[238] Of course the doctrine of contradiction always holds good in the sense that a pair of real contradictories cannot both be true or both false; and similarly with the other doctrines. The doctrines that we have to consider are not these, but whether SaP and SoP are really contradictories irrespective of the existential interpretation of the propositions, whether SaP and SeP are really contraries, and so on.
It should be added that, throughout the discussion, the propositions are supposed to be interpreted assertorically, as has always been the custom with the traditional schedule. The necessity for this proviso will from time to time be pointed out.
(1) Let every proposition be interpreted as implying the 228 existence both of its subject and of its predicate and also of their contradictories.[239]
[239]It would be quite a different problem if we were to assume the existence of S and P independently of the affirmation of the given proposition. A failure to distinguish between these problems is probably responsible for a good deal of the confusion and misunderstanding that has arisen in connexion with the present discussion. But it is clearly one thing to say (a) “All S is P and S is assumed to exist,” and another to say (b) “all S is P,” meaning thereby “S exists and is always P.” In case (a) it is futile to go on to make the supposition that S is non-existent; in case (b), on the other hand, there is nothing to prevent our making the supposition, and we find that, if it holds good, the given proposition is false.
On this supposition, if either the subject or the predicate of a proposition is the name of a class which is unrepresented in the universe of discourse or which exhausts that universe, then that proposition is false; for it implies what is inconsistent with fact. It follows that a pair of contradictories as usually stated, and also a pair of sub-contraries, may both be false. For example, All S is P and Some S is not P both imply the existence of S in the universe of discourse. In the case then in which S does not exist in that universe, these propositions would both be false.
If a concrete illustration is desired, we may take the propositions, None of the answers to the question shewed originality, Some of the answers to the question shewed originality, and assume that each of these propositions includes as part of its implication the actual occurrence of its subject in the universe of discourse. Then our position is that if there were no answers to the question at all, the truth of both the propositions must be denied. The fact of there having been no answers does not render the propositions meaningless; but it renders them false, their full import being assumed to be, respectively, There were answers to the question but none of them shewed originality, There were answers to the question and some of them shewed originality.
We must not of course say that under our present supposition true contradictories cannot be found; for this is always possible. The true contradictory of All S is P is Either some S is not P, or else either S or not-S or P or not-P is non-existent. Similarly in other cases. The ordinary doctrines of subalternation and contrariety remain unaffected.
229 (2) Let every proposition be interpreted as implying the existence of its subject.
For reasons similar to those stated above, the ordinary doctrines of contradiction and sub-contrariety again fail to hold good. The true contradictory of All S is P now becomes Either some S is not P, or S is non-existent. The ordinary doctrines of subalternation and contrariety again remain unaffected.
(3) Let no proposition be interpreted as implying the existence either of its subject or of its predicate.
(a) The ordinary doctrine of subalternation holds good.
(b) The ordinary doctrine of contradiction does not hold good. All S is P, for example, merely denies the existence of any S’s that are not P’s; Some S is not P merely asserts that if there are any S’s some of them are not P’s. In the case in which S does not exist in the universe of discourse we cannot affirm the falsity of either of these propositions.[240]
230 (c) The ordinary doctrine of contrariety does not hold good. For if there is no implication of the existence of the subject in universal propositions we are not actually precluded from asserting together two propositions that are ordinarily given as contraries. All S is P merely denies that there are any S not-P’s, No S is P that there are any SP’s. We may, therefore, without inconsistency affirm both All S is P and No S is P ; but this is virtually to deny the existence of S.[241]
(d) The ordinary doctrine of sub-contrariety remains unaffected.
[240] Dr Wolf (Studies in Logic, p. 132) denies the validity of this reasoning. He admits apparently that the existential propositions SPʹ = 0 and Either S = 0 or SPʹ > 0 are not contradictories; but he denies that on the supposition under discussion SaP and SPʹ = 0 are equivalent. His main ground for taking this view is that SaP carries with it the implication If there are any S’s they are all P’s, while SPʹ = 0 does not carry with it any such implication. This position has been already criticized in section [157]. Dr Wolf relies partly upon concrete examples, but in so doing he complicates the discussion by introducing modal forms of expression. Thus for the proposition “Some successful candidates do not receive scholarships,” we find substituted in the course of his argument “If there are any successful candidates then some of them do not (or need not) receive scholarships,” and the insertion of the words in brackets yields a proposition which, although an inference from the original proposition, is not really equivalent to it, unless the original proposition is itself interpreted modally. Later on Dr Wolf explicitly alters the whole problem by assuming that what is under consideration is a modal schedule of propositions. Thus he goes on to say, “What SaP and SeP really express severally is the necessity and the impossibility of S being P”; and for the purpose of contradicting SaP and SeP, “SiP and SoP need mean no more than S may be P and S need not be P.” The question how far SaP and SeP should be interpreted modally is discussed elsewhere. All I would point out here is that it is a distinct question from that raised in the text, which is a question relating to the traditional schedule of propositions interpreted assertorically. The whole question of existential import is indeed one that cannot be discussed to any purpose until the character of the schedule of propositions under consideration has been defined. From the mixing up of schedules and interpretations nothing but confusion can result. In the following section the opposition of modals will be briefly considered in connexion with their existential import.