According to this system, universal propositions are to be regarded as not necessarily implying the existence of their terms; and therefore, instead of giving them a positive form, they are translated into symbols that express what they deny.  For example, 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 in favour of the symbolic method.  But, as an introduction to philosophy, the common Logic must hold its ground. (Venn:  Symbolic Logic, c. 7.)