11/9 AN INTRODUCTION TO CYBERNETICS 



But always, by algebraic necessity, 



H{R,E) < H{R) + H{E) 

 so H{D) + Hj,{R) < H{R) + H{E) 



i.e. H{E) > H{D) + Hj,{R) - H{R). 



Thus the entropy of the £"s has a certain minimum. If this minimum 

 is to be affected by a relation between the D- and i?-sources, it can 

 be made least when H^iR) = 0, i.e. when R is a determinate function 

 of D. When this is so, then //(£)'s minimum is H(D) - H{R), a 

 deduction similar to that of the previous section. It says simply 

 that the minimal value of £"s entropy can be forced down below 

 that of D only by an equal increase in that of R. 



11/9. The theorems just established can easily be modified to give 

 a worth-while extension. 



Consider the case when, even when R does nothing (i.e. produces 

 the same move whatever D does) the variety of outcome is less than 

 that of D. This is the case in Table 11/4/1. Thus if R gives the 

 reply a to all D's moves, then the outcomes are a, b or J— a variety 

 of three, less than Z)'s variety of five. To get a manageable calcula- 

 tion, suppose that within each column each element is now repeated 

 k times (instead of the "once only" of S. 11/5). The same argument 

 as before, modified in that kn rows may provide only one outcome, 

 leads to the theorem that 



Fo>K^-log/v'- V,,, 



in which the varieties are measured logarithmically. 



An exactly similar modification may be made to the theorem in 

 terms of entropies, by supposing, not as in S.l 1/8 that 



Hr(E) > Hr{D), but that 

 HAE) > H^{D) - K. 



H{Eys minimum then becomes 



HiD) - K- H(R), 

 with a similar interpretation. 



11/10. The law states that certain events are impossible. It is 

 important that we should be clear as to the origin of the impossibility. 

 Thus, what has the statement to fear from experiment? 



It has nothing to do with the properties of matter. So if the law 

 is stated in the form "No machine can . . .", it is not to be overthrown 



208 



