598 PROFESSOR BOOLE ON THE COMBINATION 



I regard the elements of a problem relating to probability as logical, when its data 

 and its queesitum are the probabilities of events. The reason for this appellation 

 will shortly be seen. In expression, events may be distinguished as simple or com- 

 pound. A simple event, i.e., an event simple in expression, is one which is expressed 

 by a single term or predication; a compound event, one which is formed by 

 combining the expressions of simple events. " It rains," — " it thunders," would 

 be simple events; "it rains and thunders,' 1 — "it either rains or thunders," &c, 

 would be compound events. The constructions by which such combinations are 

 expressed, although they belong to language, have their foundations in Logic. 

 Thus the conjunctions and, either, or, &c, express merely certain operations of 

 the faculty of Conception, the entire theory of which belongs to the science of 

 Logic. The calculus of Logic, to which I shall have occasion to refer, is a deve- 

 lopment of that science in mathematical forms, in which letters represent things, 

 or events, as subjects of Conception, and signs connecting those letters reprepent the 

 operations of that faculty, the laws of the signs being the expressed laws of the 

 operations signified. It is simply a mistake to regard that calculus as an attempt 

 to reduce the ideas of Logic under the dominion of number. Such are the grounds 

 upon which the class of problems to which I have referred are said to involve 

 logical elements. The description is, however, not entirely appropriate, for the 

 problems, as they are concerned with probabilities, in the mathematical accepta- 

 tion of that term, involve numerical as well as logical elements ; but it is by the 

 latter that they are distinguished, and of them only is account taken in the no- 

 menclature. 



Thus, as an illustration of what has been said, that problem would be com- 

 posed of logical elements, which, assigning for its numerical data the probabilities 

 of the throwing an ace or six with each single die, should propose to determine the 

 probability that the issue of a throw with two dice should be two aces, or that it 

 should be an ace and a six, or that it should be either two aces or an ace and a 

 six ; and so on for any conceivable throw with any number of dice. 



3. In the above example, the events whose numerical probabilities are given are 

 simple events, of which the event whose probability is sought is a logical combina- 

 tion. But it might happen that the former events were themselves combinations 

 of simple events. For instance, the data might be the probabilities that certain 

 meteorological phenomena, as rain, thunder, hail, &c, would occur in certain de- 

 finite combinations, and the object sought might be the probability that they 

 would occur in certain other combinations ; all these combinations being, such 

 as it is within the province of language to express by means of conjunctions, 

 and of the adverb not. Now this would still be a problem whose elements are 

 logical. 



4. But there are questions universally recognised as belonging to the theory 

 of probabilities, whose elements cannot, in their direct significance, be regarded 



