1 Aj)peiidix. 



8. (Fact, Event.) — "Wc may observe, in passing, tliat while the word 

 *' fact," like the word " event," is, according to its etymology, applicable only 

 to a clmnge in things, it may, however, be allowably used to signify any | 

 reality. It is, for example, allowable to speak of the brightness of the star 

 Sirius as a fact. Any question, indeed, respecting a fact, in this wide sense 

 of the term, is resolvible into a question respecting an event or events. For 

 any evidence, whatsoever it may be, of an existence must be evidence of the 

 existent thing having been, directly or mediately, perceived ; and a perception 

 is an event. We may, therefore, in treating of proofs and probabilities, 

 employ ad libitum the occurrence of events as representing all reality of 

 things. 



9. (Numerical Notation of Prohahility.) — In the mathematical calculation 

 of probabilities, the number 1 or unity is employed to signify that the data 

 necessitate the truth of the given proposition j or zero signifies that the data 

 necessitate its falsity ; and the intermediate fractions are used to denote the 

 various amounts of 'probability. Thus, if a probability be stated as J, this 

 denotes that the truth and the falsity of the proposition are equally probable ; 

 if we estimate the probability of a proposition to be f, we estimate the 

 probability of its falsity to be \ ; and so on. 



Now, to employ an arithmetical fraction in any real computation implies 

 that we contemplate something or other as divisible into so many equal parts 

 as there are units in the " denominator," and that of those equal parts we take 

 so many as there are units in the "numerator." Thus, if we speak of j of a 

 mile, or of a pound, or of X, we mean that if the mile, the pound, or X, be 

 divided into 4 equal parts, then of those 4 our proi^osition pertains to 3. 

 What, then, it behoves us to inquire, are the equal parts which a fraction of 

 probability denominates? 



10. (Probability and Belief ) — It has become usual to assume that Pro- 

 bability is the same as Belief; and that, accordingly, the amount of a 

 proposition's probability is neither more nor less than the amount of belief 

 given to that proposition. Thus Professor De Morgan says (Formal Logic, 

 chap. 9): " By degree of probability we really mean, or ought to mean, degree 

 of belief. It is true that we may, if we like, divide probability into ideal and 

 objective, and that we must do so, in order to represent common language." 

 And he adds : '' I throw away objective probability altogether, and consider 

 the word as meaning the state of the mind with respect to an assertion, a 

 coming event, or any other matter on which absolute knowledge does not 

 exist." And he broadly avers, that even when an error of computation is 

 incurred in deducing the amount of a probability, still the conclusion 

 erroneously arrived at constitutes the probability to the computator ; so that 

 precisely the same data may render dilTorcnt probabilities. A similar defi- 



