OPINION AND PROBABILITY 275 



the person, however, to claim that he had, previously to playing, written down 

 the contents of a certain hand, and that he had been actually dealt that hand, 

 we should probably hesitate to receive his statement just as much as if he told 

 us he had been dealt thirteen trumps ; for the previous defining of the hand 

 would have made the odds against it as apparent as in the other case. When 

 we hesitate to believe the statement of such coincidences it is because we feel 

 that the odds against the occurrence were antecedently very great, and we 

 balance that with the odds in favour of the credibility of the witness. If we do 

 not doubt his credibility we receive his statement in spite of its antecedent im 

 probability, for to assume that the extremely improbable is impossible is to fall 

 into a dangerous fallacy.&quot; 1 



The data for probable knowledge furnish us with some certain 

 knowledge ; without some certitude there could be no probability. 

 What we are certain of in regard to such data may be expressed 

 in the form of a disjunctive proposition : e.g. &quot; The result of 

 throwing a die will be to turn up either the one, or the two, or 

 the three, or the four, or the five, or the six &quot; ; or symbolically, 

 &quot; A is either X-^ or X^ or X 3 ... or X where A represents a 

 particular throw, and X lt X 2 , etc., the possible alternative results. 

 We do not know whether the result of the throw A will be X or 

 not ; whether the definite judgment &quot; A is X will prove true. 

 But it may be true ; we are entitled to entertain some degree of 

 rational expectation that it will be true ; and the question now is 

 whether we have any means of measuring the likelihood, the 

 rational expectation we may entertain, of its truth. The mathe 

 matical calculus of probability places such a means at our dis 

 posal. It aims at expressing probability by means of fractions. 



When the chances for and against an event are equal, we are 

 left in absolute doubt about its happening ; the probability is then 

 expressed by the fraction . When the chances are all for and 

 none against, the fraction has grown to unity, and the probability 

 becomes a certainty that the event will happen, or has happened. 

 When the chances are none for and all against, the fraction has 

 decreased to zero, and represents certainty against the event. 

 But those are limits towards which the chances may tend in 

 definitely without ever absolutely reaching them. The event is 

 said to be &quot;probable&quot; or &quot;improbable&quot; according as the fraction 

 is greater t&amp;gt;r less than . Close approximation to the positive 

 and negative limits to unity or to zero are spoken of as 

 &quot;moral certainties&quot; and &quot;bare possibilities&quot; respectively. 



We have now to examine briefly, in their logical aspect, the 



1 WELTON, op. cit., pp. 169-70, 178-80. C/. BORDEN P. BOWNE, Theory of 

 Thought and Knowledge, pp. 187-8. 



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