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

 

[CellCoree]

In logic, the statement "P→Q" means "if P, then Q." This is known as an implication, where the truth of one statement (P) leads to the truth of another statement (Q). In the case where P is false and Q is true, the statement "if P, then Q" is still true because there is no contradiction. While P may be false, the statement is still logically valid because Q is true, thus fulfilling the condition of the implication. In other words, the implication is not dependent on the truth values of P and Q individually, but rather on the logical relationship between them.

Similarly, when both P and Q are false, the statement "if P, then Q" is still true because there is no contradiction. In this case, since both P and Q are false, the statement cannot be disproved and therefore is considered true. This may seem counterintuitive, but in logic, a statement is only considered false if it can be proven false. Since there is no evidence to disprove the statement "if P, then Q," it is considered true.

In summary, the truth value of an implication is not determined by the individual truth values of P and Q, but rather by the logical relationship between them. In both cases, there is no contradiction, and therefore the implication is considered true.