(2) It may be held that every proposition should be interpreted as implying simply the existence of its subject. This is Mill’s view (as regards real propositions); for he holds that we cannot give information about a non-existent subject.[224] This is no doubt the view that, at any rate on a first 220 consideration of the subject, appears to be at once the most reasonable and the most simple.
[224] “An accidental or non-essential affirmation does imply 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).
(3) It may be held that we should not regard propositions as necessarily implying the existence either of their subjects or of their predicates. On this view, the full implication of All S is P may be expressed by saying that it denies the existence of anything that is at the same time S and not-P. Similarly No S is P implies the existence neither of S nor of P, but merely denies the existence of anything that is both S and P. Some S is P (or is not P) may be read Some S, if there is any S, is P (or is not P). Here we neither affirm nor deny the existence of any class absolutely;[225] the sum total of what we affirm is that if any S exists, then something which is both S and P (or S and not-P) also exists. On this interpretation, therefore, particular propositions have a hypothetical and not a purely categorical character.
[225] Jevons lays down the dictum that “we cannot make any statement except a truism without implying that certain combinations of terms are contradictory and excluded from thought” (Principles of Science, 2nd edition, p. 32). This is true of universals (though somewhat loosely expressed), but it does not seem to be true of particular propositions, whatever view may be taken of them.
(4) It may be held that universal propositions should not be interpreted as implying the existence of their subjects, but that particular propositions should be interpreted as doing so.[226] On this view All S is P merely denies the existence of anything that is both S and not-P; No S is P denies the existence of anything that is both S and P ; Some S is P affirms the existence of something that is both S and P ; Some S is not P affirms the existence of something that is both S and not-P. Thus, universals are interpreted as having existentially a negative force, while particulars have an affirmative force. This hypothesis will be found to lead to certain paradoxical results, but it will also be shewn to lead to a more satisfactory and symmetrical treatment of logical problems than is otherwise possible.[227]
[226] Dr Venn advocates this doctrine with special reference to the operations of symbolic logic; but there is no reason why it should not be extended to ordinary formal logic.
[227] The hypothesis in question has been already provisionally adopted in the scheme of logical equivalences given in section [108], and also in the symbolic scheme of propositions given on page [193].
221 157. Reduction of the traditional forms of proposition to the form of Existential Propositions.—Without at present attempting to decide between the different possible suppositions as to the existential import of the traditional forms of proposition, we may enquire how on the different suppositions they may be reduced to existential form. It will be assumed throughout that both the traditional forms and the existential forms are interpreted assertorically. In the case of each of the traditional forms it will suffice to deal with the two fundamental suppositions, namely, that it does and that it does not imply the existence of its subject.
The universal affirmative. (1) If SaP is interpreted as not carrying with it any existential implication in regard to its separate terms, it is equivalent to the existential proposition SPʹ = 0. Dr Wolf denies this on the ground that SaP contains further the implication “If there are any S’s, they must all be P’s”; and hence that, while on the supposition in question SPʹ = 0 is an inference from SaP, it is not equivalent to it. It is of course a very elementary truth that inferences are not always the exact equivalents of their premisses. But in the above argument Dr Wolf has apparently overlooked the fact that SPʹ = 0, equally with SaP, contains the implication “If there are any S’s they are all P’s.”[228] By the law of excluded middle, every S (if there are any S’s) must be P or not P, and since SPʹ = 0, the above inference clearly follows. SPʹ = 0 carries with it in fact the two implications If S > 0 then P > 0, If P > 0 then Sʹ > 0. These may also be written in the forms Either S = 0 or P > 0, Either Pʹ = 0 or Sʹ > 0.
[228] Dr Wolf perhaps draws a distinction between the proposition “If there are any S’s they must all be P’s” and the proposition “If there are any S’s they are all P’s,” giving to the former an apodeictic, and to the latter a merely assertoric, force. But if so, then the former is implied by All S is P, only if this proposition is apodeictic, not if it is merely assertoric. The argument is in this case irrelevant so far as the position which I take is concerned, since it is only the assertoric SaP that I regard as equivalent to SPʹ = 0. Dr Wolf can hardly maintain that all propositions of the form All S is P are apodeictic. His whole treatment of the subject with which we are now dealing appears, however, to be valid only if it relates to a modal schedule of propositions. At the same time he nowhere clearly indicates a limitation of this kind, and many of the doctrines which he criticises are intended by those who adopt them to apply only to an assertoric schedule.