Note 299 — Chapter X

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.”