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.
If, then, the disjunctive A is either B or C (B and C being contraries) implies that both alternatives cannot be true, it can only be adequately rendered in hypotheticals by the two forms—(1) If A is B, it is not C, and (2)If A is not B, it is C. But if the disjunctive A is either B or C (B and C not being contraries) implies that both may be true, it will be adequately translated into a hypothetical by the single form, If A is not B, it is C. We cannot translate it into—If A is B, it is not C, for, by our supposition, if 'A is B' is true, it does not follow that 'A is C' must be false.
Logicians are also divided in opinion as to the function of the hypothetical form. Some think it expresses doubt; for the consequent depends on the antecedent, and the antecedent, introduced by 'if,' may or may not be realised, as in If the sky is clear, the night is cold: whether the sky is, or is not, clear being supposed to be uncertain. And we have seen that some hypothetical propositions seem designed to draw attention to such uncertainty, as—If there is a resisting medium in space, etc. But other Logicians lay stress upon the connection of the clauses as the important matter: the statement is, they say, that the consequent may be inferred from the antecedent. Some even declare that it is given as a necessary inference; and on this ground Sigwart rejects particular hypotheticals, such as Sometimes when A is B, C is D; for if it happens only sometimes the connexion cannot be necessary. Indeed, it cannot even be probably inferred without further grounds. But this is also true whenever the antecedent and consequent are concerned with different matter. For example, If the soul is simple, it is indestructible. How do you know that? Because Every simple substance is indestructible. Without this further ground there can be no inference. The fact is that conditional forms often cover assertions that are not true complex propositions but a sort of euthymemes ([chap. xi. § 2]), arguments abbreviated and rhetorically disguised. Thus: If patience is a virtue there are painful virtues—an example from Dr. Keynes. Expanding this we have—
Patience is painful;
Patience is a virtue:
∴ Some virtue is painful.
And then we see the equivocation of the inference; for though patience be painful to learn, it is not painful as a virtue to the patient man.