How to explain that $A \implies B$ is true when $A$ is false I'm teaching my little sister propositional logic per her request.
I was trying to explain to her why $A \implies B$ holds whenever $A$ is false, and I didn't succeed with that.
I referred her to the definition: $A \implies B$ is true if whenever $A$ true, $B$ is true. If $A$ is false, then it is "never" true. If we want to test whether the implication is true, we need to check if $B$ follows when $A$ is true, but $A$ is never true, so we need not check anything, hence the implication is true. 
Is my reasoning even correct? I forgot how to do these stuff anyway. 
Please help me explain this idea. It would be nice if you can give an intuitive way to think about it as well. 
 A: Yes, your reasoning is correct. It may help to present a situation close to the student's range of interests. Ask her to say "if I steal, then I don't get a cookie". Then ask her if she stole anything, to which, presumably, she will answer "no, I did not steal". Now give her a cookie and accuse her of lying in the first place when she said "if I steal, then I don't get a cookie" and have her defend her honour. Repeat the process, but this time do not give her a cookie and accuse her again of lying. If she gets convinced that the claim "if I steal, then I don't get a cookie" is not a lie in the situation where she did not steal, then she must accept the statement is true (regardless of whether or not she has a cookie). This is precisely the situation of $A\implies B$ where $A$ is "I steal" and $B$ is "I don't get a cookie", which is true if $A$ is false. 
A: Assuming your kitchen is entirely within your house you can make the true statement "When you are in the kitchen you are in your house" - clearly it is still true even if you are not in the kitchen, the statement isn't saying anything about where you are, it's all about the fact that the kitchen is in your house.
This could lead to a discussion around Venn diagrams and amusing flights of fancy about what it would be like to live in a house where the kitchen was partly inside and partly outside or even entirely outside your house ( which probably is actually the case for a lot of people.)
A: You can think of $A \Rightarrow B$ as a promise; if you do A, then you'll get B.
BUT if you don't do A, I'm not breaking my promise. 
For example, the gravity promises you that if you drop and object, then it will fail. 
But what if I dont drop it? Well, the gravity isn't breaking his promise. The only way that gravity will break his promise is if you drop and object and it doesn't fall.
A: When you write $A \implies B$, you mean "$A$ implies $B$". And you want to say that $A$ does not imply $B$ only if $A$ is true and $B$ isn't. Meaning that "$A \implies B$" is false only if $A$ is true and $B$ is false. If $A$ is false, we're not in this situation, so automatically $A \implies B$ is true.
A: In a not very rigourous, but perhaps expressive way, I would say $A\Rightarrow B$ is true when $A$ is false because you cannot produce a counter-example, i.e. a situation in which $A$ is true and $B$ false.
A: You could try making some statements involving something that is
obviously false. For example, pick up a spoon made entirely of metal, and say,
"If this spoon is made of wood, then I'm a monkey's uncle."
Or say,
"If this spoon is made of wood, then I'll eat my hat."
People say things like this sometimes specifically to emphasize that
the part before "then" is false.
That is, while both of these statements use the language of logical implication,
$A \implies B$,
what the speaker of such a sentence is really trying to communicate
is that $A$ is false.
After all, if I thought it were possible that the metal spoon was actually
made of wood, I would not want to say the first sentence, because I do
not want people to think I'm a monkey's uncle;
and likewise I would not utter the second sentence, because I certainly
do not want to risk having to eat my hat!
And while the mathematical logic of implication does not always work the
same way as colloquial speech, in this case it does.
My knowledge that the spoon is not made of wood allows me to safely say
that anything at all is implied if the spoon is wooden;
I know I can never be held to account for having said the $B$ part
of $A \implies B$, because the $A$ part will never be true.
By the way, another example of the "monkey's uncle" argument appears in
this answer to a related question.
