Logic - clarification needed about implication

autodidude

If P→Q, and P is false but Q is true, then why is P→Q true? To me, it seems as though we shouldn't be able to do proceed because there isn't enough information. Same goes when P and Q are both false, how does that suggest P→Q is true?

mfb

"If it rains, the street gets wet"
This statement is true, even if I spill water on the street (without rain).
More general: It cannot be false, if it does not rain. It just does not give any information about the street in that case.

AlephZero

Another reason for those definitions is so that logic "works" the way it should, for every combination of "true" and "false".

For example, "P implies Q" means the same (in ordinary English) as "if P is true, then Q is true", which means the same as "if Q is false, then P is false".

So the truth table for P→Q must be the same as for (not Q)→(not P),

That means P→Q must be defined as true, when P and Q are both false.

You can create a similar argument to show how P→Q must be defined with P is false and Q is true.

Bacle2

See the bottom half of :

http://en.wikipedia.org/wiki/Material_conditional

i.e., the section on philosophical problems.

And

http://en.wikipedia.org/wiki/Strict_conditional