Mn. ELLIS, ON THE FOUNDATIONS OF THE THEORY OF PROBABILITIES. 3 



it will take place more frequently. But if this is admitted, Bernouilli's theorem is unnecessary. 

 It leaves the matter just where it was before, and introduces no new element into the question. 



(8.) Thus, both by an appeal to consciousness, and by the impossiljility of dispensing 

 with such an admission, we are led to recognize the principle, that when an event is 

 expected rather than another, we believe it will occur more frequently on the long run. And 

 thus we perceive that we are in the habit of forming judgments as to the comparative fre- 

 quency of recurrence of different possible results of similar trials. These judgments are founded, 

 not on the fortuitous and varying circumstances of each trial, but on those which are per- 

 manent — on what is called the nature of the case. They involve the fundamental axiom, that 

 on the long run, the action of fortuitous causes disappears. Associated with this axiom is the 

 idea of an average among discordant results, &c. 



I conceive this axiom to be an a j)riori truth, supplied by the mind itself, which is ever 

 endeavouring to introduce order and regularity among the objects of its perceptions. 



(9.) With a view to conciseness, I omit several interesting points which here present them- 

 selves — namely, the connection between the axiom just stated, and the inductive principle; 

 the real utility of Bernouilli's theorem ; and what seems to me to be the true definition of 

 probability, founded on a reference to the ratios developed on the long run. 



I proceed to illustrate what has been said by a few passages from Laplace's " JEssai 

 Philosophique sur les Probabilitis.'''' 



(10.) It seems obvious that no mathematical deduction from premises which do not relate to 

 laws of nature, can establish such laws. Yet it is beyond doubt that Laplace thought Bernouilli's 

 theorem afforded a demonstration of a general law of nature, extending even to the moral world. 



At p. xtii. of the Essay, prefixed as an Introduction to the tliird edition of the Theorie des 

 Probabilites, after giving some account of the theorem of James Bernouilli, Laplace proceeds : 

 " On pent tirer du theorcme precedent cette consequence qui doit etre regardee comnie une loi 

 generale, savoir que les rapports des effets de la nature, sont a fort peu pres constants, quand 



ces effets sont consideres en grand nombre Je n'excepte pas de la loi precedente, les effets 



dus aux causes morales." 



It appears not to have occurred to Laplace, that this theorem is founded on the mental phe- 

 nomenon of expectation. But it is clear that expectation never could exist, if we did not believe 

 in the general similarity of the past to the future, i. e. in the regularity of nature, which is here 

 deduced from it. 



A little further on,.,." II suit encore de ce theoreme que dans une serie d'evenemens inde- 

 finiment prolongee. Taction des causes regulieres et constantes doit Temporter a la longue, sur 



celle, des causes irregulieres Ainsi des chances favorables et nombreuses etant constamment 



attachees a Tobservation des principes eternels de raison de justice et d'humanite, qui fondent 

 et qui maintiennent les societes ; il y a un grand avantage a se conformer a ces principes, et 

 de graves inconveniens a s'en ecarter. Que Ton consulte les histoires, et sa propre experience on 

 y verra tons les faits venir a Tappui de ce resultat du calcul." Without disputing the truth of 

 the conclusion, we may doubt whether it is to be considered as a " resultat du calcdl." 



The same expression occurs immediately afterwards in another passage, in which the writer 

 seems to allude to the history of his own times, and to the ambition of the great chieftain whom he 

 at one time served. 



Indeed it would seem as if to Laplace all the lessons of history were merely confirmations 

 of the "resultats du calcul." We are tempted to say with Cicero — "hie ab artificio suo non 

 recessit." 



(11.) The results of the theory of probabilities express the number of ways in which a 

 given event can occur, or the proportional number of times it will occur on the long run : they 

 are not to be taken as the measure of any mental state ; nor are we entitled to assume that the 

 theory is applicable wherever a presumption exists in favour of a proposition whose truth is un- 

 certain. 



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