AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 399 



Authority apart, the odds for the conclusion are 1—6 to 1 — a. When both arguments are of 

 great force, or b and a both near to unity, the ratio of the small quantities 1—6 and 1 — a, which 

 determines the probability for the conclusion, cannot be distinctly apprehended. When, then, there 

 is something as near to demonstration on both sides as can be found in a subject which does 

 not admit of absolute demonstration, the mind ought not to arrive at any conclusion more favour- 

 able to one side than the other. We constantly see the refusal of human nature to acquiesce in this 

 reasonable rule, and always with a determination to find out weakness in the argument on one side or 

 the other. It must be sometimes true that false conclusions shall be the exceptional cases, in which 

 arguments of the highest probability fail. 



It also appears that moral demonstration on one side is not enough, if there be anything 

 resembling it on the other. All controversialists admit this in fact, by the stress which they lay on 

 answering the arguments of the opposite side. But they frequently do this as if it were a kind of 

 surplusage, a charitable (but not in any other sense necessary) allowance for the weakness of those 

 who do not see the force brought forward on their side of the question. Whereas it appears that it 

 may be perfectly necessary to answer an opponent who admits all they say to the full extent which 

 is demanded for it, supposing that to be anything short of absolute demonstration. 



PuoB. 5. To ascertain the manner in which the inconclusiveness of the arguments is divided 

 by the authorities between the probabilities of the truth and falsehood of the conclusion. 



If we find X from the equation, 



„.(1_6)(1 -«)(!_ 6) (!-/')/. 



\ - ab I - ab (l - 6),^ + (1 - a) (l - m) ' 



(1 + a)iu - o 



we find \ = 



I - \ = 



(1 - 5)m + (1 - a)(l -/x) 

 (1 + 6) (1 - m) - 6 



(1 -6),. + (!-«) (I -,>.) 

 From this it appears that \ is negative only when ft. is less than , and I — \ when n is greater 



than . In the former case we see that unless the testimony of authority to the conclusion be 



1 + 6 ■ ' 



greater than the success of the argument for the conclusion against a counter-argument of equal 

 strength, the probability of the conclusion is less than that of the validity of the joint arguments. 

 If there be no authority, or if /i = 1, we have 



I - a ^ I -ft 



A = , 1 - \ = 7 , 



1-6 + 1— « l-h + l-fl 



a result which demonstrates the unmeaning character of the result of Problem ."!. For the incon- 

 clusiveness is divided between the truth and falsehood of ihe conclusion in the proportion of the final 

 probability of its falsehood to that of its truth. Or the more likely the conclusion is to he false, 

 the larger proportion of the inconclusiveness does its truth get. 

 But we find 



(1 - b),i 1-6 _(!-«) (1-6) 2,1 - 1 



(1 - 6) ju + (1 -a) (1 -m) ~ 1 - 6 + 1 - a ~ 1 - 6 + f- a ' (1 - 6) ,u + (1 - o) (1 - n)' 

 which shews the addition made to the probability of the conclusion in passing from the case of argu- 

 ments without authority to that of arguments backed by the authority i/m - I. In the case of 



arguments of equal strength, this is /ii - ^, as it ought to be. When ~^ = » <»■ when the 



