(3) Mathematical Probability. This infers that A is connected either with B or C or D, and asks the degree of probability. I. e.: A woman is brought to bed either with a boy or a girl: therefore the probability that a boy will be born is one-half. Of these forms of probability the first two are of equal importance to us, the third rarely of value, because we lack arithmetical cases and because probability of that kind is only of transitory worth and has always to be so studied as to lead to an actual counting of cases. It is of this form of probability that Mill advises to know, before applying a calculation of probability, the necessary facts, i.e., the relative frequency with which the various events occur, and to understand clearly the causes of these events. If statistical tables show that five of every hundred men reach, on an average, seventy years, the inference is valid because it expresses the existent relation between the causes which prolong or shorten life. A further comparatively self-evident division is made by Cournot, who separates subjective probability from the possible probability pertaining to the events as such. The latter is objectively defined by Kries[149] in the following example: “The throw of a regular die will reveal, in the great majority of cases, the same relation, and that will lead the mind to suppose it objectively valid. It hence follows, that the relation is changed if the shape of the die is changed.” But how “this objectively valid relation,” i.e., substantiation of probability, is to be thought of, remains as unclear as the regular results of statistics do anyway. It is hence a question whether anything is gained when the form of calculation is known. Kries says, “Mathematicians, in determining the laws of probability, have subordinated every series of similar cases which take one course or another as if the constancy of general conditions, the independence and chance equivalence of single events, were identical throughout. Hence, we find there are certain simple rules according to which the probability of a case may be calculated from the number of successes in cases observed until this one and from which, therefore, the probability for the appearance of all similar cases may be derived. These rules are established without any exception whatever.” This statement is not inaccurate because the general applicability of the rules is brought forward and its use defended in cases where the presuppositions do not agree. Hence, there are delusory results, e.g., in the calculation of mortality, of the statements of witnesses and judicial deliverances. These do not proceed according to the schema of the ordinary play of accident. The application, therefore, can be valid only if the constancy of general conditions may be reliably assumed.