NOTES AND REFERENCES. 
597 
we thus have 
£ + + v + y + ? + ¿7 + <r‘ 
=1, 
f+ ? + ?+?' 
= a , 
1 + 1' + V + v' 
= ß, 
? + £ 
= a P> 
£ + v 
= ßq, 
Ç + V + Ç 
= u, 
six equations for the determination of the eight quantities f, rj, rj', £", a-', and u. 
For the determination of u, it is therefore necessary to find or assume two more 
equations: in my solution this is in effect done by giving to f, rj, rj', £ cr' the 
values in the fourth column, values which satisfy the six equations, and establish the 
two additional relations 
l_l f + r_g+r 
/ / > , / / } 
7] <J 7] + 7] (7 
or, as these may be written, 
ABE' AB'E' A B _AB' 
A'BE' ~ A'B'E' ’ A'B ~ A'B’ ’ 
these then are assumptions implicitly made in my solution; they amount to this, that the 
events A, B are treated as independent, first in the case where E does not happen; 
secondly in the case where it is not observed whether E does or does not happen. 
Boole in his solution introduces what he calls logical probabilities (but what these 
mean, I cannot make out): viz. these are Prob. A = x, or say simply A — x; and 
similarly, B = y, AE = s, BE = t; then in the case ABE we have A, B, AE, BE, and 
the logical probability is taken to be xyst; and we obtain in like manner the other 
terms of the third column. And then taking f', rj, rj', g, £', cr' to be proportional 
to the terms of the third column, say F£ = xyst, &c. and substituting in the six 
equations, we have six equations for the determination of x, y, s, t, V, u, and we thus 
arrive at the value of the required probability u. 
But the assumed values of £, &c. give further 
\ K ¥ r . A BE AB'E ABE' AB'E' 
that 1S A'BE ~ A'B'E' and A'BE'~ A'B'E" 
which are assumptions made in Boole’s solution. Wilbraham remarks that the second 
of these assumed equations, though perfectly arbitrary, is perhaps not unreasonable : 
it asserts that in those cases where E does not happen, the relation of independence 
exists between A and B, that is, provided E does not happen, A is as likely to 
happen whether B happens or does not happen. But that the first of these equations 
appears to him not only arbitrary, but eminently anomalous: no one (he thinks) can 
contend that it is either deduced from the data of the problem, or that the mind 
by the operation of any law of thought recognises it as a necessary or even a reasonable 
assumption.