But if it is true that we know definitely the relative extent of subject and predicate, and if some is used strictly in the sense of some but not all, we should have but five propositional forms instead of eight, namely,—All S is all P, All S is some P, Some S is all P, Some S is some P,[192] No S is any P.
[192] Using some in the sense here indicated, the interpretation of the proposition Some S is some P is not altogether free from ambiguity. The interpretation I am adopting is to regard it as equivalent to the two following propositions with unquantified predicates, namely, Some but not all S is P and Some but not all P is S. It then necessarily implies the Hamiltonian propositions Some S is not any P and No S is some P.
We have already seen (in section [126]) that the only possible relations between two terms in respect of their extension are given by the following five diagrams,—
These correspond respectively to the five propositional forms given above;[193] and it is clear that on the view indicated by Dr Baynes the eight forms are redundant.[194]
[193] Namely U, A, Y, I, E. O and η cannot be interpreted as giving precisely determinate information; O allows an alternative between Y and I, and η between A and I. For the interpretation of ω see note [2] on page 206.
[194] Compare Venn, Symbolic Logic, chapter I.
It is altogether doubtful whether writers who have adopted the eightfold scheme have themselves recognised the pitfalls 201 surrounding the use of the word some. Many passages might be quoted in which they distinctly adopt the meaning—some but not all. Thus, Thomson (Laws of Thought, p. 150) makes U and A inconsistent. Bowen (Logic, pp. 169, 170) would pass from I to O by immediate inference.[195] Hamilton himself agrees with Thomson and Bowen on these points; but he is curiously indecisive on the general question here raised. He remarks (Logic, II. p. 282) that some “is held to be a definite some when the other term is definite,” i.e., in A and Y, η and O: but “on the other hand, when both terms are indefinite or particular, the some of each is left wholly indefinite,” i.e., in I and ω.[196] This is very confusing, and it would be most difficult to apply the distinction consistently. Hamilton himself certainly does not so apply it. For example, on his view it should no longer be the case that two affirmative premisses necessitate an affirmative conclusion; or that two negative premisses invalidate a syllogism.[197] Thus, the following should be regarded as valid:
| All P is some M, | |
| All M is some S, | |
| therefore, | Some S is not any P. |
| No M is any P, | |
| Some S is not any M, | |
| therefore, | Some or all S is not any P. |
[195] “This sort of inference,” he remarks, “Hamilton would call integration, as its effect is, after determining one part, to reconstitute the whole by bringing into view the remaining part.”