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Explicitly teach the contrapositive

The contrapositive is a a crucial part of logic but is often overlooked. For a statement of the form A\implies B the contrapositive is the statement “not B \implies not A“. For example, the contrapositive of “If I am Winston Churchill, then I am English” is “If I am not English, then I am not Winston Churchill”. Put like that it is clear that both statements are true. In fact, the contrapositive and the original statement are equivalent.

This is is of course extremely useful as a method of proof. Want to prove some assumptions imply that some number is non-zero? Just assume the number is zero and proceed directly to show that the assumptions are false. It’s easier to use equalities than equalities in equations and so assuming the number is zero is much more tractable.

Hence, we would like our students to understand the contrapositive and use it in their work. There are, however, at least three problems with the contrapositive.

1. Lecturers often learnt the contrapositive implicitly from repeated exposure as students. Colleagues have even told me that they didn’t know that the concept had a name — it was something they `just know’. Their contemporaries who didn’t pick it up implicitly didn’t become lecturers and so the cycle is repeated with students not explicitly taught about the contrapositive. I believe we should give the concept a name and teach it. (Once you name it you can tame it!)

2. The contrapositive is confused with the method of contradiction (both begin with contra — that doesn’t help). The example above about non-zero numbers looks very like a proof by contradiction. (I don’t believe it is as we are not assuming at the same time that the assumptions are true and the conclusion is false.) Nonetheless, there is a very close connection between contrapositive and contradiction. For example, the proof that the contrapositive is equivalent to the original statement relies on contradiction. Explicitly naming both methods gives the opportunity for them to be distinguished in a student’s mind.

3. The contrapositive is counterintuitive. It doesn’t feel like a statement and its contrapositive should be logically equivalent. I shall discuss the consequences of this tomorrow with the Wason Test.

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