So far as Dr Wolf’s argument is independent of the above concrete example, it appears to depend upon an identification of the proposition Some S is P with the proposition S may be P. The latter is a modal form, and is undoubtedly consistent with the existence of S and the non-existence of P. But I venture to think that the identification of the two forms runs entirely counter to the current use of language. I am quite prepared to admit that if All S is P is interpreted as an unconditional universal, meaning S as such is P, its true contradictory is S may be P, not Some S is P. But this is just because I do not think that Some S is P would be understood to express merely the abstract compatibility of S and P. Certainly Dr Wolf’s own concrete example, referred to above, cannot bear this interpretation. For some further observations on modals in connexion with existential import, see sections [160] and [163].
[220] Jevons remarks that he does not see how there can be in deductive logic any question about existence, and observes, with reference to the opposite view taken by De Morgan, that “this is one of the few points in which it is possible to suspect him of unsoundness “ (Studies in Deductive Logic, p. 141). It is, however, impossible to attach any meaning to Jevons’s own “Criterion of Consistency,” unless it has some reference to “existence.” “It is assumed as a necessary law that every term must have its negative. Thence arises what I propose to call the Criterion of Consistency, stated as follows:—Any two or more propositions are contradictory when, and only when, after all possible substitutions are made, they occasion the total disappearance of any term, positive or negative, from the Logical Alphabet” (p. 181). What can this mean but that although we may deny the existence of the combination AB, we cannot without contradiction deny the existence of A itself, or not-A, or B, or not-B? This assumption regarding the existential implication of propositions runs through the whole of Jevons’s equational logic. The following passage, for example, is taken almost at random: “There remain four combinations, ABC, aBC, abC, abc. But these do not stand on the same logical footing, because if we were to remove ABC, there would be no such thing as A left; and if we were to remove abc there would be no such thing as c left. Now it is the criterion or condition of logical consistency that every separate term and its negative shall remain. Hence there must exist some things which are described by ABC, and other things described by abc” (p. 216).
218 155. The Existential Formulation of Propositions.—We may define an existential proposition as one that directly affirms or denies existence (or occurrence) in the universe of discourse (or portion of reality) to which reference is made. Such propositions are of course met with in ordinary forms of speech: for example, God exists, It rains, There are white hares, It does not rain, Unicorns are non-existent. There is no rose without a thorn. Sometimes the affirmation or denial of existence takes a less simple form, but is none the less direct: for example, The assassination of Caesar is an historical event, D’Artagnan is not an imaginary person, The centaur is a fiction of the poets, The large copper butterfly is extinct.
In the formal expression of existential propositions it will be convenient to make use of certain symbols described in the preceding chapter. Thus, the affirmation of the existence of S may be written in the form S > 0, and the denial of the existence of S in the form S = 0. We shall then have an existential schedule of propositions if we reduce our statements to one or other of these forms or to a conjunctive or disjunctive combination of them. The relation between the traditional schedule and an existential schedule of this kind will be discussed in the next [section] but one.
It may here be pointed out that since the universe of discourse is itself assumed to be real and hence cannot be entirely emptied of content, any denial of existence involves also an affirmation of existence. For if we deny the existence of S, we thereby implicitly affirm the existence of not-S, since by the law of excluded middle everything in the universe of discourse must be either S or not-S. It follows that every proposition contains directly or indirectly an affirmation of existence.[221]
[221] In an article in Baldwin’s Dictionary of Philosophy and Psychology, Mrs Ladd Franklin points out that the proposition All S is P is equivalent to the proposition Everything is P or not-S, and hence necessarily implies the existence of either P or not-S. Write x for not-S and y for P, so that the original proposition becomes All but x is y ; it then implies, as its minimum existential import, the existence of either x or y.
156. Various Suppositions concerning the Existential Import of Categorical Propositions.—Several different views may be 219 taken as to what implication with regard to existence, if any, is involved in categorical propositions of the traditional type. The following may be formulated for special discussion:—[222]
[222] The suppositions that follow are not intended to be exhaustive. We might, for instance, regard propositions as implying the existence both of their subjects and their predicates, but not of the contradictories of these; or we might regard universals as always implying the existence of their subjects, but particulars as not necessarily implying the existence of theirs (see note [3] on p. 241); or affirmatives as always implying the existence of their subjects, but negatives as not necessarily implying the existence of theirs. This last supposition represents the view of Ueberweg. Still another view is taken by Lewis Carroll, who regards all categorical propositions, except universal negatives, as implying the existence of their subjects. “In every proposition beginning with some or all, the actual existence of the subject is asserted. If, for instance, I say ‘all misers are selfish,’ I mean that misers actually exist. If I wished to avoid making this assertion, and merely to state the law that miserliness necessarily involves selfishness, I should say ‘no misers are unselfish,’ which does not assert that any misers exist at all, but merely that, if any did exist, they would be selfish” (Game of Logic, p. 19). It would take too much space, however, to give a separate discussion to suppositions other than those mentioned in the text.
(1) It may be held that every categorical proposition should be interpreted as implying the existence both of objects denoted by the terms directly involved and also of objects denoted by their contradictories; that, for example, All S is P should be regarded as implying the existence of S, not-S, P, not-P. This view is implied in Jevons’s Criterion of Consistency mentioned in the [note] on page 217. It is also practically adopted by De Morgan.[223]
[223] “By the universe (of a proposition) is meant the collection of all objects which are contemplated as objects about which assertion or denial may take place. Let every name which belongs to the whole universe be excluded as needless: this must be particularly remembered. Let every object which has not the name X (of which there are always some) be conceived as therefore marked with the name x meaning not-X” (Syllabus, pp. 12, 13). Compare, also, De Morgan’s Formal Logic, p. 55.