Why do we need to define the value of $p \implies q$ when $p$ is false?

Question

Is there any problem if we don't define the value of $p \implies q$ when $p$ is false?

I didn't learn the value of $p \implies q$ when I was a high-school student.
But I didn't have a problem at all.

Why do mathematicians define the value of $p \implies q$ when $p$ is false?

I cannot understand that.

I know if we define the value of $p \implies q$ correctly, we can prove for example $\emptyset \subset A$ for any set $A$.
Maybe it is convenient.
But do we need to define the value of $p \implies q$ when $p$ is false just for a convenience?

In introductory courses, the truth table for IMPLIES is usually given as The Definition. See https://en.wikipedia.org/wiki/Material_conditional#Truth_table From this truth table, we see that whenever $p$ (the antecedent) is false, the implication is true (lines 3 and 4). Each of the entries in the truth table, however, can be derived from more "primitive" rules of logic. See my blog posting on this topic at https://www.dcproof.com/IfPigsCanFly.html — Dan Christensen, Jul 13 '19 at 18:16

score 0 · Answer 1 · answered Jul 13 '19 at 06:18

Most of the statements in mathematics are of the form ''If ..., then ...''. If the premise is true, then the conclusion must also be true. My impression is that the mathematicians do not care about the case when the premise is false. Then by definition, the whole statement is true.

On the other hand, $p\Rightarrow q$ is logically equivalent to the contraposition $\neg q\Rightarrow \neg p$. And in some situations, the contraposition is proved (as it might be easier) rather than the original implication.

Why do we need to define the value of $p \implies q$ when $p$ is false?

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