322 COBBESPONDENCE. b or c'. But this is by no means the valid conclusion from the premisses of the problem. Mr. Venn's error lies in the fact that his symbolic statement of the first premiss may stand equally well for 2045 other hypothetical propositions, looking at it from his own point of view. Likewise his statement of the second premiss may stand equally well for 32j765 other hypothetical pro- positions. Lastly, his symbolic conclusion dabc = may stand for 32,766 different hypothetical propositions when interpreted according to his peculiar method. One of these hypotheticals is the proper conclusion from the premisses of the problem, since his symbolic premisses imply the premisses. It would have been remarkable if Mr. Venn had been able to tell which of the 32,766 is the'correct conclusion. It is still more remark- able that he did not read dale = as a hypothetical at all, but gives the Boolian interpretation of it. My conclusion, whose accuracy he does not impeach, shows how he might have read dabc = so as to get the proper conclusion from the premisses of the problem, viz., by dividing dabc into the two logical factors ab and dc, and equating to zero the negative of ab as the antecedent and equating to zero dc = as the consequent. Thus we have ' If ab = then rfc = 0, ! that is, ' If there is no a nor b, then all d is c,' or ' Either a or & exists, or all d is c '. In Mr. Venn's reprint of my solution there is a typographical error. The term dx should read dx. O. H. MITCHELL. On the main point in question, viz., the inadequacy of the expression y8(l a/3) to express the proposition ' If no a is J3 then y is 8,' I must quite admit the effectiveness of Mr. Mitchell's remarks. But I shall be glad, with the Editor's permission, to explain more fully on another occasion in what sense I proposed it, and to what extent I still think that it may serve such a purpose. I -will only remark here that I never in any pub- lished work claimed the expression as exactly expressing the customary signification of such hypotheticals, though I certainly thought that it might, in certain cases, express more of such signification than I now see that it could do. On the fundamental principle which underlies the interpretation of these hypotheticals, viz., the impossibility of expressing particular proposi- tions on a 1 and scheme, I judge that I am entirely at one with Mr. Mitchell. I have urged as against Boole and Jevons as strongly as I could that such propositions demand some third symbol, whether it stand in the place of copula or predicate, and cannot be represented with the ordinary notation for universals. J. VENN. [Miss Ladd (Mrs. Fabian Franklin), author of another paper in the Studies in Logic, also writes, with reference to Mr. Venn's Critical Notice in MIND XXXII., that the second sentence of the quotation on p. 589, which he found obscure, should have had the words "of the product" inserted after the words "partial inclusion". EDITOR.]
