210 THE PRINCIPLES OF SCIENCE. [CHAP. 



ment of a conclusion and their probabilities be p, q, r, &c., 

 the probability of the conclusion on the ground of these 



premises is p x q x r x This product has but a small 



value, unless each of the quantities p, q, &c., be nearly 

 unity. 



But it is particularly to be noticed that the probability 

 thus calculated is not the whole probability of the con- 

 clusion, but that only which it derives from the premises 

 in question. Whately's l remarks on this subject might 

 mislead the reader into supposing that the calculation is 

 completed by multiplying together the probabilities of the 

 premises. But it has been fully explained by De Morgan 2 

 that we must take into account the antecedent probability 

 of the conclusion ; A may be C for other reasons besides 

 its being B, and as he remarks, "It is difficult, if not 

 impossible, to produce a chain of argument of which the 

 reasoner can rest the result on those arguments only." 

 The failure of one argument does not, except under special 

 circumstances, disprove the truth of the conclusion it is 

 intended to uphold, otherwise there are few truths which 

 could survive the ill-considered arguments adduced in their 

 favour. As a rope does not necessarily break because one 

 or two strands in it fail, so a conclusion may depend upon 

 an endless number of considerations besides those imme- 

 diately in view. Even when we have no other informa- 

 tion we must not consider a statement as devoid of all 

 probability. The true expression of complete doubt is a 

 ratio of equality between the chances in favour of and 

 against it, and this ratio is expressed in the probability \. 



Now if A and C are wholly unknown things, we have 

 no reason to believe that A is C rather than A is not- C. 

 The antecedent probability is then ^. If we also have the 

 probabilities that A is B, | and that B is C, \ we have no 

 right to suppose that the probability of A being C is re- 

 duced by the argument in its favour. If the conclusion is 

 true on its own grounds, the failure of the argument does 

 not affect it ; thus its total probability is its antecedent 

 probability, added to the probability that this failing, the 

 new argument in question establishes it. There is a pro- 



1 Elements of Logic, Book III. sections n and 18. 



2 Encyclopaedia Metropolitana, art. Probabilities, p. 400. 



