122 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



The objective impossibility has disappeared before thought and experience. All who either 

 think or try are now satisfied that repetition of trials will produce any event, however rare. 

 Buffon, in 2048 trials of the Petersburg problem, found only one case in which the first head 

 was deferred as late as the 9th throw. In the other instance of as many trials, mentioned in 

 my Formal Logic, p. 185, head was once deferred till the 16th throw. A third trial of 2048 

 sets was made by a gentleman in the country, who communicated the results to me : he also 

 had one case in which head was deferred to the 16th throw. Both these cases are extraordinary : 

 for, a priori, more than 22,000 trials must be undertaken, to have an even chance of seeing 

 head deferred beyound the fifteenth throw. 



It is sometimes replied that any given order of head and tail in 100 throws is just as unlikely 

 as a hundred consecutive heads : and is therefore objectively as impossible. But this reply 

 does not attend to the circumstance that the fulfilment of precedent conditions is the extra- 

 ordinary event counted upon as impossible. 



And it is to be noted that, in looking out for the way of judging the probability of what 

 may take place in 100 throws, the primary distribution of our minds does not take in 2 100 

 cases, on the one hand ; neither do we divide into the case of 100 heads — and others. We group 

 the cases in thought into something between these two extremes : in fact, we imagine collections 

 of the same degree of remarkability. Different minds will do this in different ways : if we 

 could stamp the weights in some one mind, we might find perhaps only 2 5 groups, many of 

 them involving each more than 2 95 cases. We should thus have ordinary events, each of a 

 probability of 2 ~ 5 nearly ; and cases more or less extraordinary, varying in probability down 

 to 2" 100 . 



Those who have not attended to arithmetic would make the primary distribution in a 

 manner essentially wrong. Nothing would be called ordinary which very much differs from 

 alternate head and tail : so that they would be surprised at almost any thing that might 

 happen. 



This method of distribution into groups, the members of each group being of equal 

 notability, is that which prevails in our methods of judging : and it serves to explain the 

 conclusions which we actually, though unconsciously, draw on the subject of evidence. 



The preceding remarks may now be applied to the case of an authority asserting a universal 

 proposition. 



If we suppose n - 1 to be the number of Xs in existence, and if we admit into our 

 thoughts all the varieties of the numerically definite proposition, from • Every X is Y' through 

 ' Every X but one is V &c. down to 'no AT is Y : ' then, if all these n cases be of equal 

 probability a priori, that is, before any consideration whatever of the connexion of the terms, 

 it follows that the credit of the witness is neither raised nor lowered by the assertion of any 

 one of them. And the same, if we make groups of equal notability, m in number, provided 

 that the probability of the universal a priori, is looked upon as being exactly or nearly l -r m. 

 The two universals would each form a group, the idea of necessity, as distinguished from that 

 of contingency, not only securing them this character, but perhaps giving them rather a 

 higher share of probability than deliberate consideration would approve of. I do not profess 



