CONDORCET. 357 



and 



jiijn I jni^jn ^-^-^ ^.tnjn > jn^jn 



See his page 10. 



668. The student of Condorcet's work must carefully dis- 

 tinguish between the probability of the correctness of a decision 

 that has been given when we know the numbers for and against, 

 and the probability when we do not know these numbers. Con- 

 dorcet sometimes leaves it to be gathered from the context which 

 he is considering. For example, in his Preliminary Discourse 

 page XXIII. he begins his account of his first Hypothesis thus : 



Je considere d'abord le cas le plus simple, celui ou le nombre des 

 Votans etant impair, on prononce simplemeBt a la plurality. 



Dans ce cas, la probabilite de ne pas avoir uue decision fausse, celle 

 d'avoir une decision vraie, celle que la decision rendue est conforme a la 

 verite, sont les memes, puisqu'il ne peut y avoir de cas oii il n'y ait 

 pas de decision. 



Here, although Condorcet does not say so, the words celle que 

 la decision rendue est conforme a la verite mean that we know 

 the decision has been given, but we do not know the numbers 

 for and against. For, as we have just seen, in the Essay Con- 

 dorcet takes the case in which we do know the numbers for and 

 against, and then the probability is not the same as that of the 

 correctness of a decision not yet given. Thus, in short, in the 

 Preliminary Discourse Condorcet does not say which case he takes, 

 and he really takes the case which he does not consider in the 

 Essay, excluding the case which he does consider in the Essay; 

 that is, he takes the case which he might most naturally have 

 been supposed not to have taken. 



669. We will now proceed to Condorcet's second H3rpothesis 

 out of his eleven ; see his page 14. 



Suppose, as before, that there are 2q -\-l voters, and that a 

 certain plurality of votes is required in order that the decision 

 should be valid ; let 2q + 1 denote this plurality. 



Let </) (q) denote the terms obtained from the expansion of 

 (v + ey'"-', from v'^"-' to the term which involves t*^'-^^^ e^'^, both 

 inclusive. Let yfr (^) be formed from </> (q) by interchanging e 

 and V. 



