CHANCE AND FAITH.
MIL BALFOUR in his Gifford Lectures has flattered plain men by making them feel that perhaps they have too long ordered themselves lowly and reverently before the highly exclusive cults of the philosophers, logicians, and mathematicians. At all events, when they find so brilliant an intellect as Mr. Balfour's defiantly breaking away from the set paths which pierce the jungle and cheerfully thrusting its way in directions which are said to lead nowhere, they are prepared on the spot to bear him company and see what happens. On Monday Mr. Balfour classified beliefs as "inevitable" and "probable," and criticized the mathematical theory of probability. Traditional logical theory, he said, had confined itself to the special kind of probability called mathematical, and though the mathematical statement of chances had yielded results of the first importance, both for science and for practical life, it did not cover the whole `ground. It had not distinguished clearly between the different kinds of probabilities, and in particular it had ;failed to give any account of those numerous beliefs which he described as "probable." What are "probable" beliefs ? To 'Mr. Balfour's mind, they are beliefs which are neither inevit- able nor axiomatic, and yet are, nevertheless, an actual part of our basis of knowledge. In other words, they are such beliefs, often incalculable and vague, as cannot be taken into account by the hard formulae of mathematicians, and yet had a very real existence for suck a man as Joseph Butler, who laid it down in his immortal Analogy that probability is the guide of life.
We feel that Mr•. Balfour is right to open up this field. And yet it is obvious that the mathematicians, when, as he says, they are more concerned with their conclusions than careful in stating their premisses, are only abiding by the terms of their craft. The mathematician is not concerned with the worth of premisses; his business is to tell you what the result will be under the given conditions. He may or may not approve of the conditions. The fact is that he does not trouble himself about them. Suppose that a mathe. matician were asked to calculate what length of paper 'would be produced by a complicated paper-making plant in a certain time. If all the data were given to him be would be able to tell you with accuracy, but he would not consider it part of his duty to say that the materials put into the machines were of poor quality and that the product would not be the best attainable. It is inevitable, we think, much as we like Mr. Balfour's insouciant challenge, that the philosophic premisses from which mathematioians have produced their theory of probability should be much less accurate than the conclusion itself. A glance at the nature of the mathematical theory will show the comparatively limited extent of its field. The theory deals with well-known phenomena which are employed as measures of credibility. The results yielded by tossing a coin or drawing a card from a pack are the familiar instruments of the mathematicians, and Mr. Balfour, of course, turned to these. "It is possible," he said humorously, "to argue a priori that the chances are one in two that the tossing of a penny will result in heads, and it is equally possible to say on purely a priori grounds that the chances are much against a visitor's leaving Menlo Carlo with as much money as he had on entering." We can fancy Mr. Balfour letting his mind run on along this line of thought and setting before us questions which plain men have often asked them- selves without suspecting that they were revolting in the spirit of the inventor of "probable" beliefs. For example, suppose the visitor• to Monte Carlo of Mr. Balfour's imagina- tion is playing at the tables, and has noticed that red has turned up fifteen times in succession. We do not know what the longest recorded run on one colour is, but fifteen is probably very near the "record." The plain man who is guided, not by the mathematical theory, but by something in the nature of a "probable" belief, tells himself that it is highly improbable that the run will be continued, and puts his money with some confidence on black. The mathematicians tell us that the chance of red turning up again is still an even chance. The plain man, however, persists in feeling that the strong probability that the extraordinary run on one colour will not be continued further has somehow impinged upon the even chance us between the red and the black and affected its quality. At all events, this feeling is strong enough to be an absolute and dominant motive. We should all of us probably not in the same way if we wanted to choose a colour.
Human beings can seldom act upon a certainty. We believe that some statement•, which irrevocably determines our action, is true because it is more likely to be true than not true. But we have no proof. Probability is our guide; and it may be that what constitutes probability is nothing more than such an immeasurable quantity as the fact that a man who made the statement to us was reputed to be trust- worthy by other men who, as we happened to know, were trustworthy. An event must either happen or not happen, and the chances that it will happen added to the chances that it will not happen must necessarily make unity. We always have this familiar principle in mind, without perhaps remembering it, but we often " believe" without in the least knowing by what ratio the chances of our expectation being, fulfilled exceed the chances of its not being fulfilled. Fre- quently we are guided by pure instinct. Our expectation may be distorted, magnified, or atrophied if we are looking forward to something that we earnestly desire to see fulfilled. A temperamental error is introduced. Some men dare not believe what they ardently desire; others persuade them- [wives that what they reckon upon is bound to come true. A man, again, will often accept a small chance of winning a large prize rather than put up with a strong chance of winning a small prize. All such motives of notion are not covered by probability mathematically expressed, though, as we said, we are not disposed to blame the mathematicians for not working in materials which belong to other departments of science. "I am confirmed in my view," said Mr. Balfour, " that there aro probable beliefs to which we are inclined but not dr•iven. They vary in degree of coercive power, but are capable of being detected throughout the whole of scientific knowledge." The conclusion of Mr. Balfour's remarks on probable beliefs was summarized in the Times as follows "These beliefs had not received sufficient treatment from philosophers, either of the critical school or of the empirical. %ant and Mill alike had thought more of the grounds of belief than of the actual content of belief, and Mr. Balfour pleaded for as impartial an investigation into what men of science had actually believed as had been given to outworn philosophical creeds."
Butler in his Analogy, after all, provided a warrant for Mr. Balfour by what was a magnificent " probable " belief. As mankind reveals a supreme conscience, be argued, so nature reveals moral government acting through conscience. Men live under a system which, by discouraging vice and reward- ing virtue perceptibly yet only partially, implies the proba- bility that there is a future state beyond this world where justice will be completely satisfied. Let us give ourselves the pleasure of quoting from that splendid piece of English, Butler's introduction, some observations on the sufficiency of slight probabilities "Probable evidence is essentially distinguished from demonstra- tive by Ms, that it admits of degrees ; and of all variety of them, from the highest moral certainty, to the very lowest presumption. We cannot indeed say a thing is probably true upon one very slight presumption for it; because, as there may bo probabilities on both sides of a question, them may be sores against it; and though there be not, yet a slight presumption does not beget that degree of conviction which is implied in saying a thing is probably true. But that the slightest possible presumption is of the nature
of a probability, appears from hence; that such low presumption often repeated, will amount even to moral certainty. Thus a man's having observed the ebb and flow of the tide to-day, affords some sort of presumption, though the lowest imaginable, that it may happen again to-morrow ; but the observation of this event for so many days, and months, and ages together, as it has been observed by mankind, gives us a full assurance that it will. . . . Probable i evidence, in its very nature, affords but an imperfect kind of in- formation; and is to be considered as relative only to beings of limited capacities. For nothing which is the possible object of knowledge, whether past, present, or future, can be probable to an infinite Intelligence; since it cannot but be discerned absolutely as it is in itself—certainly true, or certainly false. But to us, probability is the very guide of life. From these things it follows, that in questions of difficulty, or such as are thought so, where more satisfactory evidence cannot be had, or is not seen ; if the result of examination be, that there appears upon the whole, any the lowest presumption on one side, and none on the other, or greater presumption on one side, though in the lowest degree greater ; this determines the question, even in matters of specula- tion ; and in matters of practice, will lay us under an absolute and formal obligation, in point of prudence and of interest, to act upon that presumption or low probability, though it be so low as t leave the mind in very great doubt which is the truth."
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