r", and that
certain pairs are related as "creditor and debtor", "father and
son", "master and servant", "persecutor and victim", "uncle and
nephew".]

9.

"Jack Sprat could eat no fat:
His wife could eat no lean:
And so, between them both,
They licked the platter clean."

Solve this as a Sorites-Problem, taking lines 3 and 4 as the Conclusion
to be proved. It is permitted to use, as Premisses, not only all that is
here _asserted_, but also all that we may reasonably understand to be
_implied_.

pg195

NOTES TO APPENDIX.

(A) [See p. 167, line 6.]

It may, perhaps, occur to the Reader, who has studied Formal Logic that
the argument, here applied to the Propositions I and E, will apply
equally well to the Propositions I and A (since, in the ordinary
text-books, the Propositions "All xy are z" and "Some xy are not z" are
regarded as Contradictories). Hence it may appear to him that the
argument might have been put as follows:--

"We now have I and A 'asserting.' Hence, if the Proposition 'All xy are
z' be true, some things exist with the Attributes x and y: i.e. 'Some x
are y.'

"Also we know that, if the Proposition 'Some xy are not-z' be true the
same result follows.

"But these two Propositions are Contradictories, so that one or other of
them _must_ be true. Hence this result is always true: i.e. the
Proposition 'Some x are y' is _always_ true!

"_Quod est absurdum._ Hence I _cannot_ assert."

This matter will be discussed in Part II; but I may as well give here
what seems to me to be an irresistable proof that this view (that _A_
and _I_ are Contradictories), though adopted in the ordinary text-books,
is untenable. The proof is as follows:--

With regard to the relationship existing between the Class 'xy' and the
two Classes 'z' and 'not-z', there are _four_ conceivable states of
things, viz.

(1) Some xy are z, and some are not-z;
(2) " " none "
(3) No xy " some "
(4) " " none "

Of these four, No. (2) is equivalent to "All xy are z", No. (3) is
equivalent to "All xy are not-z", and No. (4) is equivalent to "No xy
exist."

Now it is quite undeniable that, of these _four_ states of things, each
is, _a priori_, _possible_, some _one must_ be true, and the other three
_must_ be false.

Hence the Contradictory to (2) i