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4. Proof of "Association," $\scriptstyle {p}$$\scriptstyle {q/r}$$\scriptstyle {.\supset .q}$$\scriptstyle {p/s}$.
5. Theorems equivalent to the definitions of $\scriptstyle {p.q}$, $\scriptstyle {p\supset q}$ in Principia.

Proof of Identity, $\scriptstyle {t{\mathsf {I}}t|t}$.

As this first proof from a single formal premiss stands in a unique position I shall, without in any way obscuring the precise play of the symbols, expound it after a more heuristic order than is usually followed.

We start with the Prop. $\scriptstyle {P{\mathsf {I}}\pi |Q}$, and the Rule enabling us to pass from the truth of $\scriptstyle {P}$ to that of $\scriptstyle {Q}$; and we have to prove $\scriptstyle {\pi }$. This can only be reached through some proposition of the form $\scriptstyle {A{\mathsf {I}}B|\pi }$, where $\scriptstyle {A}$ is a truth of logic^[1]. The proof will thus consist in passing from $\scriptstyle {P{\mathsf {I}}\pi |Q}$ to $\scriptstyle {A{\mathsf {I}}B|\pi }$ by some permutative process.

A simple two-terms permutative law $\scriptstyle {s|q}$$\scriptstyle {q|s,}$ we do not yet possess. Our Prop. yields only a roundabout three-terms permutation, $\scriptstyle {s|q}$$\scriptstyle {p|s}$, subject to the condition of $\scriptstyle {p{\mathsf {I}}q|r}$ being a truth of logic^[1]. This, however, is enough for our purpose.

In the Prop., write $\scriptstyle {t}$ for $\scriptstyle {p}$, $\scriptstyle {q}$, $\scriptstyle {r}$:

(a)	$\scriptstyle {\pi {\mathsf {I}}\pi |Q_{1}}$,

$\scriptstyle {Q_{1}}$ being $\scriptstyle {s|t}$$\scriptstyle {t|s}$. Write now $\scriptstyle {\pi }$ for $\scriptstyle {p}$, $\scriptstyle {q}$; $\scriptstyle {Q_{1}}$ for $\scriptstyle {r}$: then by (a) and the Rule,

(b)	$\scriptstyle {s|\pi }$$\scriptstyle {\pi |s}$.

From (b) in the same manner,

(c)	$\scriptstyle {u|\pi /s}$$\scriptstyle {s/\pi |u.}$

This enables us to pass, by the Rule, from $\scriptstyle {P{\mathsf {I}}\pi |Q}$ to

(d)	$\scriptstyle {Q|\pi {\mathsf {I}}P}$.

In order to complete the proof of $\scriptstyle {\pi }$, we need only find some expression which: (α) can be a value for $\scriptstyle {P}$, i.e. is a case of $\scriptstyle {p{\mathsf {I}}q|r}$, and (β) is implied in some truth of logic, say $\scriptstyle {T}$. For, by $\scriptstyle {T{\mathsf {I}}P|P}$, the Prop., and the Rule, as above,

(e)	$\scriptstyle {s|P}$$\scriptstyle {T|s}$.

In (e), write $\scriptstyle {Q|\pi }$ for $\scriptstyle {s}$: first by (d) and the Rule, then by $\scriptstyle {T}$ and the Rule, we obtain $\scriptstyle {T{\mathsf {I}}Q|\pi }$, and so

(f)	$\scriptstyle {\pi }$.

1. This use of the Rule by anticipation, with still undetermined $\scriptstyle {P}$'s and $\scriptstyle {Q}$'s, is in truth contrary to the nature of a non-formal rule, which must never be used to build up the structure of an argument. It must always be possible to dispense with all such 'anticipated' assertions in the final form of a proof. This will be seen to be very easy in the present case.