ILLUSTRATIONS OF THE LOGIC OF SCIENCE. 7 o 7 



we have to believe that it has taken place, which is conceptualistic ; 

 but shortly after they state that it is the ratio of the number of cases 

 favorable to the event to the total number of cases favorable or 

 contrary, and all equally possible. Except that this introduces the 

 thoroughly unclear idea of cases equally possible in place of cases 

 equally frequent, this is a tolerable statement of the materialistic 

 view. The pure conceptualistic theory has been best expounded by 

 Mr. De Morgan in his " Formal Logic : or, the Calculus of Infer- 

 ence, Necessary and Probable." 



The great difference between the two analyses is, that the con- 

 ceptualists refer probability to an event, while the materialists make 

 it the ratio of frequency of events of a species to those of a genus 

 over that species, thus giving it two terms instead of one. The oppo- 

 sition may be made to appear as follows : 



Suppose that we have two rules of inference, such that, of all the 

 questions to the solution of which both can be applied, the first yields 

 correct answers to ^ w , and incorrect answers to the remaining T ^ ; 

 while the second yields correct answers to -j 8 ^, and incorrect answers 

 to the remaining -j-J^-. Suppose, further, that the two rules are en- 

 tirely independent as to their truth, so that the second answers cor- 

 rectly T 9 ^ of the questions which the first answers correctly, and also 

 T 9 ^o of the questions which the first answers incorrectly, and answers 

 incorrectly the remaining yL- of the questions which the first answers 

 correctly, and also the remaining yig- of the questions which the first 

 answers incorrectly. Then, of all the questions to the solution of 

 which both rules can be applied 



both answer correctly of or 93 X 81 



100 100' 100 X 100 ' 



the second answers correctly and the first incorrectly of-^- or 93 ^ * 9 



100 100' 100X100' 



the second answers incorrectly aud the first correctly ' * 81 7 X 81 



100 100' 100 X 100 ' 



and both answer correctly ' of-^- r>r "* ^ 



100 100' 100X100 ' 



Suppose, now, that, in reference to any question, both give the 

 same answer. Then (the questions being always such as are to be an- 

 swered by yes or no), those in reference to which their answers agree 

 are the same as those which both answer correctly together with those 



93 X 81 7 X 19 



which both answer falsely, or 4- - 1- of all. The pro- 



" 100 X 100 100 X 100 F 



portion of those which both answer correctly out of those their an- 

 swers to which agree is, therefore 

 93 X 81 



100 X 1 00 93 X 81 



93 X 81 7 X 19 r (93 X 81) + (7 X 19). 

 100 X 100 + 100 X 100 



