INDUCTIVE LOGICAL PROBLEM. 123 



The first of these pairs gives the combinations in the ioth 

 column ; but it may readily be shown that the two pro- 

 positions all A's are B's, and all not A's are not B's, are 

 but equivalent to the single proposition A=B, or all A's 

 are all B's. In the same way the second pair of pro- 

 positions gives the combinations of the 7th column, and 

 are equivalent to the single proposition A =-b, or all A's 

 are all not B's. There remains but a single case, that in 

 the 1 6th column; but as all the combinations are present, 

 there can be no condition except the laws of thought. 



We have now effected a complete solution of the inverse 

 logical problem of two terms; we have found that two 

 terms can manifest themselves only in seven series of com- 

 binations, and the corresponding laws are as below : — 



Series of combinations. Laws. Equivalent laws. 



12th A=AB b — ab 



8th A=Ab B = «B 



15th a =aB b = Ab 



14th a —ab B = AB 



A=AB 



ioth A=B a =b , 



a —ab 



7th A = b a=B\ 



[a = a& 



16th No law 



We also learn from the above investigation that there is 

 no possible logical relation between two terms which may 

 not be expressed in a proposition either of the general 

 type A=AB or of A = B. For the logical relation must 

 manifest itself in some series of combinations of natural 

 qualities ; but every series of combinations which is pos- 

 sible, according to the very laws of thought, has been 

 included in our investigation. Thus every such logical 

 relation must either be expressed in one of the six laws, 

 or must be equivalent to one of them. The general result 



