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By Add., Syll., $\scriptstyle {\vdash }$(1),

$\scriptstyle {q.\supset :q/p}$$\scriptstyle {p/p}$

(2)

The right side of (2) implies, by Syll.,

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

(3)

By Id., Perm., Add.$\scriptstyle {\frac {p/p|p,\quad q}{\quad p,\qquad q}}$,

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

(4)

By Syll. twice, $\scriptstyle {\vdash }$(2), $\scriptstyle {\vdash }$(3), $\scriptstyle {\vdash }$(4),

$\scriptstyle {q\supset :q.\supset .q/p|p}$, i.e. $\scriptstyle {q|L/L}$.

(b) By lemma to Syll., $\scriptstyle {q/q}$$\scriptstyle {s/q}$; by Perm. and Syll., $\scriptstyle {q/q}$$\scriptstyle {q/s}$. Hence, $\scriptstyle {q/q|L/L}$; by Perm., $\scriptstyle {L/L|q/q}$.

Now, by Syll.:

$\scriptstyle {L/L|q/q.\supset :q|L/L.\supset .L/L|L/L}$.

By $\scriptstyle {\vdash }$b, $\scriptstyle {\vdash }$a, and Taut.$\scriptstyle {\frac {L}{p}}$, result. We can now complete the proof of 'Association.'

Association,

$\scriptstyle {p}$$\scriptstyle {q/r}$$\scriptstyle {q}$$\scriptstyle {p/r}$

Dem: By Syll., $\scriptstyle {p}$$\scriptstyle {q/r}$$\scriptstyle {.\supset :q/r|r.}$$\scriptstyle {.p/r}$
By Syll. twice, $\scriptstyle {\vdash }$ Lemma, result.

Summation,

$\scriptstyle {q\supset r.\supset :p\vee q.\supset .p\vee r}$

Dem.: By Syll., Assoc.,

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

(1)

By (1) $\scriptstyle {\frac {s/s,\quad q,\quad p/p}{s,\quad \;\;r,\quad p\,}}$, result.

Theorems Equivalent to the Definitions of $\scriptstyle {p\supset q,p.q}$, in Principia.

$\scriptstyle {p\supset q.\supset .\sim p\vee q}$, and reciprocal theorem.

That is,

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

Dem.: Taut., and Syll.

Reciprocal theorem by Add.$\scriptstyle {\frac {s/s,\quad p}{\;s,\;\quad p}}$, and Syll.

$\scriptstyle {p|q.\supset .\sim p\vee \sim q}$, and reciprocal theorem.

That is,

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