The main question at issue in regard to the import of the hypothetical proposition is whether it is merely assertoric or is modal. The contrast may be simply put by asking whether, when we say If A then C, our intention is merely to deny the actuality of the conjunction of A true with C false or is to declare this conjunction to be an impossibility.
The contrast between these two interpretations can be brought out most clearly by asking how the proposition If A then C is to be contradicted. If our intention is merely to deny the actuality of the conjunction of A true with C false, then the contradictory must assert the actuality of this conjunction; if 262 our intention is to deny the possibility of the conjunction, then the contradictory will merely assert its possibility. In other words, on the assertoric interpretation the contradictory will be A is true but C is false ;[280] on the modal interpretation it will be If A is true C may be false.[281]
[280] We may look at it in this way. Let AC denote the truth of both A and C, ACʹ the truth of A and the falsity of C, and so on. Then there are four à priori possibilities, namely, AC, ACʹ, AʹC, AʹCʹ, one or other of which must hold good, but any pair of which are mutually inconsistent. The proposition If A then C merely excludes ACʹ, and still leaves AC, AʹC, AʹCʹ, as possible alternatives. In denying it, therefore, we must definitely affirm ACʹ, and exclude the three other alternatives. Hence the contradictory as above stated.
[281] A certain assumption is necessary, in order that this result may be correct. The opposition of hypotheticals on the modal interpretation will be discussed in more detail in section [179].
Hypotheticals intended to be interpreted assertorically are to be met with in ordinary discourse, but they are unusual. There appear to be two cases: (a) When we know that one or other of two propositions is true but do not know (or do not remember) which, we may express our knowledge in the form of a hypothetical, If X is not true then Y is true, and such hypothetical will be merely assertoric. For example, If the flowers I planted in this bed were not pansies they were violets. Here the intention is merely to deny the actuality of the flowers being neither pansies nor violets. (b) We may deny a proposition emphatically by a hypothetical in which the proposition in question is combined as antecedent with a manifestly false consequent; and such hypothetical will again be merely assertoric. For example, If what you say is true, I’m a Dutchman ; If that boy comes back, I’ll eat my head (vide Oliver Twist); I’m hanged if I know what you mean. In these examples the intention is to deny the actuality (not the possibility) of the conjunctions,—What you say is true and I am not a Dutchman ; That boy will come back and I shall not eat my head ; I am not hanged and I know what you mean; and since the elements of the conjunctions printed in italics are admittedly true, the force of the propositions is to deny the truth of the other elements, that is to say, to affirm,—What you say is not true, That boy will not come back, I do not know what you mean. Similarly 263 we may sometimes employ the hypothetical form of expression as an emphatic way of declaring the truth of the consequent (an antecedent being chosen which is admittedly true); for example, If he cannot act, at any rate he can sing. Here once more the hypothetical is merely assertoric.
It cannot, however, be maintained that any of the above are typical hypotheticals; and the claim that our natural interpretation of hypotheticals is ordinarily modal may be justified on the ground that we do not usually consider it to be necessary to affirm the antecedent in order to be able to deny a hypothetical. We have seen that, in order to deny the assertoric hypothetical If A then C, we must affirm A and deny C ; but we should usually regard it as sufficient for denial if we can shew that there is no necessary connexion between the truth of A and that of C, whether A is actually true or not.
We shall then in the main be in agreement with ordinary usage if we interpret hypotheticals modally, and the adoption of such an interpretation will also give hypotheticals a more distinctive character. In what follows the hypothetical form will accordingly be regarded as modal, except in so far as an explicit statement is made to the contrary.[282]
[282] Either C is true or A is not true is usually regarded as the disjunctive equivalent of the hypothetical If A is true then C is true. The relation between these two propositions will be discussed further later on. It is, however, desirable to point out at once that, if the equivalence is to hold good, both the propositions must be interpreted assertorically or both modally. There is a good deal to be said for differentiating the two forms by regarding the hypothetical as modal and the disjunctive as merely assertoric. This method of treatment is explicitly adopted by Mr McColl. He writes (using the symbolism, a : b for If a then b, a + b for a or b, aʹ for the denial of a)—“The expression a : b may be read a implies b or If a is true, b must be true. The statement a : b implies aʹ + b. But it may be asked are not the two statements really equivalent; ought we not therefore to write a : b = aʹ + b? Now if the two statements are really equivalent their denials will also be equivalent. Let us see if this will be the case, taking as concrete examples: ‘If he persists in his extravagance he will be ruined’; ‘He will either discontinue his extravagance or he will be ruined.’ The denial of a : b is (a : b)ʹ and this denial may be read—‘He may persist in his extravagance without necessarily being ruined.’ The denial of aʹ + b is abʹ which may be read—‘He will persist in his extravagance and he will not be ruined.’ Now it is quite evident that the second denial is a much stronger and more positive statement than the first. The first only asserts the possibility of the combination abʹ ; the second asserts the certainty of the same combination. The denials of the statements a : b and aʹ + b having thus been proved to be not equivalent, it follows that the statements a : b and aʹ + b are themselves not equivalent, and that, though aʹ + b is a necessary consequence of a : b, yet a : b is not a necessary consequence of aʹ + b” (see Mind, 1880, pp. 50 to 54; one or two slight verbal changes have been made in this quotation).
264 Some writers who adopt the modal interpretation of hypotheticals speak of the consequent as being an inference from the antecedent. There are no doubt some hypotheticals to which this description accurately applies. Thus, we may have hypotheticals which are formal in the sense in which that term has been used in section [31], the consequent being, for instance, an immediate inference from the antecedent, or being the conclusion of a syllogism of which the premisses constitute the antecedent. The following are examples,—If all isosceles triangles have the angles at the base equal to one another, then no triangle the angles at whose base are unequal can be isosceles ; If all men are mortal and the Pope is a man, then the Pope must be mortal.
But more usually the consequent of a hypothetical proposition cannot be inferred from the antecedent alone. The aid is required of suppressed premisses which are taken for granted, the premiss which alone is expressed being perhaps the only one as to the truth of which any doubt is regarded as admissible. It would, therefore, be better to speak of the consequent as being the necessary consequence of the antecedent, than as being an inference from it. When we speak of C as being an inference from A, there is a suggestion that A affords the complete justification of C, whereas when we speak of it as a necessary consequence, this suggestion is at any rate less prominent.[283]