ment of a conclusion and their probabilities be p, q, r, &c., the probability of the conclusion on the ground of these premises is p x qxrx...... This product has but a small value, unless each of the quantities p, q, &c., be nearly unity. 1 But it is particularly to be noticed that the probability thus calculated is not the whole probability of the conclusion, but that only which it derives from the premises in question. Whately's 1 remarks on this subject might mislead the reader into supposing that the calculation is completed by multiplying together the probabilities of the premises. But it has been fully explained by De Morgan 2 that we must take into account the antecedent probability of the conclusion; A may be C for other reasons besides its being B, and as he remarks, "It is difficult, if not impossible, to produce a chain of argument of which the reasoner can rest the result on those arguments only." The failure of one argument does not, except under special circumstances, disprove the truth of the conclusion it is intended to uphold, otherwise there are few truths which could survive the ill-considered arguments adduced in their favour. As a rope does not necessarily break because one or two strands in it fail, so a conclusion may depend upon an endless number of considerations besides those immediately in view. Even when we have no other information we must not consider a statement as devoid of all probability. The true expression of complete doubt is a ratio of equality between the chances in favour of and against it, and this ratio is expressed in the probability. Now if A and C are wholly unknown things, we have no reason to believe that A is C rather than A is not C. The antecedent probability is then. If we also have the probabilities that A is B, and that B is C, we have no right to suppose that the probability of A being C is reduced by the argument in its favour. If the conclusion is true on its own grounds, the failure of the argument does not affect it; thus its total probability is its antecedent probability, added to the probability that this failing, the new argument in question establishes it. There is a pro 1 Elements of Logic, Book III. sections 11 and 18. bability that we shall not require the special argument; a probability that we shall, and a probability that the argument does in that case establish it. Thus the complete result is +(1 × 1)or §. In general language, if a be the probability founded on a particular argument, and c the antecedent probability of the event, the general result is 1 − (1 − a) (1 − e), or a + c ac. We may put it still more generally in this way :-Let a, b, c, &c. be the probabilities of a conclusion grounded on various arguments. It is only when all the arguments fail that our conclusion proves finally untrue; the probabilities of each failing are respectively, I a, I — b, I — c, &c.; the probability that they will all fail is (1 − a)(1 − b) (I c)... ; therefore the probability that the conclusion will not fail is I − (1 − a)(1 − b)(1 − c)... &c. It follows that every argument in favour of a conclusion, however flimsy and slight, adds probability to it. When it is unknown whether an overdue vessel has foundered or not, every slight indication of a lost vessel will add some probability to the belief of its loss, and the disproof of any particular evidence will not disprove the event. We must apply these principles of evidence with great care, and observe that in a great proportion of cases the adducing of a weak argument does tend to the disproof of its conclusion. The assertion may have in itself great inherent improbability as being opposed to other evidence or to the supposed law of nature, and every reasoner may be assumed to be dealing plainly, and putting forward the whole force of evidence which he possesses in its favour. If he brings but one argument, and its probability a is small, then in the formula I · (1− a)(1 − c) both a and are small, and the whole expression has but little value. The whole effect of an argument thus turns upon the question whether other arguments remain, so that we can introduce other factors (Ib), (d), &c., into the above expression. In a court of justice, in a publication having an express purpose, and in many other cases, it is doubtless right to assume that the whole evidence considered to have any value as regards the conclusion asserted, is put forward. To assign the antecedent probability of any proposition, may be a matter of difficulty or impossibility, and one = with which logic and the theory of probability have little concern. From the general body of science in our possession, we must in each case make the best judgment we can. But in the absence of all knowledge the probability should be considered , for if we make it less than this we incline to believe it false rather than true. Thus, before we possessed any means of estimating the magnitudes of the fixed stars, the statement that Sirius was greater than the sun had a probability of exactly; it was as likely that it would be greater as that it would be smaller; and so of any other star. This was the assumption which Michell made in his admirable speculations.1 It might seem, indeed, that as every proposition expresses an agreement, and the agreements or resemblances between phenomena are infinitely fewer than the differences (p. 44), every proposition should in the absence of other information be infinitely improbable. But in our logical system