Page:Observations on Man 1834.djvu/245

fact, &c. be dependent on each other, so that the first is required to support the second, the second to support the third, &c. i.e. if a failure of any one of the evidences renders all the rest of no value, the separate probability of each evidence must be very great, in order to make the proposition credible; and this holds so much the more, as the dependent evidences are more numerous. For instance, if the value of each evidence be ${\tfrac {1}{a}}$, and the number of evidences be in n, then will the resulting probability be ${\tfrac {1}{a^{n}}}$. I here suppose absolute certainty to be denoted by 1; and consequently, that a can never be less than 1. Now it is evident, that ${\tfrac {1}{a^{n}}}$ decreases with every increase both of a and n.

Secondly, If the evidences brought for any proposition, fact, &c. be independent on each other, i.e. if they be not necessary to support each other, but concur, and can, each of them, when established upon its own proper evidences, be applied directly to establish the proposition, fact, &c. in question, the deficiency in the probability of each must be very great, in order to render the proposition perceptibly doubtful; and this holds so much the more, as the evidences are more numerous. For instance, if the evidences be all equal, and the common deficiency in each be ${\tfrac {1}{a}}$, if also the number of evidences be n as before, the deficiency of the resulting probability will be no more than ${\tfrac {1}{a^{n}}}$, which is practically nothing, where a and n are considerable. Thus if a and n be each equal to 10, ${\tfrac {1}{a^{n}}}$ will be ${\tfrac {1}{10,000,000,000}}$, or only one in ten thousand millions; a deficiency from certainty, which is utterly imperceptible to the human mind.

It is indeed evident, without having recourse to the doctrine of chances, that the dependency of evidences makes the resulting probability weak, their independency strong. Thus a report passing from one original author through a variety of successive hands loses much of its credibility, and one attested by a variety of original witnesses gains, in both cases, according to the number of successive reporters, and original witnesses, though by no means proportionably thereto. This is the common judgment of mankind, verified by observation and experience. But the mathematical method of considering these things is much more precise and satisfactory, and differs from the common one, just as the judgment made of the degrees of heat by the thermometer does from that made by the hand.

We may thus also see in a shorter and simpler way that the resulting probability may be sufficiently strong in dependent evidences, and of little value in independent ones, according as