# Can we change the premise by using the contrapositive?

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 $$\neg A$$.

• If $A$ implies $Q$ and $B$, and you can show that at least one of $Q$ and $B$ is false, then clearly $A$ must be false. So you missed out the "not". Also it should be "deduce" not "deduct" (which means subtract). Commented Sep 22, 2019 at 5:50

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.

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.