AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 403 



except when » = 1 (an absurdity in this case), in which they taive the form -. But we get this 



result, that if (1 — 6)h-(I - «) = p-, 'hen the more nearly the arguments become demonstration, the 

 more nearly is it certain that either the proposition or its contrary must be true, the probabilities 

 for one and the other being as ^ (1 - i/) /u and (1 — ju) v. This is a singular result : for, since of 

 two exceedingly strong arguments, one on each side, one must be invalid, it is not easy to explain 

 from a priori notions why there is so great a probability that one or the other must be valid. 

 That it is so appears from the probabilities of the validities* of the two arguments, and of the 

 invalidity of both, namely, 



a(l -b) 6(1 - a) (1 - n) (1 -h) 



n(\-h)+b{l-a) + {\-a){l-b) a{\-b) + h{\- a) + (l- a){l-b) a(l- 6) +fc(l-a) + (l-a)(l- 6) ' 



in which n t 1 is the limiting ratio for the probabilities of the two validities. The same remark 

 mav be made with reference to the authorities : when two very high authorities affirm contraries, the 

 higher the authorities the more likely is it that one or the other is right. 



When there is no argument for the contrary, or 6 = 0, the three expressions become 



(l--)/. (l-a)(l-^> (l-a)(l-M)(]--) 



(\..■)^^^,l-a){l-l^)|'+{l-a){\-^)i^-'') (l-.')M+(l-a)(l-/")i'+(l-a)(l-f.)(l-./) (l-.')/.+(l-aXl-/')''+(l-a)(l-''Xl-'') ' 



when there are no authorities these become 



1 \ — a I — a 



3 - 2a 3 - 2ffl 3 - 2n 



or when an argument is proposed, simply, the chance thereby given to the contrary is the same as 

 that of neither being true. 



It will seem strange at first, that the probability for the conclusion is not : for it will be 



said, an argument and none for the contrary, is precisely the same position as an argument and none 

 for the contradictory. But the suppositions as to authority are different. Looking to authorities 

 only the chances of the three cases are 



m(J-i/) 1^(1 -m) ('-^')(1-»') 



m(I-^) + ^(1-/«) + (1-m)(1-i/)' /a(l-p) + ./(I-M) + (l-^')(l->')' ti^(\-v)+v{l-,x) + {l-p){\-vy 

 and in the case of no authorities, there is the chance ^ for each of the cases. Now in treating the 

 contradictory the testimony of no authority is J. 



Let us now suppose that there is authority for each of the three cases, and also argument, or 

 generally, let us take the following problem : 



PnoB. 7' Let there be a dilemma of any number of horns, one or other of which, but 

 only one, must be true ; required the probabilities of the several horns, arguments and autho- 

 rities being given for each. 



Let a, b, c, &c. be the probable validities of the several arguments, /i, v, ^, &c. the testimonies of 

 authority. This problem, treated as before, gives the following result. Let 



v = (, _/,)(, _o)...,.(i -0(1 -D + (1 -")(' -«)...(■ -m) 1/(1 -0 ■■• 



+ (1 -«)(1-6)...(1 -,.)(! -v)l... 



then the probabilities of the several horns containing the truth are 



' I nhnulil have made thi« remark before, in regard to the contradictories, but lor luiving wriitcii tlu' iltnnminaior in the iran»fornic 

 ihape I — af>. I have always found the best lule to be, never perform operations in (lcnoniinator>. 



3 F 2 



