Note 409 — Chapter VII

Using other letters, this is the example given by Mill, Logic, ii. 2, § 1, note, and quoted by Herbert Spencer, Principles of Psychology, II. p. 88. The more general problem of which the above is a special instance is as follows: Given that there are n M’s in existence, and that a M’s are S while b M’s are P, to determine what is the least number of S’s that are also P’s. It is clear that we have no conclusion at all unless a + b > n, i.e., unless there is ultra-total distribution of the middle term. If this condition is satisfied, then supposing the (n − b) M’s which are not-P are all of them found amongst the MS’s, there will still be some MS’s left which are P’s, namely, a − (n − b). Hence the least number of S’s that are also P’s must be a + b − n.