PEOl'ESSOE BOOLE ON THE THEOET OE PEOB ABILITIES. 
229 
AVe are now able to explain more clearly the nature of the analytical investigation 
which will follow. Let pi, P 21 • • • Pn represent the probabilities given in the data. As 
these will in general not be the probabilities of unconnected events, they will be subject 
to other conditions than that of being positive proper fractions, viz. to other conditions 
beside 
p2>0 . . 
J 
Those other conditions will, as will hereafter be shown, be capable of expression by 
equations or inequations reducible to the general form 
being numerical constants which differ for the different conditions in 
question. These, together with the former, may be termed the conditions of possible 
experience. When satisfied they indicate that the data may have, when not satisfied 
they indicate that the data cannot have, resulted from actual observation. On the other 
hand, the ideal events are regarded as independent, and their probabilities, which enter 
as auxiliary quantities into the process of solution, are subject to no other condition 
than that of being positive proper fractions. It is the general object of the analytical 
investigation to establish the two following conclusions, viz., — 
1st. The probabilities of the ideal independent events, as involved in the method 
under consideration, will in the process be determinable, without ambiguity, as positive 
proper fractions whenever the data satisfy the conditions of possible experience, and not 
otherwise. 
And, as a consequence of the above, 
2ndly. The probability determined by the method will have such a value as it con- 
sistently might have had if, instead of being calculated from the data, it had been deter- 
mined by observation under the same experience as the data. 
These conclusions rest upon the ground of certain analytical theorems relating to 
functional determinants, and to the possible solutions of simultaneous algebraic equa- 
tions, which will be demonstrated in this paper. But, in order to explain the appli- 
cation of those theorems, it will be necessary to show, first, how the “ conditions of 
possible experience ” in problems in the Theory of Probabilities may be determined ; 
secondly, what the analytical method in question for the solution of such problems is. 
Determination of the Conditions of possible Experience. 
The method for determining the conditions of possible experience given in the ‘ Laws 
of Thought,’ chap, xix., may be advantageously replaced by the following one, which is 
taken from the ‘ Memoir on the Combination of Testimonies and of Judgments,’ already 
referred to. 
Let the events in the data be resolved into the ultimate possible alternatives which 
they involve, and let the unknown probabilities of these alternatives be represented by 
X, V, See.; then, as the probability of each event in the data is equal to the sum of the 
