652 PEOFESSOB, BOOLE ON THE COMBINATION 



and also because the validity of the rule in question has been made the subject of 

 recent controversy, I design to offer a few remarks upon the subject here. The rule 

 is contained in the following extract. " As, in the case of two probable premises, 

 the conclusion is not established except on the supposition of their being both true, 

 so, in the case of two (and the like holds good with any number) distinct and inde- 

 pendent indications of the truth of some proposition, unless both of them fail, the 

 proposition must be true : we therefore multiply together the fractions indicating 

 the probability of failure of each, — the chances against it ; — and the result being 

 the total chances against the establishment of the conclusion by these arguments, 

 this fraction being deducted from unity, the remainder gives the probability for it. 

 E.g., A certain book is conjectured to be by such and such an author, partly, 1st, 

 from its resemblance in style to his known works, partly {2dly\ from its being at- 

 tributed to him by some one likely to be pretty well informed : let the probability 

 of the Conclusion, as deduced from one of these arguments by itself, be supposed 

 |, and, in the other case f ; then the opposite probabilities will be, respectively, 

 | and | ; which multiplied together give ||, as the probability against the Con- 

 clusion; i.e., the chance that the work may not be his, notwithstanding those rea- 

 sons for believing that it is : and consequently the probability in favour of that 

 Conclusion will be §§, or nearly §." 



A confusion may here be noted between the probability that a conclusion is 

 proved, and the probability in favour of a conclusion furnished by evidence which 

 does not prove it. In the proof and statement of his rule, Archbishop Whately 

 adopts the former view of the nature of the probabilities concerned in the data. 

 In the exemplification of it, he adopts the latter. He thus applies the rule to 

 a case for which it was not intended, and to which it is in fact inapplicable. 



The rule is given, and the conditions of its just application are assigned in 

 Professor De Morgan's Formal Logic, p. 201. Its origin may be thus explained. 

 Let there be two independent causes, A and B, either of which, when present, neces- 

 sarily produces an effect E. Let a be the probability that A is present, b the probabi- 

 liy that B is present; then 1— a is the probability that A is absent, 1 — b the proba- 

 bility that B is absent, (1 — a) (1 — b) the probability that they are both absent; 

 finally, l-(l— a) (1 — b) the probability that they are not both absent. This, then, 

 is the probability that one at least of the causes is present ; and therefore it is 

 the probability that the event E, so far as it is dependent upon these causes, will 

 occur. In its special application to arguments vieAved as causes of belief or ex- 

 pectation, it would lead to the following theorem. If there are two independent 

 arguments in favour of a conclusion which the premises of either, if granted, are 

 sufficient to establish, the doubt only existing as to the truth of the premises, and if 

 the probability that the premises of the first argument are true is a, the probability 

 that the premises of the second argument are true b, then the probability that 

 the conclusion is established is l-(l-a)(l-b). Interpreted, this formula gives 



