720 Miss Wrinch and Dr. H. Jeffreys on some 



1. To each combination o£ proposition and data corresponds 



one and only one number. 



2. If in one combination the proposition is more probable 



relative to the data than in another, the number 

 corresponding to the first is greater than that corre- 

 sponding to the second. 



3. If two propositions referred to the same data are 



mutually exclusive, the number corresponding to the 

 proposition that one of them is true is the sum of those 

 corresponding to the two original propositions. 



4. The greatest and least numbers correspond to those 



combinations and only those in which the data imply 

 that the proposition is true or untrue respectively. 



Several writers have defined probability as " quantity of 

 belief/' or somewhat better, " quantity of knowledge " ; these 

 are somewhat vague terms, and the transition from these 

 expressions to the number series has usually been carried out 

 without any explanation. Yet the use of numbers for the 

 comparison of probabilities at all was perhaps the greatest 

 advance ever made in the theory. 



The above assumptions are independent ; they are involved 

 implicitly in every theory of probability yet introduced ; and 

 with their aid it is possible to make some progress with a 

 logical theory. In the first place, we can show that the 

 number corresponding to a proposition incompatible with the 

 data is zero. For let a datum be that #=1 ■ then on this- 

 datum the propositions a; = 2 and a? = 3 are both false, and each 

 corresponds to the number a, where a is the least possible 

 number of those involved in the correspondence. Further, 

 x = 2 is incompatible with x = 3; hence by axiom 3 the. 

 probability that one of them is true is 2 a. But the proposi- 

 tion " # = 2 or .? = 3 " is incompatible with the datum " #=l/ r 

 and therefore corresponds to a. Thus 2 a is equal to a, and 

 a is therefore zero. If, then, probabilities are to be repre- 

 sented by numbers, zero must be the least number involved ; 

 but adjustments could be made in our assumptions which 

 would allow any other number to represent the minimum on 

 the scale. 



If we divide all numbers of the series by that corresponding 

 to a proposition implied by the data, all the above axioms will 

 apply equally to the numbers of the new series. We shall 

 henceforth use the notation P (p:q) to denote the number of 

 this series corresponding to the proposition p on the data q ; 

 we have ^(qiq) — 1, and ~P(not-q:q) = 0. P (p : q) may b& 

 read " the probability of p given q." 



Consider two propositions p and q which are not mutually 



