XI. Of the probability of chances. and probability? The likelihood and probability of SECT. chances is a superior number of equal chances; and consequently, when we say 'tis likely the event will fall on the side which is superior, rather than on the inferior, we do no more than affirm, that where there is a superior number of chances there is actually a su perior, and where there is an inferior there is an inferior, which are identical propositions, and of no consequence. The question is, by what means a superior number of equal chances operates upon the mind, and produces belief or assent, since it appears that 'tis neither by arguments derived from demonstration, nor from probability. In order to clear up this difficulty, we shall suppose a person to take a dye, formed after such a manner as that four of its sides are marked with one figure, or one number of spots, and two with another; and to put this dye into the box with an intention of throwing it: 'tis plain, he must conclude the one figure to be more probable than the other, and give the preference to that which is inscribed on the greatest number of sides. He in a manner believes that this will lie uppermost; though still with hesitation and doubt, in proportion to the number of chances which are contrary: and according as these contrary chances diminish, and the superiority increases on the other side, his belief acquires new degrees of stability and assurance. This belief arises from an operation of the mind upon the simple and limited object before us; and therefore its nature will be the more easily discovered and explained. We have nothing but one single dye to contemplate, in order to comprehend one of the most curious operations of the understanding. This dye formed as above, contains three circum PART Of knowledge and stances worthy of our attention. First, certain causes, such as gravity, solidity, a cubical figure, &c. which determine it to fall, to preserve its form in its fall, and to turn up one of its sides. Secondly, a certain numprobability. ber of sides, which are supposed indifferent. Thirdly, a certain figure inscribed on each side. These three particulars, form the whole nature of the dye, so far as relates to our present purpose; and consequently are the only circumstances regarded by the mind in its forming a judgment concerning the result of such a throw. Let us therefore consider gradually and carefully what must be the influence of these circumstances on the thought and imagination. First, we have already observed, that the mind is determined by custom to pass from any cause to its effect, and that upon the appearance of the one, 'tis almost impossible for it not to form an idea of the other. Their constant conjunction in past instances has produced such a habit in the mind, that it always conjoins them in its thought, and infers the existence of the one from that of its usual attendant. When it considers the dye as no longer supported by the box, it cannot without violence regard it as suspended in the air; but naturally places it on the table, and views it as turning up one of its sides. This is the effect of the intermingled causes, which are requisite to our forming any calculation concerning chances. Secondly, 'tis supposed, that though the dye be necessarily determined to fall, and turn up one of its sides, yet there is nothing to fix the particular side, but that this is determined entirely by chance. The very nature and essence of chance is a negation of causes, and the leaving the mind in a perfect indifference perfect indifference among those events which are supposed contingent. When, there fore, the thought is determined by the causes to consider the dye as falling and turning up one of its sides, the chances present all these sides as equal, and make us consider every one of them, one after another, as alike probable and possible. The imagination passes from the cause, viz. the throwing of the dye, to the effect, viz. the turning up one of the six sides; and feels a kind of impossibility both of stopping short in the way, and of forming any other idea. But as all these six sides are incompatible, and the dye cannot turn up above one at once, this principle directs us not to consider all of them at once as lying uppermost, which we look upon as impossible: neither does it direct us with its entire force to any particular side; for in that case this side would be considered as certain and inevitable; but it directs us to the whole six sides after such a manner as to divide its force equally among them. We conclude in general, that some one of them must result from the throw: we run all of them over in our minds: the determination of the thought is common to all; but no more of its force falls to the share of any one, than what is suitable to its proportion with the rest. "Tis after this manner the original impulse, and consequently the vivacity of thought arising from the causes, is divided and split in pieces by the intermingled chances. We have already seen the influence of the two first qualities of the dye, viz. the causes, and the number, and indifference of the sides, and have learned how they give an impulse to the thought, and divide that impulse into as many parts as there are units in the number of sides. We must now consider the effects of the third particular, viz. the figures inscribed on each side. 'Tis evident, that where several sides have the SECT. XI. Of the probability of chances. PART Of same figure inscribed on them, they must concur in their influence on the mind, and must unite upon one image or idea of a figure all those divided impulses, knowledge and that were dispersed over the several sides, upon which probability. that figure is inscribed. Were the question only what side will be turned up, these are all perfectly equal, and no one could ever have any advantage above another. But as the question is concerning the figure, and as the same figure is presented by more than one side, 'tis evident that the impulses belonging to all these sides must re-unite in that one figure, and become stronger and more forcible by the union. Four sides are supposed in the present case to have the same figure incribed on them, and two to have another figure. The impulses of the former are therefore superior to those of the latter. But as the events are contrary, and 'tis impossible both these figures can be turned up; the impulses likewise become contrary, and the inferior destroys the superior, as far as its strength goes. The vivacity of the idea is always proportionable to the degrees of the impulse or tendency to the transition; and belief is the same with the vivacity of the idea, according to the precedent doctrine. SECTION XII. OF THE PROBABILITY OF CAUSES. WHAT I have said concerning the probability of chances, can serve to no other purpose than to assist us in explaining the probability of causes; since 'tis XII. commonly allowed by philosophers, that what the vul- SECT. gar call chance is nothing but a secret and concealed That species of probability, therefore, is what cause. we must chiefly examine. The probabilities of causes are of several kinds; but are all derived from the same origin, viz. the association of ideas to a present impression. As the habit which produces the association, arises from the frequent conjunction of objects, it must arrive at its perfection by degrees, and must acquire new force from each instance that falls under our observation. The first instance has little or no force: the second makes some addition to it: the third becomes still more sensible; and 'tis by these slow steps that our judgment arrives at a full assurance. But before it attains this pitch of perfection, it passes through several inferior degrees, and in all of them is only to be esteemed a presumption or probability. The gradation therefore from probabilities to proofs is in many cases insensible; and the difference betwixt these kinds of evidence is more easily perceived in the remote degrees, than in the near and contiguous. 'Tis worthy of remark on this occasion, that though the species of probability here explained be the first in order, and naturally takes place before any entire proof can exist, yet no one, who is arrived at the age of maturity, can any longer be acquainted with it. 'Tis true, nothing is more common than for people of the most advanced knowledge to have attained only an imperfect experience of many particular events; which na turally produces only an imperfect habit and transition: but then we must consider, that the mind, having formed another observation concerning the connexion of causes and effects, gives new force to its reasoning Of the probability of causes. |