Studies and Exercises in Formal Logic

by John Neville Keynes

[²⁴¹] Of course on the view under consideration we ought not to continue to speak of these two propositions as contraries.

(4) Let particulars be interpreted as implying, while universals are not interpreted as implying, the existence of their subjects.
(a) The ordinary doctrine of subalternation does not hold good. Some S is P, for example, implies the existence of S, while this is not implied by All S is P.
(b) The ordinary doctrine of contradiction holds good. All S is P denies that there is any S that is not-P; Some S is not P affirms that there is some S that is not-P. It is clear that these propositions cannot both be true; it is also clear that they cannot both be false. Similarly for No S is P and Some S is P.
(c) The ordinary doctrine of contrariety does not hold good. All S is P and No S is P are not inconsistent with one another, but the force of asserting both of them is to deny that there are any S’s.[242] This follows just as in the case of our third supposition.[243]
231 (d) The ordinary doctrine of sub-contrariety does not hold good.[244] Some S is P and Some S is not P are both false in the case in which S does not exist in the universe of discourse.

[²⁴²] If, however, we are given No S is P and also Some S is P, then we are able to infer that All S is P is false. The second of these propositions affirms the existence of S, and therefore destroys the hypothesis on which alone the first and third can be treated as compatible.

[²⁴³] 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.

[²⁴⁴] It may be worth observing that, given (b), (d) might be deduced from (c) or vice versâ.

The relation between contradictories is by far the most important relation with which we are concerned in dealing with the opposition of propositions, and it will be observed that the last of the above suppositions is the only one under which the ordinary doctrine of contradiction holds good.

160. The Opposition of Modal Propositions considered in connexion with their Existential Import.—The propositions discussed in the preceding sections have been the propositions belonging to the traditional schedule interpreted assertorically. Turning now to the corresponding modal schedule, we may briefly consider how the doctrine of opposition is affected, if at all, on the supposition that the propositions included in the schedule are not interpreted as implying the existence of their 232 subjects. We find that on this supposition S as such is P and S need not be P are true contradictories.

S as such is P (interpreted as not necessarily implying the existence of S) does more than deny the actual occurrence of the conjunction S not-P, it denies the possibility of such a conjunction; and all that is necessary in order to contradict this is to affirm the possibility of the conjunction. This is done by the proposition S need not be P (also interpreted as not necessarily implying the existence of S). On the same supposition, S as such is P, S as such is other than P, are true contraries.