608 THE POPULAR SCIENCE MONTHLY. 



paper. For we found that the distinction of reality and fiction de- 

 pends on the supposition that sufficient investigation would cause one 

 opinion to he universally received and all others to he rejected. That 

 presupposition involved in the very conceptions of reality and figment 

 involves a complete sundering of the two. It is the heaven-and-hell 

 idea in the domain of thought. But, in the long run, there is a real 

 fact which corresponds to the idea of prohahility, and it is that a given 

 mode of inference sometimes proves successful and sometimes not, and 

 that in a ratio ultimately fixed. As we go on drawing inference after 

 inference of the given kind, during the first ten or hundred cases the 

 ratio of successes may he expected to show considerahle fluctuations; 

 hut when we come into the thousands and millions, these fluctuations 

 hecome less and less; and if we continue long enough, the ratio will 

 approximate toward a fixed limit. We may therefore define the proh- 

 ahility of a mode of argument as the proportion of cases in which it 

 carries truth with it. 



The inference from the premise, A, to the conclusion, B, depends, 

 as we have seen, on the guiding principle, that if a fact of the class A 

 is true, a fact of the class B is true. The prohahility consists of the 

 fraction whose numerator is the number of times in which both A and 

 B are true, and whose denominator is the total number of times in 

 which A is true, whether B is so or not. Instead of speaking of this 

 as the probability of the inference, there is not the slighest objection 

 to calling it the probability that, if A happens, B happens. But to 

 speak of the probability of the event B, without naming the condition, 

 really has no meaning at all. It is true that when it is perfectly ob- 

 vious what condition is meant, the ellipsis may be permitted. But 

 we should avoid contracting the habit of using language in this way 

 (universal as the habit is), because it gives rise to a vague way of 

 thinking, as if the action of causation might either determine an event 

 to happen or determine it not to happen, or leave it more or less free 

 to happen or not, so as to give rise to an inherent chance in regard to 

 its occurrence. It is quite clear to me that some of the worst and 

 most persistent errors in the use of the doctrine of chances have arisen 

 from this vicious mode of expression. 1 



IV. 



But there remains an important point to he cleared up. Accord- 

 ing to what has been said, the idea of probability essentially belongs 

 to a kind of inference which is repeated indefinitely. An individ- 

 ual inference must be either true or false, and can show no effect of 

 probability; and, therefore, in reference to a single .case considered in 



1 The conception of probability here set forth is substantially that first developed by 

 Mr. Venn, in his " Logic of Chance." Of course, a vague apprehension of the idea had 

 always existed, but the problem was to make it perfectly clear, and to him belongs the 

 credit of first doing this. 



