122 PROF. W. STANLEY JEVONS ON THE 



can vary this form of proposition either by interchanging 

 A and B or substituting for one or both of them its nega- 

 tive. We should arrive altogether at eight different ex- 

 pressions, which are thus stated : — 



12. 



8. 



*5- 



14. 



A=AB, 



A=A6, 



a = aB, 



a =ab, 



b —ab, 



B = «B, 



b = kb, 



B = AB 



If we test the effect of each of these conditions by ascer- 

 taining the combinations which it negatives, it will be 

 found that each proposition in the first line gives the same 

 result as the one immediately below it. Each pair con- 

 sists then of logical equivalents. And in fact the lower 

 line contains what are called by logicians the contra- 

 positives of those in the higher line. Thus the first pair 



mean 



All A's are B's, 



All not B's are not A's, 



which have exactly the same logical force. The last pair 



mean 



All not A's are not B's, 



All B's are A's, 



which are again exactly equivalent to each other. Al- 

 though then there may be eight propositions of the form 

 A = AB, only four of these have independent logical mean- 

 ings. On trial we find that these four propositions give 

 the combinations respectively shown in the 12th, 8th, 

 15th, and 14th columns, which cases are thus referred to 

 their proper law. 



If we now join these latter four propositions two and two, 

 they will generally be found to contradict each other ; thus 

 all A's are B's contradicts all A's are not B's. There are 

 only two pairs which give consistent results, namely : — 



A = AB1 A = Ab) 



a =ab ) a =«B I 



