No-one has asked for any hints for Problem Sheet 6, so here instead are some remarks about implication. Let and be mathematical statements. By definition means: ‘If is true, then is true’.
Suppose that is true. It should be clear from the definition that is true if is true, and false if is false. However there is one point that may seem unusual. Let be the proposition ‘There are infinitely many primes’ and let be the proposition ‘‘. Then since and are both true, is true. But surely no-one would say that it is because there are infinitely many primes that .
This example shows that there are ways to use that, while logically correct, do not help to show the chain of logic in an argument. In practice when mathematicians write they mean ‘if is true then there is a fairly simple reason why is true’. If you use 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 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 to mean ‘there is a fairly simple reason why is true if is true and is true if is true’. This agrees with the uses of in Exercise 5.2 and Example 5.3.
Now suppose that is false. In the lecture, we agreed that if is also false than should be true, since is logically equivalent to the contrapositive , and we know that a true statement implies a true statement. The trickier case is when is false and is true. Then 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 .
Remember means ‘if is true, then is true’. If is false, there is no restriction on at all! Therefore if is false then 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 is not, in general, logically equivalent to the convere . 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.