190 THE PRINCIPLES OF SCIENCE. 



number we are only at the first step of an extensive 

 series of generalizations. A number is general as compared 

 with the particular things numbered, so we may have 

 general symbols for numbers, or general symbols not for 

 numbers, but for the relations between undetermined num- 

 bers. There is, in fact, an unlimited hierarchy of successive 

 generalizations. 



Numerically Definite Reasoning. 



It was first discovered by Prof, de Morgan that many 

 arguments are valid which combine logical and numerical 

 reasoning, although they could in no way be included in 

 the ancient logical formulas. He developed the doctrine 

 of the 'Numerically Definite Syllogism,' fully explained 

 in his 'Formal Logic' (pp. 141-170). Dr. Boole also 

 devoted considerable attention to the determination of 

 what he called ' Statistical Conditions/ meaning the 

 numerical conditions of logical classes. In a paper pub- 

 lished among the Memoirs of the Manchester Literary and 

 Philosophical Society, Third Series, vol. IV. p. 330 

 (Session 1869-70), I have pointed out that we can apply 

 arithmetical calculation to the Logical Abecedarium. 

 Having given certain logical conditions and the numbers of 

 objects in certain classes, we can either determine the 

 number of objects in other classes governed by those con- 

 ditions, or can show what further data are required to 

 determine them. As an example of the kind of questions 

 treated in numerical logic, and the mode of treatment, I 

 give the following problem suggested by De Morgan, with 

 my mode of representing its solution f . 



f It has been pointed out to me by Mr. A. J. Ellis, F.R.S., that my 

 solution, as given in the Memoirs of the Manchester Philosophical Society, 

 does not exactly answer to the conditions of the problem, and I therefore 

 substitute above a more satisfactory solution. 



