102 



PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



Hence the symbolic canon of inference is the same as in the system of contraries, namely, the 

 erasure of the middle parentheses gives the symbolic form of the conclusion. 



But in the cumular system, this canon of inference is modified. In the exemplar system, 

 the proposition X)(Y or 'Any one X is any one Y,' though it appear to be of the widest 

 character, is restricted by its double indefiniteness : doubly indefinite produces doubly singular. 

 This will explain the mode in which conclusions are admissible in the exemplar form which are 

 not admissible in the cumular. For example, in the former, )) )( gives )( or ' Any one X is 

 some one Y, and any one Y is any one Z,' gives ' Any one X is any one Z.' Here, if all we 

 knew were that Y and Z are identical, all we could infer would be, ' Any one X is some one Z;" 

 but because we know from Y) {Z that there is but one Z, we may say that any one X is any 

 one Z. In the cumular system, then, )) )( gives )) : and the modification of the canon of 

 inference which the cumular system requires is ; — Erase the middle parentheses, but when they 

 both turn one way, any parenthesis of indefiniteness which turns the other way must itself be 

 turned, unless it be protected by a negative point. Thus ( ))( gives ( ), but ))).( gives ).( . 



To collect the syllogisms, we may observe that there are two doubly indefinite forms A[ and 

 E', each of which has three less indefinite varieties; the first A /t A', and 7 ' ; the second, 

 0, 0', and O'. A syllogism then in which both premises are double, admits, in original 

 and weakened forms, of 16 varieties; and there are three such double syllogisms which are 

 valid. Classifying the syllogisms by the degrees of relaxation in their forms, we have the 

 following list : 



a;a;a; )()(=)( 



a;e;e; )().(=>.( 



e;a;e; ).()(=>.( 



a,a;a; )))(=)( 



a'a;a' (()(=(( 



a/a a, )())=)) 



a;a'a; )(((=)( 



a e/e; »).(=).( 



a'e;o, (()•(=(.( 



A/O'O' )().)=).) 



a: ox )((.(=).( 



oa;e; ).))(=)•( 



oa;o, (•()(=(.( 



e;ao' ).())=).) 



e;a'e; ).(((=).( 



The cases in which the cumular system requires the above-mentioned modification of the 

 law of inference have their capital letters in Italics. The three fundamental syllogisms give, 

 each of them, four cases with one relaxation, five with two, and two with three. Many different 

 rules of formation might be produced, and many analogies, but it is not necessai^y for my 

 present purpose to examine them. 



