124 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



lowers his credit. The balance of the two is perfect, on the definite supposition above : and 

 thus an indisposition to state what he thinks extraordinary, is not, by itself, always a prima facie 

 indication of a good witness to an extraordinary event. 



If instead of \, = s, we take s s = 6\ s , we find that P k = v k when 



This is the case in which the disposition to accurate statement varies as the witness's pre- 

 vious expectation of the event. But if 1 — s, = 6 (1 — X s ), that is, if the bias against inaccuracy 

 vary as the previous expectation that the event shall not happen, we find 



* (i-e)v k + e\ 



This is v k when 9=1, as before. 



Let there be two distinct sources of mistatement, by the double action of which an incorrect 

 inaccuracy may be a correct statement : as in intentional and unintentional inaccuracy. Let 

 all the preceding symbols apply to error of judgment, that is, let p q signify the chance of his 

 believing A p when A q happens ; and let p q ' represent the previous probability in the mind of 

 the receiver, that the witness will wilfully state A p when he believes A q happens. The state- 

 ment A k having been made, the possible originators of this statement are the compound events 

 of which the probabilities are thus stated. 



Let A m happen, let the witness judge that A v happened, and state that A k happened. The 

 previous probability of this combination is v m v m kj: and the denominator of the particular 

 probability P k is 22 (y m v n k v ') for all combined values of m and v. Select only those cases in 

 which m = k, and we have 2 iy k v k k^) for the numerator. Accordingly 



If s s and »/ be always p and r, as in Laplace's case, and v s = 1-r-n, and if the biases both 

 to falsehood and error of judgment be all equal, we have 



m, = (1 - p) -T- (n - 1), and m s ' = (1 - r) H- (n - l) ; 



2v k v k kJ = n~ l pr + (n - 1) {n' 1 (1 - p) (l - r) (n - J) -2 } 



the rest of 22 ^M = (n - l) {li^ (r + ^(l - r))} + ^^ (1 - r) 



= |(l-p)r + (l -r)p+^-J(l-p)(l-r)Ji, 



whence the whole denominator is 1 -f- n, and 



(1 - p) (1 - r) . . _ . 



P k = pr + £i-J , as given by Laplace. 



n — 1 



There is, however, but one case of the above which need be especially considered. When 

 we suspect intentional falsehood in an assertion, it is usually because there is some particular 



