[196] Compare Veitch, Institutes of Logic, pp. 307 to 310, and 367, 8. “Hamilton would introduce some only into the theory of propositions, without, however, discarding the meaning of some at least. It is not correct to say that Hamilton discarded the ordinary logical meaning of some. He simply supplemented it by introducing into the propositional forms that of some only.” “Some, according to Hamilton, is always thought as semi-definite (some only) where the other term of the judgment is universal.” Mr Lindsay, however, in expounding Hamilton’s doctrine (Appendix to Ueberweg’s System of Logic, p. 580) says more decisively,—“Since the subject must be equal to the predicate, vagueness in the predesignations must be as far as possible removed. Some is taken as equivalent to some but not all.” Spalding (Logic, p. 184) definitely chooses the other alternative. He remarks that in his own treatise “the received interpretation some at least is steadily adhered to.”
[197] The anticipation of syllogistic doctrine which follows is necessary in order to illustrate the point which we are just now discussing.
202 Such syllogisms as these, however, are not admitted by Hamilton and Thomson; and, on the other hand, Thomson admits as valid certain combinations which on the above interpretation are not valid. Hamilton’s supreme canon of the categorical syllogism is:—“What worse relation[198] of subject and predicate subsists between either of two terms and a common third term, with which one, at least, is positively related; that relation subsists between the two terms themselves” (Logic, II. p. 357). This clearly provides that one premiss at least shall be affirmative, and that an affirmative conclusion shall follow from two affirmative premisses. Thomson (Laws of Thought, p. 165) explicitly lays down the same rules; and his table of valid moods (given on p. 188) is (with the exception of one obvious misprint) correct and correct only if some means “some, it may be all.”
[198] The negative relation is here considered “worse” than the affirmative, and the particular than the universal.
145. The use of “some” in the sense of “some only.”—Jevons, in reply to the question, “What results would follow if we were to interpret ‘Some A’s are B’s’ as implying that ‘Some other A’s are not B’s’?” writes, “The proposition ‘Some A’s are B’s’ is in the form I, and according to the table of opposition I is true if A is true; but A is the contradictory of O, which would be the form of ‘Some other A’s are not B’s.’ Under such circumstances A could never be true at all, because its truth would involve the truth of its own contradictory, which is absurd” (Studies in Deductive Logic, 151). It is not, however, the case that we necessarily involve ourselves in self-contradiction if we use some in the sense of some only. What should be pointed out is that, if we use the word in this sense, the truth of I no longer follows from the truth of A; and that, so far from this being the case, these two propositions are inconsistent with each other.
Taking the five propositional forms, All S is all P, All S is some P, Some S is all P, Some S is some P, No S is P, and interpreting some in the sense of some only, it is to be observed that each one of them is inconsistent with each of the others, whilst at the same time no one is the contradictory of any 203 one of the others. If, for example, on this scheme we wish to express the contradictory of U, we can do so only by affirming an alternative between Y, A, I, and E. Nothing of all this appears to have been noticed by the Hamiltonian writers. Thus, Thomson (Laws of Thought, p. 149) gives a scheme of opposition in which E and I appear as contradictories, but A and O as contraries.
One of the strongest arguments against the use of some in the sense of some only is very well put by Professor Veitch, himself a disciple of Sir William Hamilton. Some only, he remarks, is not so fundamental as some at least. The former implies the latter; but I can speak of some at least without advancing to the more definite stage of some only. “Before I can speak of some only, must I not have formed two judgments—the one that some are, the other that others of the same class are not? …… The some only would thus appear as the composite of two propositions already formed…… It seems to me that we must, first of all, work out logical principles on the indefinite meaning of some at least…… Some only is a secondary and derivative judgment.” (Institutes of Logic, p. 308).
If some is used in the sense of some only, the further difficulty arises how we are to express any knowledge that we may happen to possess about a part of a class when we are in ignorance in regard to the remainder. Supposing for example, that all the S’s of which I happen to have had experience are P’s, I am not justified in saying either that all S’s are P’s or that some S’s are P’s. The only solution of the difficulty is to say that all or some S’s are P’s. The complexity that this would introduce is obvious.
146. The interpretation of the eight Hamiltonian forms of proposition, “some” being used in its ordinary logical sense.[199]—Taking the five possible relations between two terms, as illustrated by the Eulerian diagrams, and denoting them respectively by α, β, γ, δ, ε, as in section [126], we may write against each of the propositional forms the relations which are compatible with 204 it, on the supposition that some is used in its ordinary logical sense, that is, as exclusive of none but not of all:—[200]
| U | α |
| A | α, β |
| Y | α, γ |
| I | α, β, γ, δ |
| E | ε |
| η | β, δ, ε |
| O | γ, δ, ε |
| ω | α, β, γ, δ, ε |