THE THEORY OF PROBABILITY. 239 



mean to be afterwards more fully considered. It illus- 

 trates the connection between combinations and permu- 

 tations, which is exhibited in the Arithmetical Triangle, 

 and which underlies many of the most important 

 theorems of science. 



Probable Deductive Arguments. 



With the aid of the theory of probabilities, we may 

 extend the sphere of deductive argument. Hitherto we 

 have treated propositions as certain, and on the hypo- 

 thesis of certainty have deduced conclusions equally 

 certain. But the information on which we reason in 

 ordinary life is seldom or never certain, and almost all 

 reasoning is really a question of probability. We ought 

 therefore to be fully aware of the mode and degree in 

 which the forms of deductive reasoning are affected by 

 the theory of probability, and many persons might be 

 surprised at the results which must be admitted. Many 

 controversial writers appear to consider, as De Morgan 

 remarked k , that an inference from several equally pro- 

 bable premises is itself as probable as any of them, but 

 the true result is very different. If a fact or argument 

 involves many propositions, and each of them is uncertain, 

 the conclusion will be of very little force. 



The truth of a conclusion may be regarded as a com- 

 pound event, depending upon the premises happening 

 to be true ; thus, to obtain the probability of the conclusion, 

 we must multiply together the fractions expressing the 

 probabilities of the premises. Thus, if the probability is 

 -J that A is B, and also ^ that B is C, the conclusion that 

 A is C, on the ground of these premises, is ^ x i or i 

 Similarly if there be any number of premises requisite to 

 k ' Encyclopaedia Metrop.' art. Probabilities, p. 396. 



