## Mathematical implication

No-one has asked for any hints for Problem Sheet 6, so here instead are some remarks about implication. Let $A$ and $B$ be mathematical statements. By definition $A \Rightarrow B$ means: ‘If $A$ is true, then $B$ is true’.

Suppose that $A$ is true. It should be clear from the definition that $A \Rightarrow B$ is true if $B$ is true, and false if $B$ is false. However there is one point that may seem unusual. Let $A$ be the proposition ‘There are infinitely many primes’ and let $B$ be the proposition ‘$2+2 = 4$‘. Then since $A$ and $B$ are both true, $A \Rightarrow B$ is true. But surely no-one would say that it is because there are infinitely many primes that $2+2 = 4$.

This example shows that there are ways to use $\Rightarrow$ that, while logically correct, do not help to show the chain of logic in an argument. In practice when mathematicians write $A \Rightarrow B$ they mean ‘if $A$ is true then there is a fairly simple reason why $B$ is true’. If you use $\Rightarrow$ in this way your arguments will be clearer and easier to read.

(Of course what is ‘fairly simple’ to one person might seem completely non-obvious to another, so how $\Rightarrow$ is used in practice also depends on the writer and his or her intended readership. When writing answers, I suggest you imagine the reader is one of your friends who is also taking the course, or yourself in six months time.)

Similarly, you should write $A \Longleftrightarrow B$ to mean ‘there is a fairly simple reason why $A$ is true if $B$ is true and $B$ is true if $A$ is true’. This agrees with the uses of $\Longleftrightarrow$ in Exercise 5.2 and Example 5.3.

Now suppose that $A$ is false. In the lecture, we agreed that if $B$ is also false than $A \Rightarrow B$ should be true, since $A \Rightarrow B$ is logically equivalent to the contrapositive $\neg B \Rightarrow \neg A$, and we know that a true statement implies a true statement. The trickier case is when $A$ is false and $B$ is true. Then $A \Rightarrow B$ is true. I was running out of time in the Monday lecture, so I said you could regard this as a convention. This was a bit lazy: really it follows from the definition of $\Rightarrow$.

Remember $A \Rightarrow B$ means ‘if $A$ is true, then $B$ is true’. If $A$ is false, there is no restriction on $B$ at all! Therefore if $A$ is false then $A \Rightarrow B$ is always true. So a false statement implies anything. This has a parallel in everyday language, where we extrapolate an obviously false statement from a statement we believe is most likely false: ‘if everyone taking MT181 completely understands implication then pigs can fly’.

Another argument that false statement imply true statements uses the converse: we know that $A \Rightarrow B$ is not, in general, logically equivalent to the convere $B \Rightarrow A$. So the truth tables must differ. Since it is uncontroversial that true statements do not imply false statements, it must be that false statements do imply true statements.

Finally: while searching the web for variants on the expression ‘I’m the Queen of Sheba’, or ‘pigs can fly’ I found a Blog post by Timothy Gowers that discusses all the issues above much more carefully. You might find the comments make interesting reading.