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XII. On tlie Theory of Probabilities. 
By Geokge Boole, Professor of Mathematics in Queen's College., Cork. 
deceived June 19, — Bead June 19, 1862. 
This paper has for its object the investigation of the general analytical conditions of a 
Method for the solution of Questions in the Theory of Probabilities, which was proposed 
by me in a work entitled “An Investigation of the Laws of Thought” (London, Walton 
and Maberly, 1854). 
The application of this method to particular problems has been illustrated in the 
work referred to, and yet more fully in a ‘ Memoir on the Combination of Testimonies 
and of Judgments ’ published in the Transactions of the Royal Society of Edinburgh 
(vol. xxi. Part 4). Some observations, too, on the general character of the solutions to 
which the method leads, founded upon induction from particular cases, were contained 
in the original treatise, and the outlines, still in some measure conjectural, of their 
general theory were given in an Appendix to the Memoir. But the complete develop- 
ment of that theory was attended with analytical difficulties which I have only lately 
succeeded in overcoming. It involves discussions relating to the properties of a certain 
functional determinant, and to the possible solutions of a system of algebraic equations 
of peculiar form — discussions which will, I trust, be thought to possess a value, as con- 
tributions to Mathematical Analysis, independent of their present application. 
As concerns the nature of the problems to which the method is applicable, it may be 
stated that they are such that the numerical elements, both given and sought, are tlie 
probabilities of events or states of things the definitions of which, and the connexions of 
which, are capable of expression by logical propositions. There is ground for believing 
that all questions w’hatever involving probability are ultimately reducible to this general 
form. This point, however, I do not jourpose to discuss here. It has been already in 
some degree considered in the Memoir referred to. 
In order to explain more fully the necessity for the present investigation, it will be 
requisite to state the fundamental principles upon which the method in question rests. 
There are only two of them which can possibly afford matter for discussion. 
1st. The expression in language of the data of a problem in the Theory of Probabi- 
lities is to a certain extent arbitrary, because it depends upon the extent of meaning of 
the primaiy simple terms employed to express the events the conceptions of which it 
involves. But the choice of simple terms is, if we consider it with respect to our abso- 
lute power of choice, arbitrary. Any complex combination of events can be contem- 
plated as a single whole in thought, and expressed by a single term. The invention of 
new simple terms to express what was before expressed by a combination of terms is a 
normal phenomenon in the growth of language. 
MDCCCLXII. 2 H 
