Can we always derive modus ponens from modus tollens, and vice versa? I have a couple of questions about the distinction between the so-called "basic rules" and the so-called "derived rules" in logic. I have been told that there is nothing substantial about this distinction, i.e., that it is a mere convention: for instance, modus ponens is usually regarded as a basic rule, whereas modus tollens is generally regarded as a derived rule, but actually one can derive modus ponens from modus tollens, with suitable additional rule. I have two questions: (1) how exactly you derive modus ponens from modus tollens in propositional logic? (2) Is it always the case (i.e., is it true for any logic) that modus ponens and modus tollens can derive each other or is this only true for propositional logic?


How exactly you derive modus ponens from modus tollens in
propositional logic?

You will need the equivalence: $$P \to Q ~~\equiv ~~ \neg [P \land \neg Q]$$
Using a form of natural deduction, here is a proof by contradiction. (Screenshot from my proof checker)


