The Contrapositive (and Modus Tollens)

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Subsection 8.3.7 The Contrapositive (and Modus Tollens)

Now suppose that we know that some implication of this form is true:

p → q

Then its contrapositive must also be true and we have:

¬q → ¬p

So, if we know ¬q, the contrapositive gives us a straightforward way to derive ¬p using Modus Ponens. Of course, an alternative is to use the original form (p → q) and Modus Tollens, but that’s not always quite so obvious a thing to do.

Suppose that we have that all college students are literate. Then we can prove that Morgan isn’t a college student if we can show that Morgan isn’t literate.

Another way to think of a contrapositive proof is that it is a proof by contradiction. We are given p → q. We want to prove:

¬p

We assume its contradiction, p. We can immediately derive q. If we can then reason and derive ¬q, we have found a contradiction that proves ¬p.