Propositions of probability.

Propositions of probability do not have anything special about them. If Tr denotes the number of basic truth values of proposition 'r', and Trs denotes the number of basic truth values of proposition 's' that are also basic truth values of 'r', then we call the ratio Trs / Tr the probability of 'r' given 's'. In a truth table, let Tr be the number of 'T' scores in proposition r, and let Trs, be the number of T scores in proposition s in rows in which the proposition r has T. Then given proposition r, proposition s has the probability value: Trs / Tr.

Propositions with no truth value arguments in common are independent of one another, so two elementary propositions give one another a probability of 1/2. If p follows from q, then the proposition q gives proposition p a probability of 1. The certainty of logical inference is a limiting case of probability. (This can be applied to tautology and contradiction.) A proposition is neither probable nor improbable on its own: Either an event occurs or it does not; the middle is excluded.

An urn contains black and white balls in equal numbers (and none of any other kind). I draw one ball after another, and replace them. With this experiment I can establish that the number of black balls drawn and the number of white balls drawn grow closer as the draw continues. So that is is not a mathematical fact. Now, if I say: "It is equally likely that I will draw a white ball as a black one," this means: All the circumstances that I know of (including the laws of nature assumed as hypotheses) give no more probability to the occurrence of the one event than to that of the other. That is to say, they give each case the probability 1/2 as can easily be gathered from the above definitions. What I confirm by the experiment is that the occurrence of the two events, the circumstances of which I do not know in more detail, is independent.

A normal probability proposition is: Circumstances of which I have no further knowledge give a degree of probability to the occurrence of a particular event. Thus, probability is a generalization, a general description of a propositional form. We use probability only in default of certainty, when our knowledge of a fact is not complete, but we do know something about its form. (A proposition may well be an incomplete picture of a certain situation, but it is always a complete picture.) So a statement of probability can be likened to a synopsis of other propositions.