.'. If there were any y in existence, some of them
would be x".

That this Conclusion does _not_ follow has been so briefly and clearly
explained by Mr. Keynes (in his "Formal Logic", dated 1894, pp. 356,
357), that I prefer to quote his words:--

"_Let no proposition imply the existence either of its subject or of its
predicate._

"Take, as an example, a syllogism in _Darapti_:--

'_All M is P_,
_All M is S_,
_.'. Some S is P_.'

"Taking S, M, P, as the minor, middle, and major terms respectively, the
conclusion will imply that, if there is an S, there is some P. Will the
premisses also imply this? If so, then the syllogism is valid; but not
otherwise.

"The conclusion implies that if S exists P exists; but, consistently
with the premisses, S may be existent while M and P are both
non-existent. An implication is, therefore, contained in the conclusion,
which is not justified by the premisses."

This seems to _me_ entirely clear and convincing. Still, "to make
sicker", I may as well throw the above (_soi-disant_) Syllogism into a
concrete form, which will be within the grasp of even a _non_-logical
Reader.

Let us suppose that a Boys' School has been set up, with the following
system of Rules:--

"All boys in the First (the highest) Class are to do French, Greek, and
Latin. All in the Second Class are to do Greek only. All in the Third
Class are to do Latin only."

Suppose also that there _are_ boys in the Third Class, and in the
Second; but that no boy has yet risen into the First.

It is evident that there are no boys in the School doing French: still
we know, by the Rules, what would happen if there _were_ any.
pg170
We are authorised, then, by the _Data_, to assert the following two
Propositions:--

"If there were any boys doing French, all of them would
be doing Greek;
If there were any boys doing French, all of them would
be doing Latin."

And the Conclusion, according to "The Logicians" would be

"If there were any boys doing Latin, some of them would
be doing Greek."

Here, then, we have two _true_ Premisses and a _false_ Conclusion (since
we know that there _are_ boys doing Latin, and that _none_ of them are
doing Greek). Hence the argument is _invalid_.

Similarly it may be shown that this "non-existential" interpretation
destroys the validity of _