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AN INTRODUCTION TO CYBERNETICS 



The system that is specified only by states, as in the previous 

 section, contains no direct reference either to parts or to variables. 

 In such a case, "re-labelling" can mean only "re-labelling the 

 states". A system with parts or variables, however, can also be 

 re-labelled at its variables — by no means the same thing. Re- 

 labelling the variables, in effect, re-labels the states but in a way 

 subject to considerable constraint (S.7/8), whereas the re-labelhng 

 of states can be as arbitrary as we please. So a re-labelHng of the 

 states is more general than a re-labelling of the variables. 



Thus suppose a system has nine states; an arbitrary re-labeUing 

 of eight of the states does not restrict what label shall be given to 

 the ninth. Now suppose that the system has two variables, x and 

 y, and that each can take three values: Xj, X2, Xt, and >'i, y2, j'3. 

 Nine states are possible, of which two are (^2, J3) and (X2,yi). Suppose 

 this system is re-labelled in its variables, thus 



^ I V 



If now {x2,yi) is transformed to some state {a,^), and iXi,yi) is trans- 

 formed to (y,S), then, for consistency, the state (x2,yi) must transform 

 to («,S). (Draw the phase spaces and identify the values on the ^ 

 and 17 axes.) Thus the nine states now cannot be transformed 

 arbitrarily and independently. A re-labelling of the variables offers 

 less scope for change than a re-labelling of states. 



As a result, certain features that are destroyed by a re-labelling 

 of states are preserved by a re-labelling of variables. Among them 

 is the diagram of immediate effects. 



The system described by its states has, of course, no such diagram, 

 for it has in effect only one variable. A system with variables, 

 however, has a diagram of immediate effects. The phase-space 

 now has axes; and it is easily seen, after a few trials, that a one-one 

 transformation that re-labels the variables, changes the diagram of 

 immediate effects only to the extent of a "button and string" change; 

 turning, say, A into B: 



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