Logicians have long distinguished two modes of human reasoning, under the respective names of deductive and inductive reasoning. In deductive reasoning we attempt to argue from a hypothesis to its necessary consequences, which may be verifiable by observation; that is, to argue from the general to the particular. In inductive reasoning we attempt to argue from the particular, which is typically a body of observational material, to the general, which is typically a theory applicable to future experience. In statistical language we are attempting to argue from the sample to the population, from which it was drawn. Since recent statistical work has shown that this type of argument can be carried out with exactitude in a usefully large class of cases(2, 3), by means of conceptions somewhat different from those of the classical theory of probability, it may be useful briefly to restate the logical and mathematical distinctions which have to be drawn.