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Rh This leads us to substitute $$\scriptstyle{p|r/q}$$ for $$\scriptstyle{p|q/q}$$ in the "left-hand sides" of both the non-formal rule of implication and the syllogistic proposition (2) above. The reform may be further extended to the proposition (2) as a whole, which might be given in the form $$\scriptstyle{P|S/Q}$$ instead of $$\scriptstyle{P|Q/Q}$$ with the proviso, if the proposition is to remain true, that $$\scriptstyle{S}$$ must be implied in $$\scriptstyle{P}$$. Now, for $$\scriptstyle{S}$$, write the proposition (1) above, $$\scriptstyle{p|p/p}$$; for (as we at this early stage know "unofficially") a true proposition will be implied by everything.

We then have the three primitive propositions of the stroke-system: This is the non-formal rule of implication, *1·1, with the modification just explained. I shall call II "the Rule," and III "the Prop." Remarks on these Primitive Propositions. Observe $$\scriptstyle{p|r/q}$$ in II, while $$\scriptstyle{p|q/r}$$ in III. This alternance will prove essential for the working of the calculus.

In III, I shall use $$\scriptstyle{\pi}$$ for $$\scriptstyle{t|t/t}$$, $$\scriptstyle{P}$$ for $$\scriptstyle{p|q/r}$$, $$\scriptstyle{Q}$$ for $$\scriptstyle{s/q}$$ and shall speak of III as $$\scriptstyle{P|\pi/Q}$$.

$$\scriptstyle{P|\pi/Q}$$, by the Rule, yields the same result as the syllogistic proposition (2) above, when the left-hand side $$\scriptstyle{P}$$ is a truth of logic. This restriction of the syllogistic form to its categorical use with an asserted premiss is a peculiar character of the first proofs to follow, and is of some philosophical interest.

One feels inclined to think that III merely asserts together (1) and (2) above. This, view, whatever may be the amount of truth it contains, takes AND too much as a matter of course, and tends to lose sight of (α) the fact that III, as a structure, is simpler than (2) alone: for III is (2) with $$\scriptstyle{t|t/t}$$ instead of $$\scriptstyle{s/q}$$; and (β) the very real step from $$\scriptstyle{p.q}$$ to $$\scriptstyle{q}$$, together with the philosophical difference between two assertions and only one.

The main steps in the formal deduction are:  Proof of "Identity," $$\scriptstyle{t|t/t}$$.  Passage from $$\scriptstyle{P|\pi/Q}$$ to the usual implicative form $\scriptstyle{P  Elimination of the twist $$\scriptstyle{s/q}$$ in $$\scriptstyle{Q}$$, and return to the normal order $$\scriptstyle{q/s}$$. 