[256] Symbolic Logic, p. 131. Again, in such a proposition as “Some sea-serpents are not half a mile long” (meaning your so-called sea-serpents), the subject of the proposition exists in the universe to which reference is made, namely, the universe which may be described as the universe of travellers’ tales. We are here regarding the proposition as elliptical in a sense that has been already explained.
On the whole it cannot be said that the usages of ordinary speech afford a decisive solution of the problem under discussion. It has, however, been shewn (1) that we seldom or never make statements about non-existent subjects in the form Some S is P or the form Some S is not P ; (2) that, although it is also true that we do not as a rule do so in the form All S is P or the form No S is P, still there are several classes of cases in which the use of these latter forms is not to be understood as necessarily carrying with it the implication that S is existent. Hence we should be departing very little from ordinary usage if we were to decide to interpret particulars as implying the existence of their subjects, but universals as not doing so (that is, as not doing so by their bare form).
I do not, however, regard this solution as necessitated by popular usage. It is, for instance, still open to anyone to adopt the convention that, for logical purposes, the categorical form shall only be used when the implication of the existence of the subject is intended. On this interpretation, the conditional or hypothetical form must be adopted whenever the existence of the subject is left an open question. Thus, if we are doubtful about the existence of S (or, at any rate, do not wish to affirm its existence), we must be careful to say, If there is any S, then all S is P, instead of simply All S is P ; in other words, the hypothetical character of the proposition so far as the existence of its subject is concerned must be made explicit.
The problem then not being decided by considerations of popular usage alone, we must go on to enquire how the question is affected by considerations of logical convenience and suitability. Here again there is no one solution that is inevitable. Reasons can, however, be urged in favour of interpreting particulars as implying, but universals as not implying, the existence of their 240 subjects;[257] and this, as we have seen, is a solution that derives some sanction from popular usage.
[257] On this view whenever it is desired specially to affirm the existence in the universe of discourse of the subject of a universal proposition, a separate statement to this effect must be made. For example, There are S’s, and all of them are P’s. If, on the other hand, it is ever desired to affirm a particular proposition without implying the existence of the subject, then recourse must be had to the hypothetical or conditional form of statement. Thus, if we do not intend to imply the existence of S, instead of writing Some S’s are P’s, we must write, If there are any S’s, then in some such cases they are also P’s.
(1) A consideration of the manner in which the validity of immediate inferences is affected by the existential import of propositions affords reasons for the adoption of this interpretation.[258] The most important immediate inferences are simple conversion (i.e., the conversion of E and of I) and simple contraposition (i.e., the contraposition of A and of O). If, however, universals are regarded as implying the existence of their subjects, then, as shewn in section [158], neither the conversion of E nor the contraposition of A is valid, irrespective of some farther assumption; whereas, if universals are not regarded as implying the existence of their subjects, then both these operations are legitimate without qualification. On the other hand, the conversion of I and the contraposition of O are valid only if particulars do imply the existence of their subjects.[259]
[258] It has been objected that to base our view of the existential import of propositions upon the validity or invalidity of immediate inferences is to argue in a circle. “Whether,” it is said, “the immediate inferences are valid or not must be a consequence of the view taken of the existential import of the proposition and should not, therefore, be made a portion of the ground on which that view is based.” This objection involves a confusion between different points of view from which the problem of the relation between the existential import of propositions and the validity of logical operations may be regarded. In section [158] the logical consequences of various assumptions were worked out without any attempt being made to decide between these assumptions. Our point of view is now different; we are investigating the grounds on which one of the assumptions may be preferred to the others, and there is no reason why the consequences previously deduced should not form part of our data for deciding this question. The argument contains nothing that is of the nature of a circulus in probando.
[259] Thus, the table of equivalences given in section [106] is valid on the interpretation with which we are now dealing. The dependence of the table given in section [108] upon the same supposition is still more obvious. It has been already pointed out that the remaining immediate inferences based on conversion and obversion are of much less importance; see page [227].
241 Turning to immediate inferences of another kind, it is clear that if universal propositions formally imply the existence of their subjects, we cannot legitimately pass from All X is Y to All AX is Y.[260] For it is possible that there may be X’s and yet no AX’s, and in this case the former proposition may be true, while the latter will certainly be false. Again, given that A is X, B is Y, C is Z, we cannot infer that ABC is XYZ. Such restrictions as these would constitute an almost insurmountable bar to progress in inference as soon as we have to do with complex propositions.[261]
[260] It will be observed further that upon the same assumption we cannot even affirm the formal validity of the proposition All X is X. For X might be non-existent, and the proposition would then be false.