314 Royal Society : — 



illustrated in the work referred to, and more fully in a Memoir pub- 

 lished in the Transactions of the Royal Society of Edinburgh, entitled 

 " On the Application of the Theory of Probabilities to the Question 

 of the Combination of Testimonies or Judgments." The latter con- 

 tains also the foundations of the general analytical theory of the 

 method. But the complete development of that theory is attended 

 with mathematical difficulties which I have only just succeeded in 

 overcoming. 



A correspondence on the subject between Mr. Cayley and myself 

 also appears in the May Number of the ' Philosophical Maga- 

 zine,' and I owe it to Mr. Cayley that these further researches, 

 of the results of which an account will here be given, were under- 

 taken. 



I shall make but few remarks here upon* the a priori grounds of 

 the method. Generally it may be said that the solution of a question 

 in the Theory of Probabilities depends upon the possibility of men- 

 tally constructing the problem from hypotheses which appear, whether 

 as a consequence of our knowledge or as a consequence of our igno- 

 rance, to be simple and independent. When the data are the proba- 

 bilities of simple events, and no conditions are added, the problem is 

 in theory sufficiently easy, the sole difficulty consisting in the calcu- 

 lation of complex combinations. But when the data are the proba- 

 bilities of compound events, or when the events are connected by 

 absolute conditions expressible in logical propositions, or when both 

 these circumstances are present, the difficulty of the required mental 

 construction becomes greater. If we assume the independence of the 

 simple events from which the compound events according to their 

 expression in language are formed, we meet, first, the difficulty that 

 the number of equations thus formed may be greater or less than 

 that which is requisite to obtain a solution ; and, secondly, the far more 

 fundamental difficulty that the conditions under which the solution, 

 supposing it to be obtained, is analytically valid may not coincide 

 with the conditions under which the data are possible. It seems 

 indeed likely — at any rate the evidence of particular examples points 

 uniformly to the conclusion — that any attempts to construct the pro- 

 blem upon hypotheses which, while not involved in the actual data are 

 of the same nature as those data (t. e. which might conceivably have 

 resulted as facts of observation from the same experience from which 

 the data were derived), limit the problem, and lead to solutions 

 which are analytically valid under conditions narrower than those 

 under which the data are possible. 



But the processes of mathematical logic enable us, without any 

 addition to the actual data, to effect the required construction of the 

 problem formally — formally because the hypotheses which are re- 

 garded as ultimate and independent in that construction refer to an 

 ideal state of things. The nature of the conceptions employed, and 

 their connexion with the conceptions involved in the actual statement 

 of the problem, are discussed in the paper. It is sufficient to say 

 here that, whatever difficulty there may be in these conceptions as 

 conceptions, there is nothing arbitrary in the formal procedure of 



