202 mk de morgan, on the general principles of which the 



which are aggregated should be of the same kind, nor that joint action should be simultaneous 

 action. 



The four postulates are as follows : — 



1. Any two tendencies have one aggregate (0, the aggregate of counteraction, being 

 included among possible cases), and one only. 



2. The magnitude of the aggregate, and its application relatively to the applications of 

 the aggregants, depend only on the relative, and not on the absolute, applications of the 

 aggregants. 



3. The order in which tendencies are aggregated, produces no effect either on the 

 magnitude or application of the aggregate. 



4. Tendencies of the same or opposite applications are aggregated by the law of 

 algebraical addition. 



Let a superfixed cross be the symbol of aggregation, the order of aggregation being the in- 

 verse of the order of writing. Thus A^B^C signifies the aggregation of A with the aggregate 

 of B and C, and is not distinguishable from A^{B^C). The symbol J expresses both mag- 

 nitude and application. The following propositions are now deducible from the postulates. 



5. In any aggregate the result of partial aggregation may take the place of its own 

 aggregants: thus A'^B'^C'D is {A'<D)''{C''B). For, by 3, A^B^'C'D is A^'D'^CB, or 

 A^'D^C^'B), or {C^'BYA'^D, or {C^'BYiA^'D), or {A^DY^C^B). 



6. Two tendencies cannot counteract each other, that is, aggregate into the tendency 0, 

 unless they have equal magnitudes and opposite applications. Let A and B be of equal 

 magnitudes and opposite applications ; and let M be any third tendency. If then A^M give 

 0, B^A^M gives B. But, by 4, B^'A gives 0, whence B^A^M or (5) M^i^B'^A) gives M. 

 That is, by 1, B and M are identical: whence no other than B, the equal and opposite of A, 

 gives or counteracts A. Hence it follows that —A and —B, the opposites of A and B, 

 have -{A^B) for their aggregate. For A, B, -A, -B, giving A^{-A) and B^{-B), or 

 and 0, by 4, counteract each other: that is, A^B and {^—A^, —B) counteract; whence 

 {-A^, -B) and -{A^B) are identical. 



7. An aggregate has not more than one pair of aggregants, when the applications of the 

 aggregants are given, and are different. Let A and B, of given different applications, aggre- 

 gate into C', and let P and Q, severally of the same or opposite applications with A and B, 

 have C also for their aggregate. Then -P and -Q give -C, by 6: whence A^B^(-P) 

 "(-Q) give C^(-C), or 0. That is, by 3 and 4, (^-P)x (S-Q) is 0; which, by 6, A-B 

 and P - Q being of different applications, cannot be, unless A-P and B-Q be severally 0. 



