OPINION AND PROBABILITY 283 



general set of antecedents, whatever it is, may be gradually chang 

 ing in character, without our knowledge (265). Especially in 

 regard to social or partly social phenomena the frequency of 

 various crimes ; the rate of births, marriages, and deaths ; politi 

 cal or economic crises, etc. our uncertainty as to the permanence 

 of their causes and conditions must always render our probable 

 estimates about the future very precarious. Yet this seems to be 

 the only direction in which we can reasonably utilize the calculus : 

 that is, for the purpose of discovering whether or not certain 

 given conditions have only an &quot; indifferent &quot; or &quot; casual &quot; con 

 nexion with an event, or whether, on the contrary, we can elimi 

 nate chance by detecting a &quot; bias,&quot; or, in other words, a causal 

 connexion, between the event and some of those conditions. 



Curious attempts have been made, however, by various logicians, to de 

 termine mathematically our probability as to the future recurrence of an event, 

 by an appeal to the number of times we have observed the event to have 

 occurred in the past. Thus, according to Professor Welton, following Lotze, 1 

 the occurrence of an event once may be taken as one reason for expecting its 

 recurrence. Apart from that, the chances for and against its recurrence would 

 be equal. There are, therefore, two reasons altogether for expecting its recur 

 rence and one against expecting it ; so that the probability of its recurring is 

 . Generalizing from this, we see that if an event has occurred m times, the 

 two possibilities of its recurring or not, once again, make the total number of 

 alternatives m + 2, of which m + i are favourable. The probability of re 

 currence is therefore Evidently, then, uncontradicted experience 



m + 2 



should give rise rapidly to a very high probability, which would continue to 

 grow with continued experience. That the sun has risen daily for 5000 years 

 would make the probability VHHli that it will rise to-morrow. &quot; It will be 

 seen,&quot; he continues, &quot; that this calculation of probability is the true basis of 

 induction by simple enumeration. The formula shows that with wide and un 

 contradicted experience the probability that an empirical law which summarizes 

 that experience will hold good in one more case is very high. But it also 

 shows that extension of it beyond the realm of actual experience becomes 

 increasingly uncertain with increase in the width of that extension. For, if the 

 formula is written to show the probability that an event which has occurred m 



times will happen n times more, it becomes m + for m + 2 in the 



(m + n + i) 



original formula = m + i + i, where i is the number of new cases, i.e. n 

 which obviously decreases in value as n is increased. Again, another modifi 

 cation of it shows how the actual experience of the failure of the event, weakens 

 the probability of its recurrence. For if an event has occurred m times and 

 failed to occur times under circumstances where it might have been expected 

 to happen, then there are already m + n cases ; the possibilities that it may or 



1 LOTZE, Logic, 282 (5) apud WELTON, op. cit., ii., pp. 180-82. Cf. VENN, 

 Logic of Chance, pp. 196 sqq. ; 358 sqq. 



