OF TESTIMONIES OR JUDGMENTS. 609 
tions of a smaller number of symbols, except when the condition referred to in 
(3) is fulfilled, in which case, the methods are identical. 
15. It remains to show how the conditions of possible experience as above 
determined, restrict us in the choice of the hypothesis, by the aid of which the 
final solution is to be obtained. 
Taking for example the above problem of Art. 13, let us inquire whether it 
would be lawful to assume 2, y, and 2 as the primary simple events of the 
problem. 
If we make this assumption and then write 
Prob. z=a Prob. y=@ Prob. z= 
we find 
Prob. zy=a8 Prob. yz=@By Prob. za=ya 
whence comparing with (1) Art. 13— 
aB=p By=q ‘Yaar 
solving which equations we have 
a= JZ B=/2 y=/P4 “. Prob. ayz=aBby=V/pqr + none 
Now, a, @, y: being by assumption probabilities, and therefore, lying numerically 
between the limits 0 and 1, we must have 
qr=p rp=q a . F ‘ ; (5) 
as the conditions to which p, q, and 7 (beside being fractional) must be subject. 
These conditions do not, however, agree with, and are not involved in, the condi- 
tions of possible experience, determined in (6) Art. 13. We may conclude, there- 
fore, that the hypothesis upon which our solution is founded, involves elements, 
the introduction of which is unwarranted, and that the value of Prob. xyz deter- 
mined is erroneous. 
We may show, in fact, that the conditions (5) imply the conditions of possible 
experience, and something more. If gr = p then, @ fortiori, gr =p +(1—g) (1-1) 
since (1—q) (1—r) is essentially positive. Therefore, 
gr <pt+l—q-—rt+gr 
whence p>qtr-l 
which is one of the conditions (6) Art 13. In the same way the other conditions 
in that article may be deduced from (5). The reverse reduction is, however, 
impossible. 
16. The hypothesis upon which the method developed in the Laws of Thought, 
cap. xvii., for the solution of questions in the theory of probabilities whose elements 
are logical, is founded, seems to be the only one which satisfies the requirement 
referred to in Art. 12. It was not, however, upon such considerations as this, 
VOL. XXI. PART IV. 8B 
