ple, the proposition _All devils
are ugly_ need not imply that any such things as 'devils' really exist;
but it certainly does imply that _Devils that are not ugly do not
exist_. Similarly, the proposition _No angels are ugly_ implies that
_Angels that are ugly do not exist_. Therefore, writing _x_ for
'devils,' _y_ for 'ugly,' and _[y]_ for 'not-ugly,' we may express A.,
the universal affirmative, thus:

A. _x[y]_ = 0.

That is, _x that is not y is nothing_; or, _Devils that are not-ugly do
not exist_. And, similarly, writing _x_ for 'angels' and _y_ for 'ugly,'
we may express E., the universal negative, thus:

E. _xy_ = 0.

That is, _x that is y is nothing_; or, _Angels that are ugly do not
exist_.

On the other hand, particular propositions are regarded as implying the
existence of their terms, and the corresponding equations are so framed
as to express existence. With this end in view, the symbol v is adopted
to represent 'something,' or indeterminate reality, or more than
nothing. Then, taking any particular affirmative, such as _Some
metaphysicians are obscure_, and writing _x_ for 'metaphysicians,' and
_y_ for 'obscure,' we may express it thus:

I. _xy_ = v.

That is, _x that is y is something_; or, _Metaphysicians that are
obscure do occur in experience_ (however few they may be, or whether
they all be obscure). And, similarly, taking any particular negative,
such as _Some giants are not cruel_, and writing _x_ for 'giants' and
_y_ for 'not-cruel,' we may express it thus:

O. _x[y]_ = v.

That is, _x that is not y is something_; or, _giants that are not-cruel
do occur_--in romances, if nowhere else.

Clearly, these equations are, like Hamilton's, concerned with
denotation. A. and E. affirm that the compound terms x[y] and xy have no
denotation; and I. and O. declare that x[y] and xy have denotation, or
stand for something. Here, however, the resemblance to Hamilton's system
ceases; for the Symbolic Logic, by operating upon more than two terms
simultaneously, by adopting the algebraic signs of operations, +,-, x, /
(with a special signification), and manipulating the symbols by
quasi-algebraic processes, obtains results which the common Logic
reaches (if at all) with much greater difficulty. If, indeed, the value
of logical systems were to be judged of by the results obtainable,
formal deductive Logic would probably be superseded. And, as a mental
discipline, there is much to be said