§ 7. There is another and practical way of getting rid of the perplexities of modal reasoning which must be noticed here. It is the resource of ordinary reasoners rather than the decision of professed logicians,[7] and, like the first method of evasion already pointed out in this chapter, is of very partial application. It consists in treating the premises, during the process of reasoning, as if they were pure, and then reintroducing the modality into the conclusion, as a sort of qualification of its full certainty. When each of the premises is nearly certain, or when from any cause we are not concerned with the extent of their departure from full certainty, this rough expedient will answer well enough. It is, I apprehend, the process which passes through the minds of most persons in such cases, in so far as they reason consciously. They would, presumably, in such an example as that previously given (§ 4), proceed as if the premises that ‘those who take arsenic will die,’ and that ‘the man in question has taken it,’ were quite true, instead of being only probably true, and they would consequently draw the conclusion that ‘he would die.’ But bearing in mind that the premises are not certain, they would remember that the conclusion was only to be held with a qualified assent. This they would express quite correctly, if the mere nature and not the degree of that assent is taken into account, by saying that ‘he is likely to die.’ In this case the modality is rejected temporarily from the premises to be reintroduced into the conclusion.
It is obvious that such a process as this is of a very rough and imperfect kind. It does, in fact, omit from accurate consideration just the one point now under discussion. It takes no account of the varying shades of expression by which the degree of departure from perfect conviction is indicated, which is of course the very thing with which modality is intended to occupy itself. At best, therefore, it could only claim to be an extremely rude way of deciding questions, the accurate and scientific methods of treating which are demanded of us.
§ 8. In any employment of applied logic we have of course to go through such a process as that just mentioned. Outside of pure mathematics it can hardly ever be the case that the premises from which we reason are held with absolute conviction. Hence there must be a lapse from absolute conviction in the conclusion. But we reason on the hypothesis that the premises are true, and any trifling defection from certainty, of which we may be conscious, is mentally reserved as a qualification to the conclusion. But such considerations as these belong rather to ordinary applied logic; they amount to nothing more than a caution or hint to be borne in mind when the rules of the syllogism, or of induction, are applied in practice. When, however, we are treating of modality, the extent of the defection from full certainty is supposed to be sufficiently great for our language to indicate and appreciate it. What we then want is of course a scientific discussion of the principles in accordance with which this departure is to be measured and expressed, both in our premises and in our conclusion. Such a plan therefore for treating modality, as the one under discussion, is just as much a banishment of it from the field of real logical enquiry, as if we had determined avowedly to reject it from consideration.
§ 9. Before proceeding to a discussion of the various ways in which modality may be treated by those who admit it into logic, something must be said to clear up a possible source of confusion in this part of the subject. In the cases with which we have hitherto been mostly concerned, in the earlier chapters of this work, the characteristic of modality (for in this chapter we may with propriety use this logical term) has generally been found in singular and particular propositions. It presented itself when we had to judge of individual cases from a knowledge of the average, and was an expression of the fact that the proposition relating to these individuals referred to a portion only of the whole class from which the average was taken. Given that of men of fifty-five, three out of five will die in the course of twenty years, we have had to do with propositions of the vague form, ‘It is probable that AB (of that age) will die,’ or of the more precise form, ‘It is three to two that AB will die,’ within the specified time. Here the modal proposition naturally presents itself in the form of a singular or particular proposition.
§ 10. But when we turn to ordinary logic we may find universal propositions spoken of as modal. This must mostly be the case with those which are termed necessary or impossible, but it may also be the case with the probable. We may meet with the form ‘All X is probably Y.’ Adopting the same explanation here as has been throughout adopted in analogous cases, we must say that what is meant by the modality of such a proposition is the proportional number of times in which the universal proposition would be correctly made. And in this there is, so far, no difficulty. The only difference is that whereas the justification of the former, viz.
the particular or individual kind of modal, was obtainable within the limits of the universal proposition which included it, the justification of the modality of a universal proposition has to be sought in a group or succession of other propositions. The proposition has to be referred to some group of similar ones and we have to consider the proportion of cases in which it will be true. But this distinction is not at all fundamental.
It is quite true that universal propositions from their nature are much less likely than individual ones to be justified, in practice, by such appeal. But, as has been already frequently pointed out, we are not concerned with the way in which our propositions are practically obtained, nor with the way in which men might find it most natural to test them; but with that ultimate justification to which we appeal in the last resort, and which has been abundantly shown to be of a statistical character. When, therefore, we say that ‘it is probable that all X is Y,’ what we mean is, that in more than half the cases we come across we should be right in so judging, and in less than half the cases we should be wrong.
§ 11. It is at this step that the possible ambiguity is encountered. When we talk of the chance that All X is Y, we contemplate or imply the complementary chance that it is not so. Now this latter alternative is not free from ambiguity. It might happen, for instance, in the cases of failure, that no X is Y, or it might happen that some X, only, is not Y; for both of these suppositions contradict the original proposition, and are therefore instances of its failure. In practice, no doubt, we should have various recognized rules and inductions to fall back upon in order to decide between these alternatives, though, of course, the appeal to them would be in strictness extralogical. But the mere existence of such an ambiguity, and the fact that it can only be cleared up by appeal to the subject-matter, are in themselves no real difficulty in the application of the conception of modality to universal propositions as well as to individual ones.
§ 12. Having noticed some of the ways in which the introduction of modality into logic has been evaded or rejected, we must now enter into a brief account of its treatment by those who have more or less deliberately admitted its claims to acceptance.
The first enquiry will be, What opinions have been held as to the nature of modality?