272 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



bols it involves. The expression xy represents, under the former 

 condition, a concurrence of the events denoted by x and y ; under 

 the latter, the product of the numbers or quantities denoted by x 

 and y. And thus every expression denoting an event, simple or 

 compound, admits, under another system of interpretation, of a 

 meaning purely quantitative. Here then arises the question, 

 whether there exists any principle of transition, in accordance 

 with which the logical and the numerical interpretations of the 

 same symbolical expression shall have an intelligible connexion. 

 And to this question the following considerations afford an 

 answer. 



19. Let it be granted that there exists such a feeling as ex- 

 pectation, a feeling of which the object is the occurrence of events, 

 and which admits of differing degrees of intensity. Let it also 

 be granted that this feeling of expectation accompanies our 

 knowledge of the circumstances under which events are produced, 

 and that it varies with the degree and kind of that knowledge. 

 Then, without assuming, or tacitly implying, that the intensity 

 of the feeling of expectation, viewed as a mental emotion, admits 

 of precise numerical measurement, it is perfectly legitimate to 

 inquire into the possibility of a mode of numerical estimation 

 which shall, at least, satisfy these following conditions, viz., that 

 the numerical value which it assigns shall increase when the 

 known circumstances of an event are felt to justify a stronger 

 expectation, shall diminish when they demand a weaker expec- 

 tation, and shall remain constant when they obviously require an 

 equal degree of expectation. 



Now these conditions at least will be satisfied, if we assume 

 the fundamental principle of expectation to be this, viz., that the 

 laws for the expression of expectation, viewed as a numerical 

 element, shall be the same as the laws for the expression of the 

 expected event viewed as a logical element. Thus if ^ (x, y> z) re- 

 present any unconditional event compounded in any manner of 

 the events or, y, 2, let the same expression (x, y> z), according 

 to the above principle, denote the expectation of that event ; 

 x, y, z representing no longer the simple events involved, but 

 the expectations of those events. 



