5.14 If one proposition follows from another, then the latter says more than the former, and the former less than the latter.

5.141 If p follows from q and q from p, then they are one and same proposition.

5.142 A tautology follows from all propositions:  it says nothing.

5.143 Contradiction is that common factor of propositions which no proposition has in common with another.  Tautology is the common factor of all propositions that have nothing in common with one another.  Contradiction, one might say, vanishes outside all propositions:  tautology vanishes inside them.  Contradiction is the outer limit of propositions:  tautology is the unsubstantial point at their centre.

5.15 If Tr is the number of the truth-grounds of a proposition ‘r’, and if Trs is the number of the truth-grounds of a proposition ‘s’ that are at the same time truth-grounds of ‘r’, then we call the ratio Trs :  Tr the degree of probability that the proposition ‘r’ gives to the proposition ‘s’. 5.151 In a schema like the one above in

5.101, let Tr be the number of ‘T’s’ in the proposition r, and let Trs, be the number of ‘T’s’ in the proposition s that stand in columns in which the proposition r has ‘T’s’.  Then the proposition r gives to the proposition s the probability Trs :  Tr.

5.1511 There is no special object peculiar to probability propositions.

5.152 When propositions have no truth-arguments in common with one another, we call them independent of one another.  Two elementary propositions give one another the probability 1/2.  If p follows from q, then the proposition ‘q’ gives to the proposition ‘p’ the probability 1.  The certainty of logical inference is a limiting case of probability. (Application of this to tautology and contradiction.)

5.153 In itself, a proposition is neither probable nor improbable.  Either an event occurs or it does not:  there is no middle way.

5.154 Suppose that an urn contains black and white balls in equal numbers (and none of any other kind).  I draw one ball after another, putting them back into the urn.  By this experiment I can establish that the number of black balls drawn and the number of white balls drawn approximate to one another as the draw continues.  So this is not a mathematical truth.  Now, if I say, ’The probability of my drawing a white ball is equal to the probability of my drawing a black one’, this means that 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 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 is independent of the circumstances of which I have no more detailed knowledge.

5.155 The minimal unit for a probability proposition is this:  The circumstances—­of which I have no further knowledge—­give such and such a degree of probability to the occurrence of a particular event.