AND MULTIPLICATION OF LOGICAL RELATIVES. 233 



The denial of a relative of the third class belongs 

 either to the first class [e. g. '^ unequal ^^ is denied by 

 " equal ^^) or to the third {e.g. " excludent ^' or "no A 

 is B " is denied by '^ participant '' or " some A is B ^^) . 



The denial of a relative of the fourth class belongs 

 either to the second class (e. g. " indifferent " is denied 

 by " enclosure," as above) or to the fourth class (e. g. 

 " teacher" is denied by " not teacher "). 



Of our sixteen terms, the old logic recognizes only four, 

 namely 



L or inclusion, iV" or exclusion, 



n or partial inclusion, / or partial exclusion ; 



and these are respectively equivalent to the well-known 

 forms of proposition 



A, E, 



I, O. 



All our syllogisms are in the fourth figure, having the 

 minor premise first, and the middle term second in the 

 minor and first in the major. 



Of course there is an endless number of relations 

 belonging to each of our four classes. Logicians, how- 

 ever, are right in treating inclusion and exclusion as 

 the fundamental relations of the science. Inclusion and 

 exclusion [L and A'') belong respectively to the second 

 and third classes of relatives ; and it is worth while to 

 remark that the two fundamental relations of geometry, 

 namely direction and distance, belong to the same. That 

 is to say, direction is transitive but not invertible. The 

 following syllogism is valid : — 



