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LVIII. On the Failure of the Attempt to deduce Inductive Prin- 

 ciples from the Mathematical Theory of Probabilities. By 

 Sophie Bkyant, B.Sc* 



THE assumption, originally made by mathematicians, and 

 used to furnish a basis of inductive inference, namely 

 that the various probabilities which may be assigned to an 

 event are themselves equally probable, was first definitely 

 called in question by the late Dr. Boole, who proposed the 

 alternative assumption that all constitutions of the Universe 

 are, to our ignorance, equally probable. In other words, if 

 we take up the position of total ignorance respecting the 

 principles on which the Universe is ordered, any one order of 

 events is as probable as any other. This principle of the equal 

 probability of all arrangements in time, prior to the formula- 

 tion of laws declaring some arrangements to be and others not 

 to be, as a matter of fact, commends itself to common sense as 

 the faithful expression of that impartial ignorance from which 

 it is always trying to escape. This impartial ignorance, which 

 is the fundamental pre-supposition in all applications of the 

 Mathematical Theory of Probability, was supposed, however, 

 before Boole's time, and to a large extent is supposed still, to 

 be expressed in the older assumption of Quetelet and Laplace ; 

 and, though an appeal to common sense would probably decide 

 in favour of Boole's view at once, it may be worth while to 

 establish the plea for rejection of the alternative assumption 

 more firmly, on the ground of certain intrinsic inconsistencies 

 w^hich it can be shown to involve. This step I propose to take, 

 as preliminary to a deduction of various consequences from 

 Boole's hypothesis (or rather, as I prefer to call it, hypothetical 

 expression of our fundamental pre-supposition), which involve 

 a generalization of the consequence that Boole himself deduced, 

 and which illustrate in detail the same logical conclusion. 



The only proof of which hypothetical expressions of this 

 kind are susceptible is proof of their perfect internal consist- 

 ency. They can, therefore, be completely established only by 

 a complete development of results which prove to be mutually 

 consistent. The development of Boole's hypothesis which 

 will presently follow constitutes, it is believed, a partial proof 

 of this internal consistency. Complete proof is difficult to 

 obtain; but complete disproof is easy. A single case of incon- 

 sistency is all that we require. Such inconsistency the older 



* Communicated by the Author. 



