OF TESTIMONIES OR JUDGMENTS. 635 



The values of u and t hence found, in accordance with the limitations ex- 

 pressed by (12), are to be substituted in the equation 



Prob. xyz u ,q\ 



Prob. xy ' u + t 



The following special deductions may now be noted. 



1st, If either of the quantities p and q is equal to 1, the probability sought 

 is equal to 1, whatever the values of c and d may be. 



Thus let p=l. Then (2) gives £=0; the only value which satisfies the condi- 

 tions (12), in connexion with (11). The equation (1) is not satisfied by u=0, 

 whence 



— = - = 1 (4) 



u+t u v ' 



This result is obviously^correct. If, for example, of two symptoms which are pre- 

 sent, and which furnish ground of inference respecting a particular disease, one 

 be of such a nature as to make the existence of the disease a matter of certainty* 

 the fact of that existence is established, however adverse to such a conclusion the 

 presumption furnished by the other symptom, supposing it our only ground of 

 inference, would be. 



So, too, the verdict of an authority deemed infallible is consistently held to 

 annul and make void all opposing testimony or argument, however powerful 

 such testimony or argument, considered in itself, may be. 



2ndly, If either of the quantities p and q is equal to 0, the probability sought 

 reduces to 0, as it evidently ought to do. 



1 1 



ddly, If p = K and q—-^, the equations for determining u and t become identical. 



Hence u=t, and 



u 1 



Probability sought = — — — ^ ( 5 ) 



This result is quite independent of the values of c and d '. And it is obviously 

 a correct result. If the causes in operation, or the testimonies borne, are, sepa- 

 rately, such as to leave the mind in a state of equipoise as respects the event 

 whose probability is sought, united they will but produce the same effect, whatever 

 the a priori probability may be that such causes will come into operation, or 

 that such testimonies will be borne. 



4thly, If c=l, and at the same time c is not equal to 0, we find, for the equa- 

 tions determining u and t, 



(p — u) (c'q—u) (u—p — c'q + l)=u(u — c'—p + l) (u — c'q) 



(1-p-t) (c'l^q-t) (t-(fL^q+p)-t(t-c'+p) (t-c'l^q) 



These give 



u = c'q t=-c'(l — q) 



a' 



