The English language is full of ambiguities. The humorous quote above exploits this and works because the word ‘something’ has two different meanings in the sentence. With mathematical language we aim to remove such ambiguities. In this chapter we will first look at a particular problem that can occur in everyday usage with the use of the if/then structure. We shall see how this example leads us to consider what are called the inverse statement and the contrapositive statement. This shows that although using everyday examples can be very illuminating we do have to be careful as the English language can play tricks on us.

The inverse: a common mistake

The most common initial mistake that people make in logic arises from an everyday usage we learn at a young age. For example, an adult says to a child

‘If you don't tidy your room, then you won't get ice-cream.’

Our instinct, like the child and parent, is to interpret this as a contract: if the child tidies their room, then they get ice-cream. In other words

‘If you tidy your room, then you will get ice-cream.’

Yet the original statement does not say that. It only says what will happen if the child does not tidy their room.