One of the blog posts on the new Philosopher's Carnival considers how to teach the truth-table for the material conditional to students:
I tried to comment, but it didn't work for some reason. Here's how I explain the troublesome cases (where P is false, but (P -> Q) is true).
Suppose Q is true, and P false. All we really need is that Q is true:
1. Q / Q
Monotonicity gets us:
2. P, Q / Q
And then by conditional proof we derive:
3. Q / P -> Q
One final application of monotonicity:
4. ~P, Q / P -> Q
So to reject (P -> Q) when P is false an Q true, the student must reject either conditional proof or monotonicity.
Now suppose P is false and Q is also false. All we really need is that P is false:
1. ~P / ~P
Again, by monotonicity, and conditional proof:
2. ~Q, ~P / ~P
3. ~P / ~Q -> ~P
Here there are several ways to proceed (contraposition being the most obvious). Let's try Modus Tollens:
4. P / P (assumption for CP)
5. P / ~~P (double negation, 4)
6. P, ~P / ~~Q (modus tollens, 3, 5)
7. P, ~P / Q (double negation, 6)
8. ~P / P -> Q (conditional proof, 7)
9. ~P, ~Q / P -> Q (monotonicity, 8)
So to reject (P -> Q) when P and Q are both false, the student must reject conditional proof, double negation, monotonicity, or modus tollens (alternatively, the student must reject conditional proof, monotonicity, or contraposition).
Of course, as pointed out last time, modus tollens, contraposition, disjunctive syllogism, and reductio are all inter-derivable. So my explanation is really of the form: everything follows from a false premise, so get used to it.