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.


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