OF TESTIMONIES OR JUDGMENTS. 641 



tended to alter that state. By the evidence I mean, of course, that which forms 

 the basis of the judgments. 

 Making, as before, 



xz = s V z = t xyz—v 



and determining »asa developed logical function of x, y, z, s, and t, we find 



v = xyzst + (xyz s t + x zs y t + yz txs + xy zst + y xz st 

 + zx y s t + xy z s t) 



+ terms whose coefficients are ^. 

 Hence, availing ourselves of the simplification of Art. 21, we have 



xyzst + xy + xzs + x xystz + xy + ytz + y 



c ~ c' 



_ xystz + xsz xystz + ytz 



cp c'q 



xystz + xsz + ytz + z xystz 

 r u 



=xystz + xy + xsz + ytz + x + y + z + l 



If we equate the product of the third and fourth to that of the fifth and sixth 



members of the above system, we find 



Prob. xyz = — ~ 



whence by symmetry, 



ccr(i-f>)(i- g ) v 



rrob. xyz = \ — r •••{*) 



Substituting these values in (3), we have 



Prob. xyz = ffg(l-r) , & \ 



Prob. xy pq(l — r) + (l—p)(l — q)r 



Before proceeding further, it will be well to note that in this formula p and q 

 represent, not the general probabilities which the testimonies or evidences 

 upon which our judgments are founded would give to the event z, but the proba- 

 bilities which they would separately produce in a mind embued with a previous 

 expectation of the event z, the strength of which is measured by r. And there 

 are some curious confirmations of the truth of the theorem, two of which I shall 

 notice. 



If we represent the a posteriori value of Prob. z by R, and accordingly make 



pq (l-r) + (l-p)(l-q)r W 



we find, on solving the equation relatively to r, 



pq(l-R) + (l-p)(l-q)R- r ' ' ' ' W 



VOL. XXI. PAET IV. 8 K 



