Why in implication, if premise is false then implication is true?

Question

I am not able to understand implication well. I can see If in $P \implies Q$, if Pis true and Q is false then implication is false. But i cannot understand other cases where it is taken as true. Is this just a convention ? or some underlying reasoning behind it

Thanks

Answer 1

The essence of implication is that when $P\implies Q$, we cannot have $P\text{ true}$ and $Q\text{ false}$. No other case if "forbidden", i.e. creates a contradiction.

For instance, with $x>1\implies x>0$, there are no $x$ such that $x>1\land x\le0$ ($\text{true}\implies\text{false}$). But $x\le1\land x>0$ ($\text{false}\implies\text{true}$) and $x\le1\land x\le0$ ($\text{false}\implies\text{false}$) are quite possible.

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The instances such that $P$ is $\text{false}$ are usually uninteresting/meaningless and create a paradox called "ex falso quodlibet".

For instance

$$x>x+1\implies x^2<0$$ and

$$x>x+1\implies \pi=2$$ are said vacuously true implications (which are perfectly valid implications), because no $x$ can invalidate them.

Answer 2

If Q is false and P is true, this contradicts the fact that "if P then Q". Since "P is true" and "if P then Q" tells us that Q must be true. Thus, P must be false.

Answer 3

Yes, it is a definition of the implication (two argument) function.

It has a sense: we obtain the consequences of true assumptions. Similarly as in our life: If there was no gravity I could jump from the Eiffel Tower and survive.