172 The Rules of Inference in Probability. [CHAP. vn. 



of the latter is In this case it is assumed that the latter 

 mn 



is so entirely dependent upon the former that though it does 

 not always happen with it, it certainly will not happen with 

 out it ; the necessity of this assumption however may be 

 obviated by saying that what we are speaking of in the 

 latter case is the joint event, viz. both together if they are 

 simultaneous events, or the latter in consequence of the 

 former, if they are successive. 



4. The above inferences are necessary, in the sense in 

 which arithmetical inferences are necessary, and they do not 

 demand for their establishment any arbitrary hypothesis. 

 We assume in them no more than is warranted, and in fact 

 necessitated by the data actually given to us, and make our 

 inferences from these data by the help of arithmetic. In the 

 simple examples given above nothing is required beyond 

 arithmetic in its most familiar form, but it need hardly be 

 added that in practice examples may often present them 

 selves which will require much profounder methods than 

 these. It may task all the resources of that higher and more 

 abstract arithmetic known as algebra to extract a solution. 

 But as the necessity of appeal to such methods as these does 

 not touch the principles of this part of the subject we need 

 not enter upon them here. 



5. The formula next to be discussed stands upon a 

 somewhat different footing from the above in respect of its 

 cogency and freedom from appeal to experience, or to hypo 

 thesis. In the two former instances we considered cases in 

 which the data were supposed to be given under the conditions 

 that the properties which distinguished the different kinds of 

 events whose frequency was discussed, were respectively 

 known to be disconnected and known to be connected. Let 

 us now suppose that no such conditions are given to us. 



