n6 THE SCIENCE OF LOGIC. 



It is purely formal whenever the grounds of it are purely imagin 

 ary and give rise to merely hypothetical sub-classes ; it is material 

 in so far as it deals with a real class and has for its grounds certain 

 attributes found in some members of that class and not in others. 

 Porphyry s Tree (46) furnishes a good illustration of a number 

 of successive dichotomous divisions (of the successive positive 

 class concepts), all of which are material or real, and not merely 

 formal or hypothetical. 



Although dichotomy, based as it is on the principles of con 

 tradiction and excluded middle, necessarily secures sub-classes 

 which are mutually exclusive, and collectively exhaustive of the 

 class divided, it is, nevertheless, useless as a means of dividing 

 real genera into real sub-classes, since it does not guarantee the 

 existence of all its sub-classes. It does not represent as co-ordinate, 

 classes which really are co-ordinate. It fails to exhibit its sub 

 classes as so many different positive characterizations, or modes 

 of realization, of the genus divided ; for half of its sub-classes are 

 wanting in any positive specifying character, being expressed by 

 purely negative terms ; and, besides, at each step it takes as a 

 distinct fundamentum divisionis what is really only a part of one 

 single fundamentum divisionis, thus reaching by a cumbrous and 

 roundabout method a number of infimae species which could have 

 been reached equally well by one single step. If a genus is 

 seen to fall naturally into three, or four, or fourteen, co-ordin 

 ate sub-classes, on a certain basis, it should not be divided first 

 into two, and so on, dichotomously, until all the sub-classes are 

 reached. It is more natural to divide triangles into equilateral, 

 isosceles, and scalene, than into equilateral and non-equilateral, the 

 latter into isosceles and non-isosceles, and the latter again into 

 scalene and non-scalene (if any). Moreover, if we are not certain 

 whether we are including all the sub-classes into which the genus 

 falls, whether we are perhaps omitting some real sub-class and 

 this is often the case dichotomy will not help us by its ultimate 

 hypothetical negative class. Without the aid of dichotomy we 

 can, in such doubtful cases, add to our infimae species a hypotheti 

 cal class of &quot;others, if any&quot;. 



The only fruitful application of dichotomous division to real 

 classes of things is that referred to in the last chapter (49) its 

 application for the purpose of discovering and formulating the 

 definition of some class. 



It can be applied, however, in a mechanical, mathematical way, 



