THE THEORY OF SYLLOGISM, ETC. 117 



bable. If not, the assignment of their several probabilities is, as it were, equivalent to a 

 further subdivision of the cases, each of which is made to consist of several : all the individuals 

 of the ultimate resolution being supposed equally probable. This last arrangement I call the 

 primary distribution. 



Not only do a great many acknowledged errors arise from mistaken modes of making this 

 primary distribution : but it is a fair matter of inquiry whether diversity of method, without 

 error, may not be forced upon the mind by its own legitimate act. 



A little consideration will shew, with regard to the mathematical part of the theory of 

 probabilities, that the primary distribution is out of the subject, as much as the matter of a 

 premise is out of the subject in logic, or the material substance of a solid out of the subject in 

 geometry. Numerical application may be made to a false distribution, as well as to a true one. 

 The well-known mistake once* made by D'Alembert, was one of primary distribution : it 

 could not be in the power of a mathematician, as such, to convince him of his error. The 

 replies of Lacroix and Laplace both amount to nothing more than a perfectly correct denial of 

 D'Alembert's primary distribution, and the proposal of another. 



The primary distribution is a mental act. It matters nothing that the circumstances of the 

 problem appear to dictate it. When it is stated that an urn contains 100 white and 100 black 

 balls, and that therefore there is an even chance of drawing a white ball, it is the want of 

 sufficient reason for any other allotment which produces a provisional assent. Experience may 

 shew sufficient reason, and may dictate a different distribution. Thus, should it turn out that 

 2000 drawings produce 1800 black balls, that circumstance alone would demand the change. 

 Both distributions may be true : that is, true exponents of the rational result of the existing 

 knowledge of the party whose mind is addressed, at two different times. 



I was led to consider the following question; — What is the primary distribution of the 

 mind in regard to a proposition and its contradiction, antecedently to the production of any 

 evidence in favour of either. In the writings of logicians, although no formal exposition of 

 their ideas upon probability is made, I thought I had detected a leaning to the notion that 

 ' Every X is Y," 1 and ' some Xs are not YsJ are a priori of equal probability. And by 

 a priori, I mean antecedently to the production of the specific subject and predicate. Say that 

 opposite to X and Y are to be written at hazard, by two persons selected at hazard, and not 

 in communication, the verbal descriptions of two objects of thought. Which is most likely to 

 turn out true, that every X is Y, or that some Xs are not Ys ? We should pronounce with- 

 out hesitation in favour of the latter, and should even say perhaps, that its extreme case, no 

 X is Y, far exceeds in probability all the others put together. 



Nevertheless, writers on logic, in their tacit references to authority and its effects, seem to 



" 1 say once made, because, though never mentioned, it is 

 pretty certain he saw his error before he died. The second 

 edition of his Opuscules (and the first also, I believe) contains 

 the reflexions on the Theory of Probabilities prefixed to his 

 dissertation on the effects of inoculation for the small-pox. 

 Herein is contained, as cited by Lacroix and others, the cele- 

 brated argument that the probabilities for head at or before the 

 second toss are two to one, the three possible, (and according to 



D'Alembert, equally probable) cases being H, TH, and TT. 

 But in the collection of D'Alembert's works, published by 

 Bastien (1805, Paris, 18 vols. 8vo.) in the fourth volume of 

 which the memoir on inoculation and its preliminaries are 

 somewhat recast and augmented, though the paradox of the 

 Petersburg problem and some other objections are reproduced, 

 all allusion to the problem above described, and all objection to 

 its ordinary solution, are struck out. 



