34^ 



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



If a compound relation be contained in another relation, by the nature of the relations 

 and not by casualty of the predicate, the same may be said when either compionent is con- 

 verted, and the contrary of the other component and of the compound change places. That 

 is if, be Z what it may, every L of M of Z be an N of Z, say LM))N, then L~'n))m, 

 and nM-'))l. If LM))N, then n))lM' and nM-> )) IM'M"'. But an 1 of every M of 

 an M"' of Z must be aa 1 of Z : hence nM"' )) 1. Again, if LM)) N, then n )) L m, whence 

 L~'n ))L"'L^m. But an L~' of an L of none but ms of Z must be an m of Z ; whence 

 L-'n))m. 



I shall call this result theorem K, in remembrance of the office of that letter in Baroko 

 and Bokardo: it is the theorem on which the formation of what I called opponent syllogisms 

 is founded. For example, the combination in one of the mathematical* syllogisms is Every 

 deficient of an external is a coinadequate: external and coinadequate have partient and com- 

 plement for their contraries, and deficient has extent for its converse : hence every exient of 

 a complement is a partient ; which is one of the opponent syllogisms of that first given. 



Identity, in theorem K, does not give identity; as will be observed by watching the 

 demonstration. For an instance, brother of parent is identical with uncle, by mere defi- 

 nition. But non-uncle of child is not identical with non-brother: for though every non- 

 uncle of child is non-brother (as by the theorem), yet it is not true that every non-brother is 

 non-uncle of child. 



If LM be identical with N, meaning that N is an L of an M and of no other signifi- 

 cation, we have LM |i N, LMM-MINM-S L-'LMl|L-'N. Now MM-'X and L-'LX are 

 classes which contain X; so that we may affirm L )) NM"' and M))L~iN; but not 

 L!|NM-i nor M||L-'N. 



; :.! Having given LM || N, it is natural to ask whether we can deduce identical expressions 

 for L and M : the answer is that no such thing can be done. If by M we mean some one 

 particular M left vague, the form of L can be deduced, as we shall see ; but not when N is 

 a name for, and only for, every L of every M. Take, for example, the word uncle: it is 

 identical with brother of parent; either is the other. Can we now construct a definition of 

 brother out of uncle, parent, and their converses. Uncle of child of X is no definition of 

 brother of X; it includes the brothers of the other parent. Uncle of every child of X 

 will not do, for a similar reason : if X had as many wives as Solomon, and children by all, 

 nothing in logic excludes the supposition that they were all sisters. 



In mathematics we have much power of forcing NM~M|L out of LM 1| N by extension 

 of language: and in a science of truths necessary as to matter it is almost a proof of insuf- 

 ficient grasp when we find either of the forms above unaccompanied by the other. Common 



gives (. .( or (( : a complement of every complement is a 

 genus ; and vice versa. Again, ,(.) (( gives ).( : a complement 

 of none but genera is an external, &c. Also, ,(( ) ( gives ) ( : a 

 genus of none but coinadequates is a coinadequate, &c. A 

 defective account of these transformations will be found in my 

 third paper. 



* In arithmetical form thus : — Some Ys are not any Xs, 

 no Y is Z, therefore some things are neither Xs nor Zs. Deny 

 the conclusion, affirm the first premise, and we may deny the 

 second, which gives some Ys are not any Xs, everything is 

 either X or Z, therefore some Ys are Zs. 



