5.132 If p follows from q then I can infer from q to p; deduce p from q.

The nature of the inference is to be gathered only from the two propositions.

Only they themselves can justify the inference.

“Laws of inference” that – as in Frege and Russell – are supposed to justify inferences, are senseless [sinnlos], and would be superfluous.

Some clarification of what ‘sinnlos’ means for Wittgenstein here. The laws of logic cannot be put into words [cf. On Certainty?]. The attempt to do so is useless and senseless, if this is a different thing.

Ostrow (p. 111) suggests that Wittgenstein’s concern here and elsewhere in the TLP is primarily “to shift our perspective so that we no longer feel any urge to account for why, for example, “q” follows from “p” and “p → q” in the first place.”

Proops points out (pp. 80-86) that Frege and Russell seem to use “laws of inference” to mean laws of logic, including axioms, not just the rules for inference in a particular axiom system. As evidence he cites Frege’s “Foundations of Geometry: First Series” in his Collected Papers p. 319, and Russell’s (1905) “Necessity and Possibility” in his Collected Papers Volume IV, p. 515. Proops takes Wittgenstein to think that (p. 90): “far from expressing truths which lie at the bottom of all valid inferences, the laws of logic are to be viewed as expressing no facts of any kind. Secondly, even if Russell’s conception of logic were correct, even if, that is to say, the laws of logic were not sinnlos, the appeal to logical laws in the explanation of valid inference would in any case be superfluous.” The concept of entailment cannot be explained in other terms, including those of derivability in a sound system.