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The above definitions give clear expression to the symmetry between OR and AND; and this, notwithstanding the choice that we had to make between an OR-form, and an AND-form. This is of some interest, because, in general, the very symmetry forces upon us an arbitrary choice, which, in turn, quite obscures the symmetry.

I shall use $\scriptstyle {\overline {q}}$ for $\scriptstyle {q|q}$ whenever convenient. Observe that $\scriptstyle {p|{\overline {q}}}$, i.e. $\scriptstyle {p\supset q}$, forms a natural symbol   for implication, allowing of permutation $\scriptstyle {q}$$\scriptstyle {p}$. We may notice in general that the new system brings the four functions into relations far closer than those in Mr Russell's system. For instance in

$\scriptstyle {p/p|p/p.|.p/p}$

the two propositions $\scriptstyle {p\vee p.\supset .p}$ and $\scriptstyle {\sim p\vee p}$ coincide.

Every stroke-formula falls into two parts on the right and left of a central stem. It will, therefore, add to clearness to use black type instead of dots to indicate the central symbol. Further, slanting strokes are covered by straight ones: thus $\scriptstyle {p/q|p/q}$ stands for $\scriptstyle {(p|q)|(p|q)}$.

The definition of the two primitive notions of the Principia in terms of a single new one tends to reduce the number of the primitive propositions needed. But how far does this reduction actually occur? Does it extend beyond the obvious substitution of "If $\scriptstyle {p}$ and $\scriptstyle {q}$ are elementary propositions, $\scriptstyle {p|q}$ is an elementary prop." (Sheffer, p. 488) for *1·7 and *1·71, stating the same for $\scriptstyle {\sim p}$ and $\scriptstyle {p\vee q}$ respectively? The reduction goes, as we shall presently find, very much farther.

It has first to be said, in order that we may be as precise as possible, that the whole amount gained in applying the stroke-definitions cannot with complete certainty be attributed to them. For Mr Russell's system, as it now stands, has not said its last word in that matter.

Incidentally I found that *1·4, $\scriptstyle {p\vee q.\supset .q\vee p}$, can be proved by means of the other four, with the unimportant change of *1·3, $\scriptstyle {q.\supset .p\vee q}$ into $\scriptstyle {q.\supset .q\vee p}$. In "Association," *1·5, write $\scriptstyle {p}$ for $\scriptstyle {r}$:

$\scriptstyle {p\vee (q\vee p).\supset .q\vee (p\vee p)}$.

The left hand side, by the help of $\scriptstyle {q.\supset .q\vee p}$ and "Summation," will be found to be implied in $\scriptstyle {p\vee q}$. The right-hand side, likewise, by $\scriptstyle {p\vee p.\supset .p}$, and "Summation," will be found to imply $\scriptstyle {q\vee p}$. The result then follows using "Syllogism" (obtained from "Summation" with the transformation $\scriptstyle {\frac {\sim p}{p}}$^[1]) twice.

1. By $\scriptstyle {\frac {p}{q}}$ or $\scriptstyle {\frac {p,p'}{q,q'}}$ I mean (following Mr Russell) the substitution of $\scriptstyle {p}$ for $\scriptstyle {q}$ or $\scriptstyle {p}$, $\scriptstyle {p'}$ for $\scriptstyle {q}$, $\scriptstyle {q'}$. By (e.g.) $\scriptstyle {P{\frac {p}{q}}}$ I mean the result of effecting the substitution in $\scriptstyle {P}$.