Note 243 — Chapter VIII

The above doctrine has been criticized on the ground that it practically amounts to saying that neither of the given propositions has any meaning whatever, but that each is a mere sham and pretence of predication; and a request is made for concrete examples. The following example may perhaps suffice to illustrate the particular point now at issue: “An honest miller has a golden thumb”; “Well, I am sure that no miller, honest or otherwise, has a golden thumb.” These two propositions are in the form of what would ordinarily be called contraries; but taken together they may quite naturally be interpreted as meaning that no such person can be found as an honest miller. The former proposition would indeed probably be intended to be supplemented by the latter or by some proposition involving the latter, and so to carry inferentially the denial of the existence of its subject.

Another example is contained in the following quotation from Mrs Ladd Franklin: “All x is y, No x is y, assert together that x is neither y nor not-y, and hence that there is no x. It is common among logicians to say that two such propositions are incompatible; but that is not true, they are simply together incompatible with the existence of x. When the schoolboy has proved that the meeting point of two lines is not on the right of a certain transversal and that it is not on the left of it, we do not tell him that his propositions are incompatible and that one or other of them must be false, but we allow him to draw the natural conclusion that there is no meeting point, or that the lines are parallel” (Mind, 1890, p. 77 n.).

Dr Wolf (Studies in Logic, p. 140), criticizing Mrs Ladd Franklin’s concrete example, maintains that the two propositions given by her are sub-contraries (I and O), not contraries (A and E). A moment’s consideration will, however, shew that this is not the case since neither of the propositions is particular. At the same time it is true that a little manipulation is required to bring them to the forms A and E. There is also the assumption that “on the right” and “on the left” exhaust the possibilities and are therefore contradictory terms. Granting this assumption, the two propositions may be expressed symbolically in the forms No S is P, No S is not P, and it then needs only the obversion of one of them to bring them to the forms A and E.