Note 282 — Chapter IX
Reading edition
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).