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Let us, however, take Mr Russell's eight propositions in the form given in Principia. It is my object to reduce them to three—two non-formal and one formal—by means of the stroke-definitions given above.

It can be shown, as a first stage, that two formal propositions are enough, namely:

(1)⁠$\scriptstyle {p|p/p}$.
(2)⁠$\scriptstyle {p|q/q}$$\scriptstyle {s/q}$$\scriptstyle {p/s}$.

The first proposition is the form of "Identity" ($\scriptstyle {p\supset p}$) in the stroke-system. It would, at first sight, appear more natural to adopt the order $\scriptstyle {q/s}$$\scriptstyle {p/s}$ in the left-hand side of (2), since

$\scriptstyle {p|q/q.\supset .q/s}$$\scriptstyle {p/s}$

is the syllogistic principle of the stroke-system, giving "Syllogism," $\scriptstyle {p\supset q.\supset :q\supset s.\supset .p\supset s}$ when $\scriptstyle {s|s}$ is written for $\scriptstyle {s}$.

It will however be found that the inverted order, $\scriptstyle {s/q}$$\scriptstyle {p/s}$, is much more advantageous than the normal syllogistic order, $\scriptstyle {q/s}$$\scriptstyle {p/s}$. For, owing to this "twist," Identity and (2) yield "Permutation," $\scriptstyle {s/p}$$\scriptstyle {p/s}$, which now enables us to eliminate the twist in (2), and revert to the normal order. From the three propositions thus obtained, the rest follow.

This, by the way, illustrates the following fundamental fact. Which form of a given principle is the most general, and contains the maximum assertion, is a function of the symbolic system used. Thus, for instance, in Mr Russell's system,

$\scriptstyle {p.\supset .q\vee p}$(a)
is more general than⁠$\scriptstyle {p.\supset .q\supset p}$(b)

since (b) is (a) with $\scriptstyle {\sim q}$ for $\scriptstyle {q}$. In the stroke-system, on the contrary, $\scriptstyle {p}$$\scriptstyle {q/q|p/p}$, meaning the same thing as (a), is less general than $\scriptstyle {p}$$\scriptstyle {q|p/p}$, whose meaning is that of (b), since it is obtained from it by writing $\scriptstyle {q|q}$ for $\scriptstyle {q}$.

A further step has to be made in order to be left with only one formal primitive proposition. It consists in adapting to better advantage the form of the primitive propositions to the properties of the stroke-symbolism where implication is concerned. We had above

$\scriptstyle {p\supset q.=.p|q/q\quad {\text{Df.}}}$

If we look for the meaning of the general form $\scriptstyle {p|r/q}$, we find this to be $\scriptstyle {\sim p\vee \sim (\sim r\vee \sim q)}$, i.e. $\scriptstyle {p.\supset .r.q}$. We thus come to the fundamental property that, in the new system, $\scriptstyle {p\supset q}$ is a case of $\scriptstyle {p.\supset .s.q}$, whereas in Principia the contrary relation of course holds.