“p.q” is one of the propositions which assert “p” and at the same time one of the propositions which assert “q”.
Two propositions are opposed to one another if there is no significant proposition which asserts them both.
Every proposition which contradicts another, denies it.
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.
When we conclude from p∨q and ~p to q the relation between the forms of the propositions “p∨q” and “~p” is here concealed by the method of symbolizing. But if we write, e.g. instead of “p∨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”.
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.
All inference takes place a priori.
From an elementary proposition no other can be inferred.
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.
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.
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 connection 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.)
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.
If a proposition follows from another, then the latter says more than the former, the former less than the latter.
If p follows from q and q from p then they are one and the same proposition.
A tautology follows from all propositions: it says nothing.
Contradiction is something shared by propositions, which no proposition has in common with another. Tautology is that which is shared by all propositions, which have nothing in common with one another.
Contradiction vanishes so to speak outside, tautology inside all propositions.