7 o8 THE POPULAR SCIENCE MONTHLY. 



This is, therefore, the probability that, if both modes of inference 

 yield the same result, that result is correct. We may here conven- 

 iently make use of another mode of expression. Probability is the 

 ratio of the favorable cases to all the cases. Instead of expressing 

 our result in terms of this ratio, we may make use of another the 

 ratio of favorable to unfavorable cases. This last ratio may be called 

 the chance of an event. Then the chance of a true answer by the 

 first mode of inference is f ^ and by the second is -^ ; and the chance 

 of a correct answer from both, when they agree, is 



81 X 93 81 v , 93 



or X 



19 X 7 19 7, 



or the product of the chances of each singly yielding a true an- 

 swer. 



It will be seen that a chance is a quantity which may have any 

 magnitude, however great. An event in whose favor there is an even 

 chance, or \, has a probability of ^. An argument having an even 

 chance can do nothing toward reenforcing others, since according to 

 the rule its combination with another would only multiply the chance 

 of the latter by 1. 



Probability and chance undoubtedly belong primarily to conse- 

 sequences, and are relative to premises ; but we may, nevertheless, 

 speak of the chance of an event absolutely, meaning by that the 

 chance of the combination of all arguments in reference to it which 

 exist for us in the given state of our knowledge. Taken in this sense 

 it is incontestable that the chance of an event has an intimate 

 connection with the degree of our belief in it. Belief is certainly 

 something more than a mere feeling; yet there is a feeling of believ- 

 ing, and this feeling does and ought to vary with the chance of the 

 thing believed, as deduced from all the arguments. Any quantity 

 which varies with the chance might, therefore, it would seem, serve 

 as a thermometer for the proper intensity of belief. Among all such 

 quantities there is one which is peculiarly appropriate. When there 

 is a very great chance, the feeling of belief ought to be very intense. 

 Absolute certainty, or an infinite chance, can never be attained by 

 mortals, and this may be represented appropriately by an infinite be- 

 lief. As the chance diminishes the feeling of believing should dimin- 

 ish, until an even chance is reached, where it should completely vanish 

 and not incline either toward or away from the proposition. When 

 the chance becomes less, then a contrary belief should spring up 

 and should increase in intensity as the chance diminishes, and as 

 the chance almost vanishes (which it can never quite do) the contrary 

 belief should tend toward an infinite intensity. Now, there is one 

 quantity which, more simply than any other, fulfills these conditions ; 

 it is the logarithm of the chance. But there is another considera- 

 tion which must, if admitted, fix us to this choice for our thermometer. 

 It is that our belief ought to be proportional to the weight of evi- 



