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



And M is [m + n + l]^ W • W' "'here [to] means 1.2.3 ... m. If the probability required 

 be denoted by P„,, „, we have, multiplying by Mdfi and integrating from ^ = to « = 1, 



1 (n + 2)(n + l) 1 (n + 3) (n + 2)(w + 1) 2 p 

 r TO(m-l)r^ to (to - 1)(to-2) j*^ "-3, ..+ 



Now P,„_3, ,, + 3 is less than unity, so that if r be considerable, any degree of approximation may 

 be obtained by this method, carried to more terms if necessary, and if the value of m will permit. 

 Take the first three terms : then if the testimony of authority were given = j«-i- (to + n), instead of 

 being most likely to have something near that value, the approximation to P,„_ „ would then be 



n 1 n'^ I 



1 + — -. 



m r m' r- 



Subtract the second from the first, and we have 



1 3 TO M + 2 TO + ■n? 1 

 mr m^ (m — 1) r^ 



Write (to + n) ix and (to + «)(!- m) for to andw, and we have, supposing to and m + n con- 

 siderable numbers, 



-^-U-(^^U^ nearly, 

 m + « [jxr fx r \ 



Except then when fx is very small, the principle of relative testimonies is sufficiently accurate, 

 in the case above supposed, taking for the testimony of authority the most probable value of that 

 testimony. 



PitOB. 6. Given arguments and authorities for a proposition atid for its contrary, required 

 the probability for the truth of each proposition, and for the falsehood of both. 



The contrary is thus distinguished from the contradictory : both the proposition and the con- 

 trary may be false, though both cannot be true : while either the proposition or its contradictory 

 must be true. As far as the arguments alone are concerned, the problem is that of Problem 3 : for 

 either one of the arguments is valid and the other invalid, or else both are invalid. But there is a 

 difference in the meaning of authorities ; for, fx being the testimony to a proposition, 1 - |U is not 

 necessarily the testimony to its contradictory. Let fx and v be the testimonies of authority to the 

 conclusion and its contradictory, and a and 6 the probable validities of the arguments. There are 

 then five cases, two favourable to the truth of the proposition, two to that of the contrary, and one 

 to that of the falsehood of both ; 1. The argument for may be valid, in which case the proposition is 

 true, the contrary false, and the argument against invalid. 2. The argument against mav be valid, 

 in which case the contrary is true, the proposition false, and the argument for invalid. 3. Both 

 arguments may be invalid, and the proposition true. 4. Both arguments may be invalid and the 

 contrary true. 5. Both arguments may be invalid and the proposition and contrary both false. 

 Treating these in the manner in which the preceding problems have been solved, and which it is now 

 unnecessary to repeat, we have the following expressions for the probability of the proposition, of 

 its contrary, and of both being false, 



(l-6)(l-'-)M (l-a) (I-M).. (l-a)U-6)(l-'^Kl-i') 



(l-6)(l-«)M+(l-a)(l->i)"'+(l-a)(l-6)(I-M)(l-i') (l-W(l->')M+(l-a)(i-i')-+a-a)(l-*)(l-/')a-'') (l-6)(l-'')M+(l-a)(l-/<)''+(l-a)(l-6)(l-,<)(l->') 



If there be no authorities, or if /» = 1; = 1, these become 



1-6 I - a (1 - a) (1 -6) 



1 - 6 + 1 - a + (1 - o) (1 - 6) 1 - 6 + 1 - a + (1 - a) (1 — 6) 1 - 6 + 1 - a + (l - a) (1 - fc) ' 



If the arguments be of equal strength these become 



1 1 \ - a 



3 — a 3 — a 3 - a 



