344 PROF. W. STANLEY JEVONS ON 



he puts the syllogism in the following form : — " If the 

 fractions a and (3 of the Y^s be severally A^s and B^s, and 

 if a + /3 be greater than unity^ it follows that some A^s are 

 B^s. . . . The logician demands a= i or /S= i, or both; he 

 can then infer /^ These arguments are readily represented 

 in my notation as follows : — 



The premises are a . (Y) = (AY), 



0.(Y) = (BY). 

 Hence 



(a + /5) (Y) = (AY) + (BY) 



= (ABY) + {KbY) + (ABY) + (aBY), 



(a + /3)(Y)-(Y) = (ABY)-(a^Y), 

 or 



(ABY) = (a + ^-i) (Y) + («6Y). 



From this we learn that the number of A's which are B's, 

 because they are Y^s, is the fraction (a + y8— i) of the Y^s 

 together with the undetermined number {abY), which 

 cannot be negative. Hence if a4-/3>i, the second side 

 has a positive value, and there must be some A^s which 

 are B'^s. If a= i, then this number is /3 . (Y), or if /3= i, 

 it is a.{Y), since (abY) then =o. If a=i and /3=i, 

 then obviously (ABY) = (Y). 



19. In Mr. MilFs chapter '' On Chance and its Elimina- 

 tions ^^"^^ occurs a problem concerning the coexistence of 

 two phenomena, in which he asserts the general proposi- 

 tion " that, if A occurs in a larger proportion of the cases 

 where B is than of the cases where B is not, then will B 

 also occur in a larger proportion of the cases where A is 

 than of the cases where A is not.'^ 



This proposition is not proved by Mr. Mill, nor do I 

 remember seeing any proof of it ; and it is not, to my 

 mind, self-evident. The following, however, is a proof of 

 its truth, and is the shortest proof I have been able to 

 tind. 



* System of Logic, 5th ed. vol. ii. p. 54. 



