THE THEORY OP SYLLOGISM, ETC. 125 



reason why want of veracity should lead to that one particular assertion. Taking the extreme 

 case, let us suppose A k to be the only assertion which has any probability of being falsely made, 

 so that m/ = 0, except only when m = k. So far the formula is unaffected : but as it generally 

 happens that the previous bias towards falsehood is towards the assertion itself, and neai'ly or 

 altogether independent of what may really have happened ; let k k = 1, and &/ = k\ = ... = k- 

 Then we have 



p = »> {K + * - K)} 



= (* ~ K ) "A + KV k 



(1 -(t)2. !/,*, + «' 



lying, as of course might have been predicted, between v k k k -r-'2v s k s and v k . 



If there be no probability of error of judgment, k k = 1, and k, = when s is not = k. We 

 have then 



P k = V * . 



(1 - K) V k + K 



The formula for any number of witnesses may be constructed as follows : Let p q m represent 

 the previous probability that the m th witness, A having happened, will judge that A p has 

 happened ; and p q ' m , the previous probability that the m th witness, judging A q to have happened 

 will state A p . If the statements made by the several witnesses be A k , A„ A m , &c, the denomi- 

 nator of the probability thence arising in favour of A t , is the sum of all terms of the form 



v «w« k J l x x uC x y» m * x 



for all combinations of values of u, w, m, y, &c. : the numerator is the collection of all the terms 

 of the denominator in which u = t. 



If all the witnesses be of the same character in every respect, and all agree in asserting A k , 

 the coefficient of v u is 



for all combinations of values of w, x, y, &c. 



In all that has preceded, it is to be remembered that application can immediately be made 

 to the case in which the n events are not the only possible ones : that is, in which 2i/, is not 

 equal to unity. For this is but the supposition of one case more, namely, no event at all, with 

 the probability 1 - 2i/,. 



A. DE MORGAN. 



University College, London, 

 January 7, 1850. 



