[283] Miss Jones (General Logic, p. 45) divides hypotheticals into formal or self-contained hypotheticals and referential hypotheticals. In the former, “the consequent is an inference from the antecedent alone”; in the latter, “the consequent is inferred not from the antecedent alone, but from the antecedent taken in conjunction with some other unexpressed proposition or propositions.”
179. The Opposition of Hypothetical Propositions.—Regarding hypotheticals as always affirming a necessary consequence, it may reasonably be held that they do not admit of distinctions of quality. Sigwart accordingly lays it down that all hypotheticals are affirmative. “Passing to hypothetical judgments 265 containing negations, we find that the form ‘If A is, B is not’ represents the negation of a proposition as the necessary consequence of an affirmation, thus affirming that the hypotheses A and B are incompatible.”[284] The force of this argument must be admitted. There is, however, some convenience in distinguishing between hypotheticals according as they lead up, in the consequent, to an affirmation or a denial; and in the formal treatment of hypotheticals, we shall be better able to preserve an analogy with categoricals and conditionals if we denote the proposition If X is true then Y is true by the symbol Am, and the proposition If X is true then Y is not true by the symbol Em.
[284] Logic, i. p. 226.
Whether or not we decide thus to recognise distinctions of quality in the case of hypotheticals, we certainly cannot recognise distinctions of quantity. The antecedent of a hypothetical is not an event which may recur an indefinite number of times, but a proposition which is simply true or false. We have already seen that the same proposition cannot be sometimes true and sometimes false, since propositions referring to different times are different propositions.[285]
[285] This, as Mr Johnson has pointed out, must be taken in connexion with the recognition of propositions as involving multiple quantification. “Thus we may indicate a series of propositions involving single, double, triple … quantification, which may reach any order of multiplicity: (1) ‘All luxuries are taxed’; (2) ‘In some countries all luxuries are taxed’; (3) ‘At some periods it is true that in all countries all luxuries are taxed’.… with respect to each of the types of proposition (1), (2), (3).… I contend that, when made explicit with respect to time or place, etc., it is absurd to speak of them as sometimes true and sometimes false” (Mind, 1892, p. 30 n.).
Do not distinctions of modality, however, take the place of distinctions of quantity? Up to this point, we have practically confined our attention to the apodeictic hypothetical, If A then C. This proposition is denied by the proposition If A is true still C need not be true (that is to say, The truth of C is not a necessary consequence of the truth of A). Can this latter proposition be described as a problematic hypothetical? Clearly it is not a hypothetical at all if we begin by defining a hypothetical as the affirmation of a necessary consequence. There seems, however, no need for this limitation. We may define a 266 hypothetical as a proposition which starting from the hypothesis of the truth (or falsity) of a given proposition affirms (or denies) that the truth (or falsity) of another proposition is a necessary consequence thereof. But, whether or not we adopt this definition, there can be no doubt that the proposition If A then possibly C appropriately finds a place in the same schedule of propositions as If A then necessarily C. In such a schedule we have the four forms,—
If A is true then C is true ; Am
If A is true then C is not true ; Em
If A is true still C may be true ; Im
If A is true still C need not be true. Om