New System of Logical Notation. 25 



that is to say " B is a pupil of none but A," or " of A only," 

 Let 5 mean child, then 



will mean " A is a teacher of all the children of B," and its 

 converse 



B = ^'-ix i-»^x A 



will mean " B is the parent of none but pupils of A." Thus 

 I means "all," or "only" with an adjective sense:— i~^ means 

 " none but," or " only " with an adverbial sense. 



The simplest forms of this kind occur when A and B are 

 individuals. When they are classes — if for instance the A's 

 are the teachers and the B's the pupils of a particular 

 school — the proposition 



A = i2xB 



asserts only that " some A's teach B's," and is a partial 

 proposition. In the present essay, nothing more is said on 

 the theory of partial propositions. The proposition 



iA = RxB 



is singly total ; it asserts that " all A's teach B's," or, what 

 is better English, "every A teaches a B or B's." The 

 proposition 



iA=/?x iB 



asserts that " every A teaches every B," and is doubly total. 

 A doubly total proposition is defined in the system here 

 expounded as one where the two terms A and B are both 

 quantified by the coefficient i or i~\ In a singly total 

 proposition only one of them is so quantified ; in a partial 

 proposition, neither. A doubly total proposition, however, 

 as De Morgan has remarked,* is one proposition, not the 

 resultant of two propositions. " Every A teaches every B," 



' " On Ihe syllogism, No. IV., and on the logic of relations." — From the 

 transactions of the Cambridge Philosophical Society, Vol. X. Part II, 



