"false implies true" is a true statement [duplicate]

In algebra, one lesson we took was about logic. We learned that it is a true statement or logical expression to say that if a Beijing was the capital of the US then the moon existed last night, as this is convenient with a false statement implying a true one being a true statement. Agree?

As an example of why the convention 'false implies true is true' is useful, consider the sentence "if a given number is smaller than $10$ then it is also smaller than $100$". This is clearly a true statement. Therefore, if we specialize the statement by replacing the words 'any number' by a number, we should still consider it to be true. So let's look at some of these specialized cases.

Using the number $5$ gives the true statement "if $5$ is smaller than $10$ then it is also smaller than $100$". This is an example of 'true implies true'.

Using the number $500$, we get "if $500$ is smaller than $10$ then it is also smaller than $100$". This is also a true statement, of the form 'false implies false'.

Finally, if we use the number $50$, we get "if $50$ is smaller than $10$ then it is also smaller then $100$". This is an example of 'false implies true', and it still should be a true statement.

So the reason for the convention 'false implies true is true' is that it makes statements like $x < 10 \rightarrow x<100$ true for all values of $x$, as one would expect.

You want "real life", eh?

Let (P) be the statement

If the policeman sees you speeding, then you will have to pay a fine.

This is true. But it could happen you have to pay a fine because you failed to shovel the snow from your sidewalk. So you have to pay a fine even though you did not speed. But this does not mean that (P) is false.

A conditional statement $p\to q$ is false only if the hypothesis $p$ is true and the conclusion $q$ is false.

The formula $p \to q$ is logically equivalent to $q \vee \neg p$ ($q$ or not $p$ in English). As you can see, if $p$ is false, then $\neg p$ is true and $q \vee \neg p$ is also true. Thus $p \to q$ is true.