Section XI. Of the probability of chances. (cont.)
Should it be said, that tho' in an opposition of chances 'tis impossible to determine with certainty, on which side the event will fall, yet we can pronounce with certainty, that it is more likely and probable, 'twill be on that side where there is a superior number of chances, than where there is an inferior: Should this be said, I would ask, what is here meant by likelihood and probability? The likelihood and probability of 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 superior, 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 it is neither by arguments derived from demonstration, nor from probability.
In order to clear up this difficulty, we shall suppose a person to take a die, 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 die 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; tho' 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 die to contemplate, in order to comprehend one of the most curious operations of the understanding.
This die formed as above, contains three circumstances worthy of our attention. First, Certain causes, such as gravity, solidity, a cubical figure, etc. which determine it to fall, to preserve its form in its fall, and to turn up one of its sides. Secondly, A certain number of sides, which are supposed indifferent. Thirdly, A certain figure inscribed on each side. These three particulars form the whole nature of the die, 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 die 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 tho' the die 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 among those events, which are supposed contingent. When therefore the thought is determined by the causes to consider the die 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 die, 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 die 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. It is 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 die, 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 unites 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 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, that were dispersed over the several sides, upon which 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 inscribed 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.