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the value of logical systems were to be judged of by the results obtainable, formal deductive Logic would probably be superseded.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 in favour of the symbolic method.
But, as an introduction to philosophy, the common Logic must hold its ground.