Page:Mind (New Series) Volume 12.djvu/376

 362 HUGH MACCOLL : the same meaning throughout ; but then the formula (unlike mine) will no longer represent Barbara. For (A' + B) 6, being synonymous with A : B, asserts that every P in the class A is also in the class B ; whereas A' + B (or its equivalent A-< B) only asserts that a certain P of the series P T, P.,, P 3 , etc., is either excluded from A or included in B ; it makes no assertion as to the other individuals of the series. This comparison of the formulae (A : B) (B : C) : (A : C) which are erroneously supposed to be equivalent, is typical of many others. Another formula of mine that has led to misunderstandings is the formula (Proceedings of the Mathe- matical Society, Third Paper) (A : x) + (B : x) : (AB : x). Not that the validity of this formula has been called in question ; it is indeed almost self-evident ; but logicians have asserted that the symbol = might with advantage replace the symbol : before the conclusion AB : x, as (in their opinion) the converse implication is also true. Now, if my symbol A : B (like their symbol A -< B) meant A' + B, this converse implication would be true, and = might replace : before the conclusion AB : x. But this, as already explained, is not the signification of my symbol A : B, so that the substitution of = for : before the conclusion (or consequent) would destroy the validity of the formula. A geometrical illustration will make this clear. Out of the total ten points marked in the ellipse x and the two circles A, B of the accompanying figure, take a point P at random, and let A, B, x assert respectively (as propositions) that P will be in A, that P will be in B, that P will be in x. It is evident that the re- spective chances of the four propositions A, B, x, AB are fV> TTT> TO, T% 5 so tnat tDev are all variables. The implication AB : x asserts that the point P cannot be in both the circles A and B without being also in the ellipse x, which is true. The implication A : x asserts that P cannot be in A without being in x, which is false ; and B : x asserts that P cannot be in B without being in x, which is false also. Thus, the alternative (A : x) + (B : x) is false while AB : x is true, so that in this case the substitution of the symbol = for : before AB : x in my formula would be wrong. But my formula is right .in this case as in all others ; for i + i : r = I T + L T : r r = 77 + 77 : e = 77 : e