PEOFESSOE BOOLE ON THE THEOEY OF PEOB ABILITIES. 
251 
1st. The probability determined is not precisely of the same nature as the probabili- 
ties given. 
For the data are supposed to be derived from experience; and therefore, on the sup- 
position that the futm’e will resemble the past, the events of which the probabilities 
are given will in the long run recur with a frequency proportioned to their proba- 
bility. 
But the probability determined is always an intellectual rather than a material pro- 
bability. We cannot affirm that in the long run an event will occur with a frequency 
proportional to its calculated probability ; but we can affirm that it is more likely to 
occur with this than with any other precise degree of frequency ; that if it do not occur 
with this degree of frequency, the data are in some measure one-sided. 
At the same time the limits of possible deviation are determined. 
2ndly. General solutions obtained by the method do sometimes, but not always, 
admit* of being verified by other methods. I believe that this is solely because it is 
not often possible to solve the problem by other methods without introducing hypo- 
theses which are of the nature of additional data, and, in effect, limit the problem. 
Every general solution, however, admits of a number of particular verifications by neces- 
sary consequence from the theorems established in this paper. 
3rdly. It has been seen that a calculated probability is not necessarily a definite 
numerical value. It may be of the form A+cC, in which c is an arbitrary positive 
fraction. Here it is implied that the probability admits of any value between A and 
A-f-C. If, further, A=0 and C=I, it is implied that the probability may have any 
value between 0 and I, — is therefore quite indefinite. This would really arise if we 
applied the method to a case in which the event of which the probability is sought had 
absolutely no connexion with those of which the probabilities are given. 
Hence in the present theory the numerical expression for the probability of an event 
about which we are totally ignorant is not but cf. Hence, also, when all the proba- 
bilities given are measured by it is not to be concluded (upon the ground of e nihilo 
nihil) that the probability sought will also be 
4thly. While extending the real power of the theory of probabilities, the method 
tends in some cases to diminish the apparent value of its results. For all problems in 
which the data admit of logical expression can be solved by it ; but the resulting solu- 
tions, founded upon the bare data, may be of an indeterminate character, in place of 
the determinate results to which ordinary methods, aided by hypotheses not really 
involved in the data, lead. 
This is the case with the problem of the combination of different grounds of belief or 
opinion. The general solution is indefinite. In two limiting cases, however, it assumes 
a definite form ; one of these, which agrees with the formula generally accepted, repre- 
senting the extreme cumulative force of testimonies, the other the mean weight of 
* Professor Doxkix has verified a general solution (Laws of Thought, p. 362). 
t See on this subject a paper by Bishop Teeeot, Edinburgh Transactions, vol. xxi. part 3. 
