AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 397 



3. When both arguments are invalid, let iV = «,' + n/ + ... , Q = 9,' + 9/ + ... witli the same 

 connexion. 



It is now clear that the number of compatible cases, in which the argument for is valid, must be 

 m,qi + m.,q,+ ... or "S-mq. Similarly, "^np and 'En q' are the numbers of cases in which the 

 argument against is valid, and in which both arguments are invalid. Hence we have for the 

 probabilities of the three cases, namely, that the conclusion is established, that the contradictory is 

 established, and that the arguments are inconclusive, the following expressions: 



'S.mq 'S.np 'S.n'q' 



2,mq + '2np + 2'«V ' 'S.mq + 'S.np + 2w'g'' Sw? + 2wp + 'S.n'q'' 



To solve the question in the most general manner, would require that we should combine 

 the preceding results in all cases, that is, for all values, and all subdivisions, of M, JV, P, Q. 

 Without attempting such generality, I may make the following observations. From what takes 

 place in other similar questions, it is highly probable we should find the result of this combination 

 either to agree with that in which any of the M cases mav occur with any one of the Q cases, &c. 

 or to approximate to such an agreement as M, &c. are increased without limit. Next, that this 

 agreement actually takes place, when all the subdivisions are the same aliquot parts of their wholes. 

 With these presumptions, I content myself with their result, which amounts to supposing that any 

 one of the M cases may enter with any one of the Q cases, and so on. The probabilities then are, 

 for the three cases above-mentioned, 



MQ NP NQ 



MQ + NP+ NQ ' MQ + NP + NQ ' MQ + NP + NQ" 



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



1 — ab ' 1 - ab ' 1 - ab 



The third term is the chance of inconclusiveness, which necessarily renders this case indefinite : and 

 all we can say is, that the chance of the truth of the conclusion is 



where the value of \ cannot be determined from argument (for all the arguments are used in 

 determining a and b). 



When the arguments are of equal force, or a = b, we have 



a a I — a 



\ + a \ + a \ + a 



Hence o-7-(l + a), which represents the probability that a verified conclusion was derived from 

 an argument of the validity a rather than from demonstration (when it must have been one 

 or the other), also represents the success of an argument of the validity a against an argument 

 of equal force on the other side. 



So far as an argument is not demonstrative, it must rest on authority, including under that word 

 the authority of the recipient himself. Now a is in fact the testimony to the validity of the argu- 

 ment on one side, and 6 to that on the other. If these were testimonies to the truth or falsehood 

 of the conclusion, the joint testimonies to the truth and falsehood of the conclusion would then be 



0(1 - b) 6(1 - a) 



a{\ - b) + 6(1 - a) ' a(l - 6) + 6(1 - a) ' 



which, since a + b - ab must be less than unity, are necessarily greater than the two first of 

 the three expressions. Or, if we attempt to consider argument entirely without reference to any 

 authority except that for the premises, the absolute testimony to llie triitii or falsehood of the 



