Can we change the premise by using the contrapositive?

Question

Can we change the premise by using the contrapositive?

For example, if we have $A \to (Q \land B)$. Say that $\neg (Q \land B) \to \neg A$, then start with the premise $\neg (Q \land B)$ and try to deduce $\neg A$.

Answer 1

Yes, $A \implies (Q \land B)$ and $\lnot (Q \land B) \implies \lnot A$ are equivalent statements. And by de Morgan this is further equivalent to $(\lnot Q \lor \lnot B) \implies \lnot A$. So yes.

But I cannot think of a situation where I'd prove a concrete statement that way.

In a formal deduction system you'd have some extra steps with this proof route:

• You could start with an assumption $A$.
• Then a subassumption $\lnot (Q \land B)$.
• deduce $\lnot A$ somehow.
• this contradicts the first assumption so we deduced $\lnot \lnot (Q \land B)$ and withdraw the subassumption (introduction rule for $\lnot$).
• deduce $Q \land B$ by the double negation rule.

Answer 2

If you wanna prove a conditional $A \rightarrow B$ by proving the conditional $\neg B \rightarrow \neg A$, then what you wanna do is what is called a proof by contrapositive. This is a proof strategy very common among mathematicians, as Keith Devlin explain in his course Introduction to Mathematical Thinking.