n8 THE SCIENCE OF LOGIC. 



62. RULES OF LOGICAL DIVISION. I. Each act of division 

 must have only one basis. 



II. The sub-classes must be together co extensive with the divided 

 whole. 



III. In a continued division each step should divide a class or 

 sub-class into its proximate sub-classes. 



In other words, logical division must (i) not be cross-division, 

 but give results that are &quot;mutually exclusive&quot;; it must (ii) be 

 exhaustive, or give results &quot;collectively exhaustive&quot; of the de 

 notation of the divided whole ; and finally, if continuous, it must 

 (iii) be step by step : Divisio non faciat saltum. 



These rules are variously stated. Sometimes superfluous rules are given, 

 as e.g. the rule that only class terms can enter into a logical division. This 

 is involved in the very definition of division. Sometimes, also, a rule is laid 

 down to the effect that none of the dividing members must be equal in 

 extent to the divided whole .* This rule must be observed in real or material 

 division, or, as it is commonly called, classification ; for here the sub-classes 

 are supposed to be groups of really existing things, and if one sub-class be 

 coextensive with the whole, the others are non-existent, and there is no real 

 division at all ; but the rule cannot be insisted on in purely formal or dichoto- 

 mous division, in which the sub-classes are not necessarily supposed to exist, 

 but are regarded as mere hypothetical possibilities of existence class com 

 partments which may be full or empty without detriment to the formal accur 

 acy of the process. 



RULE I. A cross-division is one in which some member or 

 members fall into more than one sub-class, so that these are not 

 mutually exclusive but overlap. This cannot happen unless, in 

 the act of division, we fail to adhere to one and the same basis of 

 division. There is no danger of this when the division is dichoto- 

 mous ; and very little danger when the genus is seen to be such 

 that it must yield, on a given basis, a small number of alterna 

 tive groups, as in the examples given above (58) from mathe 

 matics and geometry. But when the immediate result of applying 

 a given ground of division would be to divide the genus into a 

 large number of co-ordinate sub-classes, there is a danger that 

 before all are set down we may inadvertently modify the ground 

 of our division, partially or totally. If we do, the result may be 

 (a) to include some individuals twice, or oftener, or (b) to leave 

 out some individuals altogether, or (c) to commit both faults ; or, 

 finally, we may (d) accidentally escape both faults and reach an 

 accurate result. For example, we commit both, if, intending to 



1 CLARKE, Logic, p. 236. Cf. KEYNES, Formal Logic (4th edit.), r p. 445. 



