Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

SECTION A]

IMMEDIATE CONSEQUENCES

107

[(3).(8).*1'11]: 'p ) q. ). *-q ) (-p ) (9)
*2.05 P ) q' —(- P)), - q)P:: ~,,(~p).. ~ p:
[L)P) T r J r ]
D:.~p q.D.~q~(~p):: ~p q.D. qDp (10)
[(6).(10).,1'11] k:. —p 3)q. ). *q 2)3 ~(~p) ):
~p. ).,q p (11)
[(9).(11).,1'11] h:.p )Dq..~q Dp
Note on the proof of *2'15. In the above proof, it will be seen that (3),
(4), (6) are respectively of the forms pi D p2, p2 ) p3, p3 ) p, where p, )p4 is
the proposition to be proved. From pi ) pa, p2 ) p3, Ps ) p4 the proposition
pi )p4 results by repeated applications of *2'05 or *2*06 (both of which are
called "Syll"). It is tedious and unnecessary to repeat this process every
time it is used; it will therefore be abbreviated into
"[Syll] F. (a). (b). (c). ). (d),"
where (a) is of the form pi D p2, (b) of the form p2 ) p, (c) of the form p3 )p4,
and (d) of the form pI DP4. The same abbreviation will be applied to a
sorites of any length.
Also where we have "i. pi" and C" F. -p 2 p," and p2 is the proposition to
be proved, it is convenient to write simply
" F.2p,.
[etc.] F p2,
where "etc." will be a reference to the previous propositions in virtue of
which the implication "pi P2" holds. This form embodies the use of
*1'11 or *1'1, and makes many proofs at once shorter and easier to follow.
It is used in the first two lines of the following proof.
*2-16.:p q. ). q P.~
Dem.
[*2'12]. q D (3 q).
[*2-05]:p q. D.p (q) (1)
[203 qq] F:p -(-q).D p (2)

[Syll]

F. (1). (2). 3 F:p 3q..,q )-p