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

 If there be only two authorities, the formula is reduced to 

 Xfi + (1 - X) — r—r, ^Ti '\ ' 



for the joint testimony, and the joint authority is 



a + a - 2Xa' (l - a) 

 1 + aa 



Had it not been for the bias asserted, the authority would have been (a + a') -h(i + aa). When 

 u - a' is positive and a' negative, the joint authority is the greater for the correction of the bias. 

 This is as it should be ; for the bias is then that of contradiction, and tends, until corrected, 

 to lessen the joint authority. I have only entered thus much into this part of the subject, merely 

 to show that the results of the preceding mode of treating the problem are confirmed by those 

 of common sense. 



Peob. 3. To determine the joint effect of a number of arguments, the validities of which 

 are given, some for a conclusion, and some for its contradictory. 



By the validity of an argument, I mean the probability that it proves its conclusion. The 

 argument being of a conclusion which is legitimately inferred from the premises, it is absolutely 

 valid, if all the premises be true: and what is here called its validity therefore means the product of 

 the probabilities of all the premises. Let a, a', a", &c. be the validities of the several arguments 

 for the conclusion, and b, b', b", &c. those of the arguments for the contradiction. If one argument 

 on either side be valid the conclusion of that argument is established. Hence the joint validity 

 of the arguments for is that of an argument whose validity is 



1 - (1 - a) (l -a') (1 - «")... or "20 - 'S.aa' + 'S.aa'a" - ... 



which is the probability that one or more of the arguments for proves its conclusion. Sinnlarly the 

 arguments against amount to an argument the validity of which is 



1 - (1 - ft)(i - 6') (1 - ft")... or 2'6-2'66'+ 2'66'6"- ... 



And having thus shown how to reduce several arguments of the same kind to one, we may now 

 proceed as with one of each sort. If the process now coming be applied to several arguments 

 of each kind, the result obtained will, as we might predict, verify the correctness of the preceding 

 compositions. 



Let there be then one argument of the validity a for, and one of the validity 6 for the contra- 

 diction, or against. Let the argument for, be as a drawing from an urn in which there are M valid 

 and N invalid cases : let that against, be as from another in which there are P valid and Q invalid 

 cases. Of course M : N :: a : \ — a and P : Q v. b : \ - b. If either argument be valid the 

 other must be invalid. Now it does not follow that if the argument for be valid, and be the case 

 marked, say 1, the invalid argument against may be any one of the cases 1, 2, 3 ... up to y. For 

 it may liappen that each particular mode of succeeding in one argument must be necessarily connected 

 with some particular mode or modes of failing in the other. To represent this, let us separate the 

 three cases, and assume as follows : 



1. When the argument for is valid and that against invalid, let it be that M = ?n, + m., + ..., 

 y = 7, + 9j+ ..., and that when the first succeeds in one of the m, ways, the second must fail in one 

 of the g, ways; and the same of m^ and g^, m^ and q^, &c. 



2. When the argument against is valid, and that for invalid, let iV = «, + w.j + ..., 

 P = /), + Pi + ... with the same connexion. 



