and on the Distribution of the Fixed Stars. 525 



that the dependence of the numerical value of the probability 

 sought upon the numerical values of the probabilities given, is 

 founded solely on the logical connexion of the propositions. Or 

 the principle may be thus stated : — The probability sought is a 

 function of the probabilities given ; but the form of that function 

 depends only upon the logical connexion of the proposition whose 

 probability is sought with those propositions whose probabilities 

 are given. There are then two conditions necessary toward the 

 constructionof a perfect method for the calculusof probabilities: — 



1 . Tlie prior construction of a general method for determining 

 the logical dependence of any proposition upon another given 

 proposition, or set of propositions. 



2. The deduction from that expression of the corresponding 

 relation among their probabihties. 



Let us consider these objects in succession. 



1 . In a hasty and (for this reason) regretted publication, en- 

 titled " The i\Iathematical Analysis of Logic," and in a paper 

 published in the Cambridge Mathematical Journal, entitled " The 

 Calculus of Logic," I have stated certain general laws of thought, 

 mathematical in their expression, and constituting, as I believe, 

 the true basis of formal logic. The actual development of those 

 laws in the works referred to is far too imperfect to meet the re- 

 quirements of the case now imder our consideration. Eut that 

 imperfection does not apply to the laws themselves. The results 

 of subsequent investigations authorize me to say that there exists 

 a general method, enabling us not only to educe any of the con- 

 sequences of a system of propositions, but also to express in a 

 scientific form and order the connexion which any proposed pro- 

 position bears to any other proposition, or system of propositions. 

 It is not needful that I should here fully explain how it is that 

 a logical dependence of this nature can exist where propositions 

 appear to be wholly or quite unconnected. But I may remark, 

 that by the very conditions of thought there arise certain rela- 

 tions necessary a priori, e.g. the so-called principle of contradic- 

 tion, "iracwv jBefiaiordni tcGv ap')(a)V*," the principle that a pro- 

 position is true or false, &c., that these are implicitly involved 

 in the fundamental laws of the method, and that they render 

 possible the expression of the relation sought. By the solution 

 thus obtained, the first object which we have in view may in all 

 cas(.'s be accomplished. 



2. The result at which we shall thus have arrived will express 

 the logical dependence of our final proposition upon those which 

 are in\olved in the premises of the inquiry. It is, as has been 

 said, an equation whose symbols are logical — they relate to 

 crcnts, not to numerical maynitudes. But at the same time they 



* Arist. Met. iii. '3. 



