[261] Hence Mrs Ladd Franklin is led to the conclusion that “no consistent logic of universal propositions is possible except with the convention that they do not imply the existence of their terms” (Mind, 1890, p. 88).
(2) We may next consider the existential import of propositions with reference to the doctrine of opposition. It has been shewn in section [159] that if particulars are interpreted as implying the existence of their subjects, while universals are not so interpreted, then A and O, E and I, are true contradictories; but that this is not the case under any of the other suppositions discussed in the same section.[262] There can, however, be no doubt that one of the most important functions of particular propositions is to contradict the universal propositions of opposite quality; and hence we have a strong argument in favour of a view of the existential import of propositions which will leave the ordinary doctrine of contradiction unaffected.
[262] A and O, E and I, will also be true contradictories if universals are interpreted as implying the existence of their subjects, while particulars are not so interpreted. It would be interesting, if space permitted, to work out the results of this supposition in detail. If the student does this for himself, he will find that this is the only supposition, under which the ordinary doctrine of opposition holds good throughout. All other considerations, however, are opposed to its adoption. It altogether conflicts with popular usage; it renders the processes of simple conversion and simple contraposition illegitimate; and whilst making universals double judgments, it destroys the categorical character of particulars altogether. In regard to this last point see page [220].
As regards the doctrines of subalternation, contrariety, and subcontrariety, our results (namely, that I does not follow from A, or O from E, that A and E may both be true, and that I 242 and O may both be false) are no doubt paradoxical. But this objection is far more than counterbalanced by the fact that the doctrine of contradiction is saved. For as compared with the relation between contradictories, these other relations are of little importance. We may specially consider the relation between A and I. Some S is P cannot now without qualification be inferred from All S is P, since the former of these propositions implies the existence of S, while the latter does not. But as a matter of fact this is an inference which we never have occasion to make. If their existential import is the same why should we ever lay down a particular proposition when the corresponding universal is at our service? On the other hand, the view that we are advocating gives Some S is P a status relatively to All S is P as well as relatively to No S is P which it could not otherwise possess; and similarly for Some S is not P. Our result as regards the relation between SaP and SiP has been described as equivalent to saying “that a statement of partial knowledge carries more real information than a statement of full knowledge; since if we only possess limited information, and so can only assert SiP, we thereby affirm the existence of S ; but if we have sufficient knowledge to speak of all S (S remaining the same) the statement of that full knowledge immediately casts a doubt upon that existence.” This way of putting it is, however, misleading if not positively erroneous. On the view in question it is incorrect to say simply that SiP and SaP give “partial” and “full” knowledge respectively, for SiP while giving less knowledge than SaP in one direction gives more in another. In other words, the knowledge which is “full” relatively to SiP is not expressed by SaP by itself, but by SaP together with the statement that there are such things as S.[263]
[263] The position taken above in regard to subalternation is very well expressed by Mrs Ladd Franklin. “Nothing of course is now illogical that was ever logical before. It is merely a question of what convention in regard to the existence of terms we adopt before we admit the warm-blooded sentences of real life into the iron moulds of logical manipulation. With the old convention (which was never explicitly stated) subalternation ran thus: No x’s are y’s (and we hereby mean to imply that there are x’s, whatever x may be), therefore, Some x’s are non-y’s. With the new convention the requirement is simply that if it is known that there are x’s (as it is known, of course, in by far the greater number of sentences that it interests us to form) that fact must be expressly stated. The argument then is: No x’s are y’s, There are x’s, therefore, There are x’s which are non-y’s.”
243 (3) There is one further point of importance to be noted, and that is, that the interpretation of A, E, I, O propositions under consideration is the only interpretation according to which each one of these propositions is resolved into a single categorical statement. For if A and E imply the existence of their subjects they express double, not single, judgments, being equivalent respectively to the statements: There are S’s, but there are no SPʹ’s ; There are S’s, but there are no SP’s ; whereas on the interpretation here proposed they simply express respectively the single judgments: There are no SPʹ’s ; There are no SP’s. On the other hand, if I and O do not imply the existence of their subjects, instead of expressing categorical judgments, they express somewhat complex hypothetical ones, being equivalent respectively to the statement: If there are any S’s then there are some SP’s ; If there are any S’s then there are some SPʹ’s ; whereas on our interpretation they express respectively the categorical judgments: There are SP’s ; There are SPʹ’s.[264]
[264] Compare sections [156], [157].
On the whole, there is a strong cumulative argument in favour of interpreting particulars, but not universals, as implying formally the existence of their subjects.[265] This solution 244 is to be regarded as partly of the nature of a convention. We arrive, however, at the conclusion that no other solution can equally well suffice as the basis of a scientific treatment of the traditional schedule of propositions, so long, at any rate, as the propositions included in the schedule are regarded as assertoric and not modal.
[265] We may briefly discuss in a note one or two objections to this view which have not yet been explicitly considered.
(а) Mill argues that a synthetical proposition necessarily implies “the real existence of the subject, because in the case of a non-existent subject there is nothing for the proposition to assert” (Logic, i. 6, § 2). In answer to this it is sufficient to point out that a non-existent thing will be described as possessing attributes which are separately attributes of existing things, although that particular combination of them may not anywhere be found, and if we know (as we may do) that certain of these attributes are always accompanied by other attributes we may predicate the latter of the non-existent thing, thereby obtaining a real proposition which does not involve the actual existence of its subject. As an argument ad hominem it may further be pointed out that Mill inclines to deny the existence of perfect straight lines or perfect circles. Would he therefore affirm that we can make no real assertions about such things?