Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/135

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

Dem.

$$\scriptstyle{\vdash:.p\supset q.\supset.q:\supset:q\supset p.\supset.p\qquad\left[*2\cdot68.\text{Perm}.*2\cdot62\frac{q,p}{p,q}\right]}$$

$$\scriptstyle{\vdash:.p\supset q.\supset:p\or q\or r.\supset.q\or r\qquad[*2\cdot621\cdot38]}$$

$$\scriptstyle{\vdash:.q\supset p.\supset:p\or q\or r.\supset.p\or r\qquad\left[*2\cdot73\frac{q,p}{p,q}.\text{Assoc}.\text{Syll}\right]}$$

$$\scriptstyle{\vdash::p\or q.\supset:.p.\or.q\supset r:\supset.p\or r\qquad\left[*2\cdot74\frac{\sim q}{q}.*2\cdot53\cdot31\right]}$$

$$\scriptstyle{\vdash:.p.\or.q\supset r:\supset:p\or q.\supset.p\or r\qquad[*2\cdot75.\text{Comm}]}$$

$$\scriptstyle{\vdash:.p.\supset.q\supset r:\supset:p\supset q.\supset.p\supset r\qquad\left[*2\cdot76\frac{\sim p}{p}\right]}$$

$$\scriptstyle{\vdash:.q\or r.\supset:\sim r\or s.\supset.q\or s}$$

Dem.

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

Dem.

$$\begin{align}\scriptstyle{\vdash:.p\or q\or r.\supset:p\or\sim r\or s.\supset.p\or}&\scriptstyle{q\or s}\\&\scriptstyle{\left[*2\cdot8.*2\cdot81\frac{q\or r,\sim r\or s,q\or s}{q,\quad r,\quad s}\right]}\end{align}$$

$$\begin{align}\scriptstyle{\vdash::p.\supset.q\supset r:\supset:.p.\supset.r\supset s:}&\scriptstyle{\supset:p.\supset.q\supset s}\\&\scriptstyle{\left[*2\cdot82\frac{\sim p,\sim q}{p,q}\right]}\end{align}$$

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

Dem.

$$\scriptstyle{\vdash:.p\supset q.\supset.p\supset r:\supset:p.\supset.q\supset r\quad\left[*2\cdot85\frac{\sim p}{p}\right]}$$