at quantitative relations wherever these can be ascertained. Success in this direction is an index of perfection, and so it is not surprising that, in proportion as a science develops, it tends to establish more and more intimate relations with mathematics. CHAPTER XI PROBABILITY In the preceding chapters we have studied the methods by which universal propositions are established and causal connections ascertained. A universal proposition concerns an entire class, since it asserts that the predicate of the proposition pertains to each member of the class. A statement of causal connection is, as a rule, less definite on the side of extension or denotation. That sunshine causes plants to grow is not universally true, but only if certain conditions of soil, moisture, etc., are realized. In other words, a proposition which affirms a causal connection implies a proviso, such as, ' under ordinary conditions,' or other things equal.' The number of strictly universal propositions that can be made is relatively small. Nearly all rules have exceptions, and statements concerning causation rarely claim universality. It is, therefore, a problem of some importance to ascertain how we can make assertions about individual facts. In so far as the individual fact comes under a known universal rule there is no occasion for difficulty. If smoke is an invariable sign of fire, it is easy, in any particular case of smoke, to draw the appropriate inference. But if the rule that we are forced to employ admits of exceptions, there appears to be no guarantee that the case before us is not one of the exceptions, and inference becomes hazardous. Thus banks are reliable institutions, as a rule, but this is small comfort to the person who has on a previous occasion reposed his trust and his money in the institution that violated the rule. What he now wishes to know is whether this particular bank is trustworthy. Stated more generally, the question is how we are to proceed in the absence of reliable universal rules. It is evident that this question is intimately concerned with our everyday conduct. If, for example, an individual expects to live another year or another decade, he may go on with his education, contract obligations extending over long periods of time, build a house according to his private notions of comfort, and do many other things that he would not do if he entertained the prospect of an early dissolution. But his expectation is clearly not based on any universal rule, since many persons of his age, station, and condition of life die at an earlier time than he allows for himself. Upon what grounds are such expectations properly based? as Based upon Judgments of Probability Classes. Since inferences based upon a universal connection cannot be obtained in the situations typified by the foregoing illustration, we must content ourselves with second best. Between complete assurance and complete ignorance lies the domain of partial assurance, ranging all the way from one extreme to the other. One way in which this partial assurance, which is known as probability, may be obtained, is by a study, not of the individual directly, but of the class to which he belongs. By a study of the class many questions may be tentatively answered, when complete knowledge is lacking. To recur to our illustration, the average individual of twenty-five or thirty years, if he is of sound health, has a strong expectation that he will live through another year, although he does not even try seriously to trace out the conditions involved. Instead of this, he is apt to refer to the fact that the death-rate is low among persons of his age and general circumstances; in other words, that the deaths constitute comparatively rare exceptions to the rule. Because there are so few that die, his chances for survival are considered very good. This type of reasoning is duplicated in many other instances. When we mail a letter we count pretty confidently upon its safe arrival. From time to time letters get lost, through carelessness, in railroad wrecks, or as a result of other causes, but the number of those which get lost is so small a proportion of the total number that we treat it as practically a negligible quantity. For the same reason we leave out of account the possibility that at some time we shall be struck by lightning, or, when we make a journey, that our train will be wrecked, or that our home will be destroyed by an earthquake, or that the stranger of whom we make inquiries as to directions will be insolent or show annoyance. If, in each case, we take the class as a whole, there is a certain percentage of results one way and a certain percentage the other way-e.g., a certain percentage of letters. arrives safely while a certain other percentage, does not. Of course, we frequently do not know the proportions with any such accuracy as is embodied in a statement of percentages. 'But this is, in a sense, merely incidental. Our inference and our conduct are determined by our conception of these proportions, in that we consider the chances' greater to the degree that the proportion preponderates in a given direction. If there is no preponderance, we say that the chances are even.' There are as many cases on the one side of the line as on the other. The statistical work which seeks to state the respective results in terms of percentages merely makes our estimate of the chances more definite than they were before. The Constancy of Classes.-If we are to determine from the study of classes the probabilities of their individual members, there must be a certain constancy in the behavior of the class as a whole. Within certain limits this requirement is met remarkably well by the facts. "The total number of crimes is approximately the same year after year; the annual death-rate, the apportionment of deaths, moreover, to the several diseases as their evident causes, the number of missent letters that reach the Dead Letter Office at Washington each year, the annual number of suicides, of divorces, all these diverse events indicate a regularity, in the long run, as regards their numerical estimate."* As another writer says, more graphically: "With astonishing precision year follows year in the assigned causes of fire [in New York City]. Carelessness with matches' always leads by the inevitable percentage, and so in sequence: Children playing with fire,' and 'Cigar and cigarette ends falling through gratings.' So, too, * Hibben, Logic, p. 338. |