Validity of conditional statement when the premise is false.
Is it just for convenience that mathematicians define that false statements imply anything?
If yes, why it would be defined like this?
Solution 1:
Well, a big reason that we chose this convention is that we are sort of short on options:
- We could say that a false premise implies that the implication is true exactly when the conclusion is true, but that would be odd because then the premise doesn't do anything.
- We could say that a false premise implies the implication is true exactly when the conclusion is false, but eww.
- We could say that a false premise implies the implication is always true, which is what most people who think about the alternatives do.
- We could say that a false premise implies the implication is always false, but if we do this then $p\to q$ has the same truth table as $p\wedge q$, which isn't bad but it seems like there is something more that we want out of an implication than a simple 'and' statement.
- We could reject the law of the excluded middle so that the implication is neither true nor false. This turns out to be a valid option, but also eww. More practical reasons to dislike this is that it destroys double-negation $ (\sim\sim\! p$ is $p)$ and therefore contrapositive and contradiction proofs.
So you have to pick one. They're a sorry lot, I admit, but we're stuck with them, and one has proven to be more realistic and pragmatic than the others.
Solution 2:
It is not entirely clear from your question what you mean so I'm adding this answer too, just for the sake of completeness. I think you might be referring to the fact that in classical logic a contradiction implies any sentence. This is not the same as saying that a false statement implies anything. That is actually not true. If $P$ is false then $P\implies Q$ is a true statement but one can't conclude that $Q$ is true.
If, however, $P$ is both true and false (i.e., is a contradiction) then since $P\implies Q$ is true one may now conclude by Modes Ponens that $Q$ is true, and thus a contradiction proves any statement. This is called the explosion principle.
There are several reasons to feel somewhat not at ease with this principle and there are ways to exclude this principle and retain a very useful logical system that is tolerant to contradictions, without rendering the entire system useless. Such logics are called paraconsistent. A nice expository article on paraconsistency can be found here: http://plus.maths.org/content/not-carrot