If Vr is the number of the truth-grounds of the proposition "r", Vrs the number of those truth-grounds of the proposition "s" which are at the same time truth-grounds of "r", then we call the ratio Vrs : Vr the measure of the probability which the proposition "r" gives to the proposition "s".
5.151 (1) Suppose in a scheme like that above in No. 5.101, Tr is the number of the "T"s in the proposition r, Trs the number of those "T"s in the proposition s, which stand in the same columns as "T" of the proposition r; then the proposition r gives to the proposition s the probability Trs : Tr.
5.152 Propositions which have no truth-arguments in common with one another we call independent.
Two elementary propositions give to one another the probability ½.
If p follows from q, the proposition q gives to the proposition p the probability 1. The certainty of logical conclusion is a limiting case of probability.
(Application to tautology and contradiction.)
5.153 A proposition is in itself neither probable nor improbable. An event occurs or does not occur, there is no middle course.
5.154 In an urn there are equal numbers of white and black balls (and no others). I draw on ball after another and put them back in the urn. Then I can determine by the experiment that the numbers of the black and white balls which are drawn approximate as the drawing continues.
So this is not a mathematical fact.
If then, I say, It is equally probable that I should d raw a white and a black ball, this means, All the circumstances known to me (including the natural laws hypothetically assumed) give to the occurrence of the one event no more probability than to the occurrence of the other. That is they give - as can easily be understood from the above explanations - to each the probability ½.
What I can verify by the experiment is that the occurrence of the two events is independent of the circumstances with which I have no closer acquaintance.
5.155 The unit of the probability proposition is: The circumstances - with which I am not further acquainted - give to the occurrence of a definite event such and such a degree of probability.
5.156 Probability is a generalization.
It involves a general description of a propositional form.
Only in default of certainty do we need probability. If we are not completely acquainted with a fact, but know something about its form.
(A proposition can, indeed, be an incomplete picture of a certain state of affairs, but it is always a complete picture.)
The probability proposition is, as it were, an extract from other propositions.