346 Mr DE morgan, ON THE SYLLOGISM, No. IV, 



a fair proportion will think that himself was included. I shall hold, for logical purposes, 

 that the predicate is included among its own convertible relatives. 



A relation is transitive when a relative of a relative is a relative of the same kind ; as 

 symbolised in LL )) L, whence LLL )) LL )) L ; and so on. 



A transitive relation has a transitive converse, but not necessarily a transitive contrary: 

 for L"^L"' is the converse of LL, so that LL))L gives L~'L~' )) L~\ From these, by 

 contraposition, and also by theorem K and its contrapositions, we obtain the following 

 results. 



L is contained in LL"", 1 1"', l"'!', Lf'L 



L-' L,L-S 11-", l7'l, L-'L' 



1 IL', L,l 



I-' L -'1-', l-'L-" 



LL is contained in L 



L-'L-' L-' 



L-'],1L-' 1 



LI-', l-'L 1-' 



I omit demonstration, but to prevent any doubt about correctness of printing I subjoin 

 instances in words: L signifies ancestor and L"' descendant. 



An ancestor is always an ancestor of all descendants, a non-ancestor of none but non- 

 descendants, a non-descendant of all non-ancestors, and a descendant of none but ancestors. 

 A descendant is always an ancestor of none but descendants, a non-ancestor of all non- 

 descendants, a non-descendant of none but non-ancestors, and a descendant of all ancestors. 

 A non-ancestor is always a non-ancestor of all ancestors, and an ancestor of none but non- 

 ancestors. A non-descendant is a descendant of none but non-descendants, and a non- 

 descendant of all descendants. Among non-ancestors are contained all descendants of non- 

 ancestors, and all non-ancestors of descendants. Among non-descendants are contained all 

 ancestors of non-descendants, and all non-descendants of ancestors. 



The mathematician forces the predicate itself among its own chain of successive relatives, 

 whether the relation be transitive or not: a;, as (p''ig, appears in the sequence... d)-'r, d>~^<if, 

 {p^x, (p^uo, (Jy'cV,,.. There is a little tendency towards the same thing in ordinary language, 

 especially when the relation is transitive. Milton, in calling Eve " the fairest of her 

 daughters," meaning female descendants in general, allowed (b"x to be a case of 0"a?. 

 Nothing but circumlocution avoids the same thing in our day, and by it language loses 

 much force, or some precision. If we say that Achilles was the strongest of all his 

 enemies, we feel both definite meaning and force : if we say that he was stronger than any one 

 ^ of his enemies, we gain an enfeebling addition of logical accuracy : if that he was stronger 

 than all his enemies, we introduce ambiguity. 



I now proceed to the syllogism, taking first the case in which the terms are individual 

 notions, units of thought. All syllogisms of second intention, whether mathematical or 

 metaphysical, come under this case; and arithmetical syllogisms are but aggregates of 

 singular syllogisms, each of which also comes under tliis case. 



The supreme law of syllogism of three terms, the law which governs every possible case, 

 and to which every variety of expression must be brought before inference can be made, is 

 this ; — any relation of X to Y compounded with any relation of Y to Z gives a relation 

 of X to Z. This is very nearly the wording of Euclid's implied definition of compound 

 ratio of magnitudes ; — The ratio of X to Z is compounded of the ratios of X to Y and 



