Conditional sentences

Question

Give an example of a true conditional sentence that has a false contrapositive.

I looked up the definition of these words but I still have no idea how to do this.

Answer 1

First, let’s take a look of some basic definitions concerning your question.

Let $P$ and $Q$ be statements. We define the conditional from $P$ to $Q$ (denoted by $P \to Q$) as the statement that is true if and only if it is never the case that $P$ is true and $Q$ is false, and it is false if this case happens.

As a variant of the statement $P \to Q$, we have the following.

Let $P$ and $Q$ be statements, and consider the conditional $P \to Q.$ The statement $\neg Q \to \neg P$ is said to be the contrapositive of $P \to Q.$

There is a great result in Propositional Logic that states the following.

$$P \to Q \iff \neg Q \to \neg P$$

(You can prove this equivalence, by showing that the statement $(P \to Q) \leftrightarrow (\neg Q \to \neg P)$ is a tautology, which can be accomplished using a truth table.)

This result tell us that $P \to Q$ is true whenever $\neg Q \to \neg P$ is true and $P \to Q$ is false whenever $\neg Q \to \neg P$ is false.

Therefore, a conditional statement and its contrapositive behave in the exactly same way.

It follows that it is impossible to have a conditional statement that happens to be true and its contrapositive happens to be false. (That would be a contradiction.)