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At first it would seem that if $p{\implies}q$ means "$p$ implies $q$", then if $p$ is false then the entire statement doesn't make sense. It looks like if we have no way of knowing whether $p$ implies $q$ or not, since $p$ was never a real thing in the first place. It would seem that the statement wouldn't have either "true" or "false" value, but rather "undecided".

Why did mathematicians decide to give this statement a "false" value?

cxzcxzcas
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  • Also see http://math.stackexchange.com/questions/70736/ – GEdgar Jun 23 '16 at 19:20
  • The meaning of the implication is that $T\implies F$ is excluded. The other combinations raise no objection, hence the expression is true, and this is harmless. Adding a third logical value would be extremely costly. –  Jun 23 '16 at 19:33

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It is a special case of something called "vacuous truth", and it makes a lot more sense if you slightly change how you think about $p \implies q$.

For instance, look at the statement

$\forall$ persons$($That person is from Holoshandra$\implies$That person speaks fluently English$)$

You could interpret it litterally, which would make the statement nonsensical, since there is no such place as Holoshandra. I choose to look at it as a promise. The statement can be rephrased into "Show me a person from Holoshandra, and I promise you that he speaks fluent English". That is a promise I am guaranteed to be able to keep (you can't give me a counterexample; there is no one from Holoshandra that doesn't speak English), and thus the statement is considered true.

Also consider the contrapositive:

$\forall$ persons$($That person doesn't speak English$\implies$That person is not from Holoshandra$)$

which is more intuitively obvious, no matter what you mean by $\implies$.

Arthur
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