The probability of an asserted fact or conclusion means, in common language, the degree of belief which we think we ought to accord to it, and it depends in each mind upon the amount of evidence offered, the degree in which that evidence is understood, the fitness of that mind to receive and be acted on by evidence, and in particular by the evidence which is offered. Evidence insufficient, or ill understood, or offered to a mind which is unduly credulous or sceptical, whether to the particular sort of evidence in question or to evidence in general, may give a low degree of probability, or none at all, to what another mind considers as indisputable ; or the contrary. In the common use of the word, a result is said to be probable only when the mind inclines more or less to believe it, and improbable in the contrary case. The mathematical use of the term is rather different ; but, till further notice, we retain the common signification.
We are going to treat belief as a QUANTITY, not only to speak of its more and less, but to submit that more and less to measurement. The phrase is unusual, but when a halfpenny is in the air, we half believe that head will fall uppermost, and wo half believe in tail; this is what we mean when wo say the chances are even. But the reader must remember that in our present subject belief does not mean that result of self-will which a person puts forward as his opinion. Without any wilful falsehood, men can contrive very frequently to disguise the real state of their minds with respect to a proposition, and to substitute their wishes for their beliefs ; not their prejudices, for real prejudice is belief, but their wishes. The conclusion of the mind is thrust into a corner ; and when, after all, it is necessary to declare it, we hear—I had all the time a lurking belief, &c. That lurking belief was tho rightful owner, but the sovereign will had put a usurper in his place. The verb to believe has only one future form ; what we shall believe, and what you will believe, are legitimate matters of conjecture. But " I will ,believe" is a declaration which is no more in our own power than " thou shalt believe " is in our own right. It has cost England cen turies of struggle to tear the second out of the grammar of statesmen and divines; but how much longer it will take to get as well rid of the first is beyond the theory of probabilities to guess.
Wo mean by evidence, in the present inquiry, not merely oral or illfritten testimony, but everything which disposes the mind, boa ever little, to adopt or reject, including even the effect of previous know kilge„ The value of evidence, that is, the extent it should go towards Inducing belief, is really the subject of inquiry in a branch of exact science known by the name of the theory of probabilities. But how can the value of evidence be made a subject of measurement I why can this be done in the case of an amount of credibility more than in that of an amount • of benevolence, courage, or talent! We ehonld aanrredly think any one must be in a curious delusion who should suppose himself to have ascertained, from the data given by Homer, that the warlike skill of Achilles was exactly 552 times that of Thee. sites, and that Shakopere had done the former n foul wrong, for that he had made it only 237 times and a fraction. But what is the difference, in the nature of the inquiry we seek to institute, between attempts to measure prowess and probability I On the mode of answering this question it depends whether we are to make our subject merely, as heretofore, a kind of artificial method of judging the chances of a game of hazard ; or a rational and exact mode of doing, when data are sufficient, that which we daily attempt to do, as well as we can, with our inaccurate appreciations of the circumstances of common life ; and a science to be used, as are others of a mathematical nature, fur accustoming ourselves to estimate or guess with something like accuracy, by habitual acquaintance with cases in which absolute accu racy is attainable.
When we consider all the circumstances which affect belief or opinion, both these which are external and those which depend on the mind which he exposed to them, it may well bewilder the imagination of a person not accustomed to the idea, when he hears of an attempt to reckon credibility in numbers, and to deduce what are called exact conclusions from hypotheses as to the force of assertions. To remove from the threshold of the subject the incredulity which must exist, and ought to exist in the first instance, let us suppose the other branches of science presented to a student not in their simple begin nings, but by a description of their ultimate physical objects. To put this beginner in a state parallel to that of readers in general with respect to the subject of probabilities, lie must be of mature age, with very little knowledge of number, none of any other branch of mathe matics, and no conception of the construction or use of any physical instrument, nor of the object and procedure of any one experi ment. He might then be addressed as follows :—" You are, without moving from this earth, to track the motions of all the heavenly bodies, to bo able to ascertain where they were or will be at any moment of time past or future, to measure their sizes, to weigh their contents, and to find the species and amount of insensible forces which, by some unknown means, they exercise on one another. You are to detect the existence of a subtle fluid which can neither be seen, heard, nor felt, to measure vibrations of which there are millions in a minute, and to trace the course of effects which travel hundreds of thousands of miles in a second. You are to weigh against each other atoms of matter of which it cannot be shown that millions put together would be visible to the eye." The person so addressed would not be less bewildered nor more disposed to treat the proposed results as fictions, than ho who hears for the first time of a numerical theory of probabili ties. But let us now reverse the method, and suppose the learner allowed to begin at the beginning. He first finds that, step by step, his rude notions of number are organised into a method of computation which enables him easily to perform more than he could have imagined the most subtle brain to have devised. From notions of the simplest kind connected with space, properties of figure become almost intuitive, of which ho could at one time not have comprehended the description, far less the demonstration. By reasoning on the simplest properties of matter, such as can be proved to his senses, ho finds no difficulty in tracing remote and complicated combinations of effects from the plainest causes, by which ho learns to invert this process, and to reduce observed combinations to their simplest elements. But if during this long and very gradual process, he were to keep continually before his mind those groat results the knowledge of which he had been promised, looking to arrive at the fulfilment of the promise by some sudden acquisition of power, his whole course would be one of dis appointment. Ile would be peeping forward a few pages in his Euclid, in the hope of seeing himself almost arrived at tho means of calcula ting an eclipse or explaining the theory of colours, and would find that he was to learn how to make a square equal to a given figure instead.