AND MULTIPLICATION OF LOGICAL RELATIVES. 



209 



true in arithmetic for all numerical values of L. In the 

 eight remaining cases the results depend on the properties 

 of the relative L and the constitution of the universe. 

 When our logical universe is the actual universe and of 

 indefinite extent^ and when the relative is neither transitive 

 nor invertible — that is to say, when neither of the equations 



L*=XandZ-'=L 



is true — the eight syllogisms which here have zero pro- 

 ducts become inconclusive ; that is to say, none of the 

 products are of the same form with any of the factors. In 

 arithmetic there is nothing analogous to an inconclusive 

 syllogism, because every number is a factor of every other 

 number. 



Let us now suppose the universe to be of indefinite 

 extent, and the relative L, and of course also its reciprocal 

 L~^, to be transitive, the table will then be as follows. It 

 ■will be seen that two of the places for products are blank; 

 this indicates that the syllogisms are inconclusive. 





L 



L° 



(L-r 



L-' 



L 



' L 



2 



L° 



3 L 



4 io 



U^ 



5 L 



6 



i° 



7 



' Zo 



(i-)° 



9 (Z-')° 



lO 



" (L-r 



'^ i- 



i- 



'3 (Z-«)° 



H 



L-' 



'5 (Z-)° 



^' i- 



These syllogisms are true if we assign any transitive 

 meaning whatever to L, such as greater, smaller, above, 

 below, before, or after. Let us, for instance, convene that 

 the cause of a cause is a cause ; and let L mean cause, 



SER. III. VOL. VII. p 



