# Contrapositive grammar

We had students the other day from Maths for Information Technology and their task was to form the contrapositive of a several statements. Given a particular statement of the form “If A, then B”, the contrapositive is “If not B, then not A”, so mathematically the problem is not actually very difficult. However grammatically the problem is much harder than it looks.

Consider this statement: “If it is raining, then there are clouds.” If we compare this to my generic example above, we see that A is “it is raining” and B is “there are clouds”, so by my own rule, the contrapositive ought to be this: “If not there are clouds, then not it is raining.” This is obviously not a grammatical English sentence! A correct version is, “If there are not clouds, then it is not raining.” By giving people a generic rule, we are getting them into trouble with their grammar. This may seem like a small thing, but there are plenty of students for whom English is not their first language, and even students whose first language is English often don’t have a very good command of the rules of grammar!

There’s no easy way around this in the way we present these generic rules, except to make them aware that they need to think about the grammar of the sentence that they write when they do this, in particular, where the word “not” has to go.

But it gets worse! Consider the statement “If f: R -> R is differentiable, then f is continuous”. According to my above rule, A is “f:R -> R is differentiable” and B is “f is continuous”, so taking into account the grammatical placement of the word not, we get the contrapositive is “If f is not continuous, then f:R ->R is not differentiable.” The students we worked with did this exact thing, and they could tell there was something odd about it, but they couldn’t quite figure out what it was.

The problem is that the part saying “:R -> R” is not technically part of the if-then construction. It could have been stated in a completely different sentence like this: “Let f: R -> R. If f is differentiable, then f is continous.” Then that lead-in sentence isn’t an if-then construction, so it isn’t part of the contrapositive.