Contradiction is the external limit of the propositions, tautology their substanceless centre.
If Tr is the number of the truth-grounds of the proposition “r”, Trs 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 Trs:Tr the measure of the probability which the proposition “r” gives to the proposition “s”.
Suppose in a schema 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”’s of the proposition r; then the proposition r gives to the proposition s the probability Trs:Tr.
There is no special object peculiar to probability propositions.
Propositions which have no truth-arguments in common with one another we call independent.
Independent propositions (e.g. any 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.)
A proposition is in itself neither probable nor improbable. An event occurs or does not occur, there is no middle course.
In an urn there are equal numbers of white and black balls (and no others). I draw one 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 draw 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.
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.
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.
The structures of propositions stand to one another in internal relations.
We can bring out these internal relations in our manner of expression, by presenting a proposition as the result of an operation which produces it from other propositions (the bases of the operation).
The operation is the expression of a relation between the structures of its result and its bases.
The operation is that which must happen to a proposition in order to make another out of it.
And that will naturally depend on their formal properties, on the internal similarity of their forms.
The internal relation which orders a series is equivalent to the operation by which one term arises from another.