[299] Obviously a disjunctive proposition is a more condensed form of expression on the exclusive than on the non-exclusive interpretation. Compare Mansel’s Aldrich, p. 242, and Prolegomena Logica, p. 288. “Let us grant for a moment the opposite view, and allow that the proposition All C is either A or B implies as a condition of its truth No C can be both. Thus viewed, it is in reality a complex proposition, containing two distinct assertions, each of which may be the ground of two distinct processes of reasoning, governed by two opposite laws. Surely it is essential to all clear thinking that the two should be separated from each other, and not confounded under one form by assuming the Law of Excluded Middle to be, what it is not, a complex of those of Identity and Contradiction” (Aldrich, p. 242). It may be added that one paradoxical result of the exclusive interpretation of alternatives is that not either P or Q is not equivalent to neither P nor Q.
A further paradoxical result is pointed out by Mr G. R. T. Ross in an article on the Disjunctive Judgment in Mind (1903, p. 492), namely, that on the exclusive interpretation the disjunctives A is either B or C and A is either not B or not C are identical in their import; for in each case the real alternants are B but not C and C but not B. Thus, to take an illustration borrowed from Mr Ross, the two following propositions are (on the interpretation in question) identical in their import,—“Anyone who affirms that he has seen his own ghost is either not sane or not telling what he believes to be the truth,” “Anyone who affirms that he has seen his own ghost is either sane or truthful.”
Mr Bosanquet and other writers who advocate the exclusive interpretation of disjunctives appear to have chiefly in view the expression in disjunctive form of a logical division or scientific classification. I should of course agree that such a division or classification is imperfect if the members of which it consists are not mutually exclusive as well as collectively exhaustive. This condition must also be satisfied when we make use of the disjunctive judgment in connexion with the doctrine of probability.[300] It will, however, hardly be proposed to confine the disjunctive judgment to these uses. We frequently have occasion to state alternatives independently of any scientific classification or any calculation of probability; and we must not regard the bare form of the disjunctive judgment as expressing anything that we are not prepared to recognise as universally involved in its use.
[300] In this connexion the further condition of the “equality” in a certain sense of the alternants has in addition to be satisfied.
It is of course always possible to express an alternative 281 statement in such a way that the alternants are formally incompatible or exclusive. Thus, not wishing to exclude the case of A being both B and C we may write A is B or bC ;[301] or, wishing to exclude that case, A is Bc or bC. But in neither of these instances can we say that the incompatibility of the alternants is really given by the alternative proposition. It is a merely formal proposition that No A is both B and bC or that No A is both Bc and bC. The proposition Every A is Bc or bC does, however, tell us that no A is both B and C ; and when from our knowledge of the subject-matter it is obvious that we are dealing with alternants that are mutually exclusive (and no doubt this is a very frequent case), we have in the above form a means of correctly and unambiguously expressing the fact. Where it is inconvenient to use this form, it is open to us to make a separate statement to the effect that No A is both B and C. All that is here contended for is that the bare symbolic form A is either B or C should not be interpreted as being equivalent to A is either Bc or bC.
[301] Where b = not-B, and c = not-C. What is contained in this paragraph is to some extent a repetition of what is given on page [278].
(2) We may pass on to consider the second main question that arises in connexion with the import of disjunctive (alternative) propositions, namely, whether such propositions are to be interpreted as modal or as merely assertoric.
In chapter 9 it was urged that the modal interpretation of the typical hypothetical proposition If A then C must be regarded as the more natural one, on the ground that we should not ordinarily think it necessary to affirm the truth of A in order to contradict the proposition, as would be necessary if it were interpreted assertorically.[302] Similarly the enquiry as to how we should naturally contradict the typical alternative propositions Every S is either P or Q, Either X or Y is true, may help us in deciding upon the interpretation of these propositions.
On the assertoric interpretation, the contradictories of the propositions in question are Some S is neither P nor Q, Neither X nor Y is true ; on the modal interpretation, they are An S need not be either P or Q, Possibly neither X nor Y is true. 282 There can be no doubt that this last pair of propositions would not as a rule be regarded as adequate to contradict the pair of alternatives; and on this ground we may regard the assertoric interpretation of alternatives as most in accordance with ordinary usage. There is also some advantage in differentiating between hypotheticals and alternatives by interpreting the former modally and the latter assertorically, except in so far as a clear indication is given to the contrary. It is not of course meant that modal alternatives are never as a matter of fact to be met with or that they cannot receive formal recognition; they can always be expressed in the distinctive forms Every S must be either P or Q, Either X or Y is necessarily true.