Implications and Converses (If and Only If)

ON BROCCOLI AND ICE CREAM

One of my favourite examples of implications and converses is “If you eat your broccoli you can have ice cream.” I will admit up front that this gets quite confusing in normal language, which is why mathematicians quickly prefer using letters and symbols to keep things clear. But let’s try with words.

A logical, literal child might start asking what other foods will imply ice cream, if they really want to avoid broccoli. Here their precision comes across as pedantry to the adult, who might say exasperatedly “You know what I mean!”, but the child is just seeking clarity and trying to find a loophole to avoid eating broccoli. (I was not that child – I have always loved broccoli. Perhaps because it was never used as a threat to me. Or maybe it was never used as a threat because I loved it.)

The child might say “How about if I eat some fish instead?” to which the answer might be “No, you have to eat your broccoli!” or “No, you get ice cream only if you eat your broccoli!” These are both examples of a converse, but it’s a bit hard to see with the broccoli example because it’s not really a logical implication – more of a bribe.

Here are the two statements the parent made. First they said

If you eat your broccoli you can have ice cream.

broccoli ⇒ ice cream

which guarantees the child that broccoli leads to ice cream. It says that if the child eats broccoli that is sufficient for earning ice cream.

The parent then said

You can eat ice cream only if you eat broccoli.

ice cream ⇒ broccoli

which guarantees the parent that the child can’t sit and try to find other ways of earning ice cream. It says that broccoli is necessary for earning ice cream, and there is no way round it. It is the converse of the first statement. (If the direction of this arrow looks strange, you can think of it as saying that if we subsequently see the child eating ice cream we can logically deduce that they must have eaten their broccoli.)

All this is to explain why “only if” is a way of expressing the converse of “if” – logic is flowing in the opposite direction. To ensure both the guarantee to the child and the guarantee to the parent, the promise technically needs to be “You can have ice cream if and only if you eat your broccoli”. The trouble is that probably only a rather pedantic mathematician would bother saying that, so we grow up with the vague sense that “only if” means the same as “if and only if”. Making the distinction in normal language is probably pedantic because it oversteps the goal of clarification. The trouble is that not making the distinction causes confusion for people when they do start thinking about logic formally. This can lead to much more serious consequences in more serious situations.

Imagine you’re trying to catch a group of bank robbers and you know the whole gang was white men. So you know

If someone you encounter is in that gang then they are a white man.

This is equivalent to

Someone you encounter can only be in that gang if they are a white man.

So we can start by looking for white men. But finding a white man does not ensure that we have found a criminal, because the converse isn’t true. The converse would be

If someone you encounter is a white man then they are in the gang.

Being a white man is a necessary condition for being in the gang, but not sufficient.

All this is very prone to get confusing and make your head spin in normal language, which is one of the reasons mathematicians reduce things to letters and symbols, because it can be easier to see patterns. Using arrows we have:

• True: gang ⇒ white
• False: white ⇒ gang