Hypotheticals, like Categoricals, have been classed according to Quantity and Quality.  Premising that the quantity of a Hypothetical depends on the quantity of its Antecedent (which determines its limitation), whilst its quality depends on the quality of its consequent (which makes the predication), we may exhibit four forms: 

    A. If A is B, C is D;
    I. Sometimes when A is B, C is D;
    E. If A is B, C is not D;
    O. Sometimes when A is B, C is not D.

But I. and O. are rarely used.

As for Disjunctives, it is easy to distinguish the two quantities thus: 

    A. Either A is B, or C is D;
    I. Sometimes either A is B or C is D.

But I. is rarely used.  The distinction of quality, however, cannot be made:  there are no true negative forms; for if we write—­

    Neither is A B, nor C D,

there is here no alternative predication, but only an Exponible equivalent to No A is B, and No C is D.  And if we write—­

    Either A is not B, or C is not D,

this is affirmative as to the alternation, and is for all methods of treatment equivalent to A.

Logicians are divided in opinion as to the interpretation of the conjunction ‘either, or’; some holding that it means ‘not both,’ others that it means ‘it may be both.’  Grammatical usage, upon which the question is sometimes argued, does not seem to be established in favour of either view.  If we say A man so precise in his walk and conversation is either a saint or a consummate hypocrite; or, again, One who is happy in a solitary life is either more or less than man; we cannot in such cases mean that the subject may be both.  On the other hand, if it be said that the author of ‘A Tale of a Tub’ is either a misanthrope or a dyspeptic, the alternatives are not incompatible.  Or, again, given that X. is a lunatic, or a lover, or a poet, the three predicates have much congruity.

It has been urged that in Logic, language should be made as exact and definite as possible, and that this requires the exclusive interpretation ‘not both.’  But it seems a better argument, that Logic (1) should be able to express all meanings, and (2), as the science of evidence, must not assume more than is given; to be on the safe side, it must in doubtful cases assume the least, just as it generally assumes a preindesignate term to be of particular quantity; and, therefore ‘either, or’ means ‘one, or the other, or both.’

However, when both the alternative propositions have the same subject, as Either A is B, or A is C, if the two predicates are contrary or contradictory terms (as ‘saint’ and ‘hypocrite,’ or ‘saint’ and ’not-saint’), they cannot in their nature be predicable in the same way of the same subject; and, therefore, in such a case ‘either, or’ means one or the other, but not both in the same relation.  Hence it seems necessary to admit that the conjunction ‘either, or’ may sometimes require one interpretation, sometimes the other; and the rule is that it implies the further possibility ‘or both,’ except when both alternatives have the same subject whilst the predicates are contrary or contradictory terms.