every term may be indifferently positive or negative, so that we express under the same form as many differences as agreements. It is impossible therefore that we should have any reason to disbelieve rather than to believe a statement about things of which we know nothing. We can hardly indeed invent a proposition concerning the truth of which we are absolutely ignorant, except when we are entirely ignorant of the terms used. If I ask the reader to assign the odds that a "Platythliptic Coefficient is positive" he will hardly see his way to doing so, unless he regard them as even. The assumption that complete doubt is properly expressed by has been called in question by Bishop Terrot, who proposes instead the indefinite symbol ; and he considers that "the à priori probability derived from absolute ignorance has no effect upon the force of a subsequently admitted probability." But if we grant that the probability may have any value between 0 and 1, and that every separate value is equally likely, then n and In are equally likely, and the average is always . Or we may take p. dp to express the probability that our 1 Philosophical Transactions (1767). Abridg. vol. xii. p. 435Transactions of the Edinburgh Philosophical Society, vol. xxi. p. 375. estimate concerning any proposition should lie beween p and p+dp. The complete probability of the proposition is then the integral taken between the limits I and o, or again. Difficulties of the Theory. The theory of probability, though undoubtedly true, requires very careful application. Not only is it a branch of mathematics in which oversights are frequently committed, but it is a matter of great difficulty in many cases, to be sure that the formula correctly represents the data of the problem. These difficulties often arise from the logical complexity of the conditions, which might be, perhaps, to some extent cleared up by constantly bearing in mind the system of combinations as developed in the Indirect Logical Method. In the study of probabilities, mathematicians had unconsciously employed logical processes far in advance of those in possession of logicians, and the Indirect Method is but the full statement of these processes. It is very curious how often the most acute and powerful intellects have gone astray in the calculation of probabilities. Seldom was Pascal mistaken, yet he inaugurated the science with a mistaken solution. Leibnitz fell into the extraordinary blunder of thinking that the number twelve was as probable a result in the throwing of two dice as the number eleven.2 In not a few cases the false solution first obtained seems more plausible to the present day than the correct one since demonstrated. James Bernoulli candidly records two false solutions of a problem which he at first thought self-evident; and he adds a warning against the risk of error, especially when we attempt to reason on this subject without a rigid adherence to methodical rules and symbols. Montmort was not free from similar mistakes. D'Alembert constantly fell into blunders, and could not perceive, for instance, that the probabilities would be the same when Montucla, Histoire des Mathématiques, vol. iii. p. 386. * Leibnitz Opera, Dutens' Edition, vol. vi. part i. p. 217. Todhunter's History of the Theory of Probability, p. 48. To the latter work I am indebted for many of the statements in the text. coins are thrown successively as when thrown simultaneously. Some men of great reputation, such as Ancillon, Moses Mendelssohn, Garve, Auguste Comte,1 Poinsot, and J. S. Mill, have so far misapprehended the theory, as to question its value or even to dispute its validity. The erroneous statements about the theory given in the earlier editions of Mill's System of Logic were partially withdrawn in the later editions. Many persons have a fallacious tendency to believe that when a chance event has happened several times together in an unusual conjunction, it is less likely to happen again. D'Alembert seriously held that if head was thrown three times running with a coin, tail would more probably appear at the next trial. Bequelin adopted the same opinion, and yet there is no reason for it whatever. If the event be really casual, what has gone before cannot in the slightest degree influence it. As a matter of fact, the more often a casual event takes place the more likely it is to happen again; because there is some slight empirical evidence of a tendency. The source of the fallacy is to be found entirely in the feelings of surprise with which we witness an event happening by chance, in a manner which seems to proceed from design. Misapprehension may also arise from overlooking the difference between permutations and combinations. To throw ten heads in succession with a coin is no more unlikely than to throw any other particular succession of heads and tails, but it is much less likely than five heads and five tails without regard to their order, because there are no less than 252 different particular throws which will give this result, when we abstract the difference of order. Difficulties arise in the application of the theory from our habitual disregard of slight probabilities. We are obliged practically to accept truths as certain which are nearly so, because it ceases to be worth while to calculate the difference. No punishment could be inflicted if absolutely certain evidence of guilt were required, and as 1 Positive Philosophy, translated by Martineau, vol. ii. p. 120. |