduction are applied, and it is then an ordinary premiss. The distinction between the two uses of principles of deduction is of some philosophical importance, and in the above proofs we have indicated it by putting the rule of inference in square brackets. It is, however, practically inconvenient to continue to distinguish in the manner of the reference. We shall therefore henceforth both adduce ordinary premisses in square brackets where convenient, and adduce rules of inference, along with other propositions, in asserted premisses, i.e. we shall write e.g.

"$\scriptstyle {\vdash .(1).(2).{\text{Syll}}.\supset \vdash .{\text{Prop}}}$"

rather than

"$\scriptstyle {[{\text{Syll}}]\vdash .(1).(2).\supset \vdash .{\text{Prop}}}$"

*2·4.⁠$\scriptstyle {\vdash :.p.\lor .p\lor q:\supset .p\lor q}$

Dem.

${\begin{array}{ll}\scriptstyle {\vdash .*2\cdot 31.\supset \vdash :.p.\lor .p\lor q:}&\scriptstyle {\supset :p\lor p.\lor q:}\\\scriptstyle {[{\text{Taut}}.*2\cdot 38]}&\scriptstyle {\supset :p\lor q:.\supset \vdash .{\text{Prop}}}\end{array}}$

*2·41.⁠$\scriptstyle {\vdash :.q.\lor .p\lor q:\supset .p\lor q}$

Dem.

${\begin{array}{ll}\scriptstyle {\left[{\text{Assoc}}{\frac {q,p,q}{p,q,r}}\right]\vdash :.q.\lor .p\lor q:}&\scriptstyle {\supset :p.\lor .q\lor q:}\\\scriptstyle {[{\text{Taut}}.{\text{Sum}}]}&\scriptstyle {\supset :p\lor q:.\supset \vdash .{\text{Prop}}}\end{array}}$

*2·42.⁠$\scriptstyle {\vdash :.\sim p.\lor .p\supset q:\supset .p\supset q~\left[*2\cdot 4{\frac {\sim p}{p}}\right]}$

*2·43.⁠$\scriptstyle {\vdash :.p.\supset .p\supset q:\supset .p\supset q\quad [*2\cdot 42]}$

*2·45.⁠$\scriptstyle {\vdash :\sim (p\lor q).\supset .\sim p\qquad [*2\cdot 2.{\text{Transp}}]}$

*2·46.⁠$\scriptstyle {\vdash :\sim (p\lor q).\supset .\sim q\qquad [*1\cdot 3.{\text{Transp}}]}$