That the truth of one proposition follows from the truth of other propositions, we perceive from the structure of the propositions.
If the truth of one proposition follows from the truth of others, this expresses itself in relations in which the forms of these propositions stand to one another, and we do not need to put them in these relations first by connecting them with one another in a proposition; for these relations are internal, and exist as soon as, and by the very fact that, the propositions exist.
5.1311
When we conclude from p v q and ~p to q the relation between the forms of the propositions "p v q" and "~p" is here concealed by the method of symbolizing. But if we write, e.g. instead of "p v q" "p | q .|. p | q" and instead of "~p" "p | p" (p | q = neither p nor q), then the inner connection becomes obvious.
(The fact that we can infer fa from (x) . fx shows that generality is present also in the symbol "(x) . fx".
5.132
If p follows from q, I can conclude from q to p; infer p from q.
The method of inference is to be understood from the two propositions alone.
Only they themselves can justify the inference.
Laws of inference, which -- as in Frege and Russell -- are to justify the conclusions, are senseless and would be superfluous.
5.133
All inference takes place a priori.
From an elementary proposition no other can be inferred.
5.135
In no way can an inference be made from the existence of one state of affairs to the existence of another entirely different from it.
5.136
There is no causal nexus which justifies such an inference.
The events of the future cannot be inferred from those of the present.
Superstition is the belief in the causal nexus.
5.1362
The freedom of the will consists in the fact that future actions cannot be known now. We could only know them if causality were an inner necessity, like that of logical deduction. -- The connexion of knowledge and what is known is that of logical necessity.
("A knows that p is the case" is senseless if p is a tautology.)
5.1363
If from the fact that a proposition is obvious to us it does not follow that it is true, then obviousness is no justification for our belief in its truth.
