What is implied by "implies"?

When lecturing recently to first-term undergraduates I asked them to vote on whether the following statement, concerning an unknown positive integer n, was true or false:

if n is both even and odd then n=17. (*)

An overwhelming majority voted that it is false. Moreover, when I tried to explain the normal mathematical convention by which it would come out true (since anything follows from a false premise) I met with considerable resistance. Surely if that statement is valid then so is the same statement with 19 instead. But n can't be both 17 and 19! Yes, I replied, but it can't be both even and odd either. Nevertheless, many people left the lecture dissatisfied, so let me try to say a bit more about mathematical implication and why the standard convention about "silly" sentences such as the example above is a good convention.

First, here is an argument I gave in the lecture. If you think that statement (*) is false then you ought to be able to provide a counterexample. What constitutes a counterexample to an if-then statement? An example where the if part is true and the then part is false. In our case we would need to provide a positive integer n that is both even and odd and yet not equal to 17. Well, we obviously can't.

Why do some people not feel convinced by this argument? It is because the mathematical use of "if ... then ..." is importantly different from the everyday use. Here are five if-then statements, all of which are somewhat peculiar even though they count as true in the mathematical sense (because they do not have a true premise followed by a false conclusion).

1. If Mount Everest shoots up into the air and lands in the Indian Ocean tomorrow, then there will be the mother of all tidal waves.

2. If unicorns exist, then our biology textbooks are seriously flawed.

3. If unicorns exist, then mice do not exist.

4. If n is both even and odd, then n is certainly even.

5. If Cambridge is in East Anglia then Paris is the capital of France.

These are my instinctive reactions to the five sentences. I hope my use of the English language is typical enough, even after years of doing mathematics, for the reactions to be representative.

1. Seems true, even if the premise is grotesquely unlikely.

2. Seems true, even though in this case the premise is false. Why? Because I can imagine the premise being true and then the consequence is a reasonable one.

3. A much more objectionable sentence. Why should unicorns and mice be incompatible? So even though the premise is false, I don't like the sentence. To put it another way, if unicorns did exist, then there is no particular reason that mice should not.

4. This conclusion seems a lot more reasonable than the conclusion that n=17, because you can see how the the evenness of n follows from the premise (if n has two properties then it has each property individually). Of course, I could equally have concluded that n is odd, but that isn't as troubling as the n=19 objection because I've actually said that n is even, whereas 17 comes out of the blue. So perhaps what's going on in my mind is this. If I imagine a number n that is both even and odd - temporarily forgetting that the two properties are incompatible - then I can certainly conclude that n is even and that n is odd, but I can't conclude something more specific such as that n=17.

5. This seems objectionable but also harmless. If somebody said it I would want to reply that Cambridge is indeed in East Anglia and Paris is indeed the capital of France, but the second fact has nothing to do with the first. The implication sounds a bit odd because it suggests that there is a connection.

These examples point to an interesting phenomenon: our use of "if ... then ..." in everyday life is intimately bound up with our ideas of causality, and this transfers to our thoughts about mathematics, making us find some implications (in the strict, truth-table sense) strange and counterintuitive and others completely natural. A related phenomenon is that we tend to blur in our minds the distinction between a straightforward conditional:

if P is true then Q is true

and a counterfactual conditional:

if P were true then Q would be true.

See for example my discussions of sentences 2, 3 and 4 above. This gives a very clear explanation for our dislike of the original statement (*). Suppose we replaced it with the following counterfactual variant:

if n were both even and odd then n would equal 17.

Now we are forced to consider a hypothetical situation where a number actually is both even and odd. Of course, this situation is impossible, but if we are forced to imagine it then we can - by ignoring some inconvenient details such as the exact meanings of the words "even" and "odd". Perhaps what we imagine is a strange planet where the mathematicians happily accept the existence of numbers that are both even and odd. Clearly on such a planet, they would agree that a number with this property was necessarily even, but there is no reason to suppose that they would agree that it had to be 17.

Now let me complicate the discussion further by giving an argument that I wish I had thought of in lectures. Let E be the set of all even positive integers and O the set of all odd ones. What is E intersect O? Obviously the empty set. Does it follow that E intersect O is a subset of {17}? Yes, of course it does, as the empty set is a subset of any set. But what does it mean for a set A to be a subset of another set B? It means that every element of A is an element of B. Is that true in our case? We would need every element of E intersect O to be an element of {17}. When is n an element of E intersect O? Answer: when n is even and n is odd. When is n an element of {17}? Answer: when n=17. So the statement that E intersect O is a subset of {17} is exactly equivalent to (*). And yet, for some reason it seems much more palatable.

This presents us with another psychological puzzle. How can it be that statement (*) is so peculiar, while another statement that says exactly the same thing is not peculiar at all?

To answer this, let us think a bit more about the seemingly obvious fact that if A is a set then the empty set is one of its subsets. To prove that, we need to establish the following:

every element of the empty set is an element of A. (+)

What is our instinctive reaction to that? Mine is to say, "Yes, that's true because there aren't any elements of the empty set, so those that there are " (and here I might give a little apologetic laugh) "are elements of A."

Now let us look at a different formulation of what we need.

If x is an element of the empty set, then x is an element of A.

I think it is possible to feel this sentence as reasonable or unreasonable. It seems reasonable if you say to yourself that it means exactly the same as (+) and unreasonable if you think of it in a more counterfactual way, as saying something like

if x were an element of the empty set then it would be an element of A.

This last sentence is another one that forces us to try to imagine an impossible situation (this time forgetting that the empty set has to be, well, empty).

Here are four more sentences to consider.

6. Every element of the empty set is an element of the set {17}.

7. If n is an element of the empty set, then n is an element of the set {17}.

8. Every element of the empty set equals 17.

9. If n is an element of the empty set, then n=17.

Of these, I find 6 the most acceptable and 9 very much the least. 7 ought to be troubling but isn't too bad because it feels as though it is saying much the same as 6 (that is, I'm not tempted to go couterfactual) while 9 does begin to make me imagine a world in which the empty set had elements. So, presumably, the reason it seems all right that E intersect O is a subset of {17} is that in our minds we feel it as more like 6 than like 9. We don't feel any impulse to say to ourselves, "Yes, but what would happen if E intersect O did contain a number n?" It is empty, and that is all there is to it.

What I hope all this illustrates, at the very least, is that ordinary language is full of odd subtleties of interpretation and would therefore be hopeless for use in a mathematical context. That is why it is better to allow a few peculiar mathematical sentences to count as true.

Two final thoughts. First, even professional mathematicians wouldn't go so far as to say that Fermat's Last Theorem is a consequence of the Riemann hypothesis (on the grounds that FLT is true, and therefore the statement RH => FLT is true). Second, here's another puzzle. Suppose I tell you that n is a positive integer. What do you make of the following sentence?

If n=n+1 then n=n-1.

Obviously in the strict mathematical sense the implication is valid, as the premise is always false. But what about in the subjective ordinary-language sense? There I want to say that just because n equals n plus one, it doesn't follow that it also equals n minus one. And yet there is a powerful argument that it does: subtract one from both sides. If you share my feeling but can't be bothered to think for yourself why, then here is my suggested explanation.