Such propositions as the following, sometimes known as exclusive propositions, may be given as examples of Y: Only S is P ; Graduates alone are eligible for the appointment; Some passengers are the only survivors. These propositions may be interpreted as being equivalent to the following: Some S is all P ; Some graduates are all who are eligible for the appointment; Some passengers are all the survivors.[204] This is, indeed, the only way of treating the propositions which will enable us to retain the original subjects as subjects and the original predicates as predicates.
[204] In these propositions, some is to be interpreted in the indefinite sense, and not as exclusive of all.
We cannot then agree with Professor Fowler that the additional forms “are not merely unusual, but are such as we never do use” (Deductive Logic, p. 31). Still in treating the syllogism &c. on the traditional lines, it is better to retain the traditional schedule of propositions. The addition of the forms 206 U and Y does not tend towards simplification, but the reverse; and their full force can be expressed in other ways. On this view, when we meet with a U proposition, All S is all P, we may resolve it into the two A propositions, All S is P and All P is S, which taken together are equivalent to it; and when we meet with a Y proposition, Some S is all P or S alone is P, we may replace it by the A proposition All P is S, which it yields by conversion.
148. The proposition η.—This proposition in the form No S is some P is not I think ever found in ordinary use. We may, however, recognise its possibility; and it must be pointed out that a form of proposition which we do meet with, namely. Not only S is P or Not S alone is P, is practically η, provided that we do not regard this proposition as implying that any S is certainly P.
Archbishop Thomson remarks that η “has the semblance only, and not the power of a denial. True though it is, it does not prevent our making another judgment of the affirmative kind, from the same terms” (Laws of Thought, § 79). This is erroneous; for although A and η may be true together, U and η cannot, and Y and η are strictly contradictories.[205] The relation of contradiction in which Y and η stand to each other is perhaps brought out more clearly if they are written in the forms Only S is P, Not only S is P, or S alone is P, Not S alone is P. It will be observed, moreover, that η is the converse of O, and vice versâ. If, therefore, η has no power of denial, the same will be true of O also. But it certainly is not true of O.
[205] We are again interpreting some as indefinite. If it means some at most, then the power of denial possessed by η is increased.
149. The proposition ω.—The proposition ω, Some S is not some P, is not inconsistent with any of the other propositional forms, not even with U, All S is all P. For example, granting that “all equilateral triangles are all equiangular triangles,” still “this equilateral triangle is not that equiangular triangle,” which is all that ω asserts. Some S is not some P is indeed always true except when both the subject and the predicate are the name of an individual and the same individual.[206] De 207 Morgan[207] (Syllabus, p. 24) observes that its contradictory is—“S and P are singular and identical; there is but one S, there is but one P, and S is P.”[208] It may be said without hesitation that the proposition ω is of absolutely no logical importance.
[206] Some being again interpreted in its ordinary logical sense. Mr Johnson points out that if some means some but not all, we are led to the paradoxical conclusion that ω is equivalent to U. We may regard a statement involving a reference to some but not all as a statement relating to some at least, combined with a denial of the corresponding statement in which all is substituted for some. On this interpretation, Some S is not some P affirms that “S and P are not identically one,” but also denies that “some S is not any P” and that “some P is not any S”; that is, it affirms SaP and PaS.
[207] De Morgan in several passages criticizes with great acuteness the Hamiltonian scheme of propositions.
[208] Professor Veitch remarks that in ω “we assert parts, and that these can be divided, or that there are parts and parts. If we deny this statement, we assert that the thing spoken of is indivisible or a unity…… We may say that there are men and men. We say, as we do every day, there are politicians and politicians, there are ecclesiastics and ecclesiastics, there are sermons and sermons. These are but covert forms of the some are not some…… ‘Some vivisection is not some vivisection’ is true and important; for the one may be with an anaesthetic, the other without it” (Institutes of Logic, pp. 320, 1). It will be observed that the proposition There are politicians and politicians is here given as a typical example of ω. The appropriateness of this is denied by Mr Monck. “Again, can it be said that the proposition There are patriots and patriots is adequately rendered by Some patriots are not some patriots? The latter proposition simply asserts non-identity: the former is intended to imply also a certain degree of dissimilarity [i.e., in the characteristics or consequences of the patriotism of different individuals]. But two non-identical objects may be perfectly alike” (Introduction to Logic, p. xiv).