398 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM, 



conclusion thus obtained is not so great as would be obtained from testimonies to the conclusion as 

 strong as those to the validities of the arguments. 



Prob. 4. Given a number of arguments for a conclusion and for its contradictory, and 

 also a number of authorities, all of given probabilities ; required the resultitig probabilities for the 

 conclusion and for its contradictory. 



Let a and b have the meaning of the last prol)lem, and let /i be the testimony which the joint 

 authorities give for the conclusion and against its contradictory. Let a and b be represented by 

 urns of tn and p valid cases, and n and q invalid ones; and let n be represented by an urn of 

 V truths and w falsehoods. Then there are mqv cases in which the argument for is valid and the 

 conclusion true; npw cases in which the argument against is valid and the conclusion false; nqv 

 cases in which both arguments are invalid and the conclusion true ; nqw in which both arguments 

 are invalid and the conclusion false. And these are all the possible combinations. Hence the 

 probability that the conclusion is true must be 



(m + n)qv (1 - 6)/i 



— or — , 



(m + n)qv + {p + q)nw (1 - V)ix + (1 - o) (1 - *i) 



and the probability that the conclusion is false must be 



(p + q)nw (I - a) (I - n) 



— ' or . 



(m + n)qv + (/) + q)nw (1 - 6)/u+ (1 - a) (1 - m) 



To show the accordance of these formulae with common notions, observe that they give the first 

 four of the following results : 



1. In an impossible conclusion (or when ix = 0) the first expression vanishes : or no argument, 



however strong, can give any probability to an impossibility. 





 If /ui = and (7 = 1, we have incompatible hypotheses, and the expressions take the form - . 



2. If a = 1, the conclusion is certain; or absolute demonstration establishes its result, in 

 spite of any amount of authority against it. 



.3. If there be no authority, or if ji = ^, then the probability of the conclusion is 



1 - 6 + 1 - a' 



and hence counter-arguments of equal strength, applied with no authority, give no authority to the 

 conclusion. 



4. If « = b, the probability of the conclusion is («; or counter-arguments of equal strength 

 leave previous authority unaffected. 



5. If o + 6 = 1, the effect of the arguments is simply that of one more authority: and that 

 independently of their inconclusiveness, which still remains. 



6. If there be no argument against, or if 6 = 0, the probability of the conclusion is not n 

 (as stated by writers on logic*, who confound it with the conclusion made valid by the argument) 



but •■or ) when there is no authority. 



M + (1 - a) (1 - /u) 2 - o 



7. When there is no opposition, and no previous authority, any unopposed argument, however 

 weak, gives some authority to the conclusion ; and every argument, however weak, increases the 

 probability derived from previous authority. 



Myself among the rest. 



