What would happen if $p\rightarrow q$ were defiend to be false if $p$ is false?

Question

What does $\rightarrow$ mean in $p \rightarrow q$

As in the above link, "The truth table of $\rightarrow$ is defined to be that $p\rightarrow q$ is false if and only if $p$ is true and $q$ is false." So $p\rightarrow q$ is defiend to be also true if $p$ is false. Is there any reason for mathematicians to have agreed to define the truth table in the above way? Would it cause any problems when $p\rightarrow q$ were defiend to be false if $p$ is false?

For example, True or false: Every Field is a UFD. any field is vacuously a UFD. If the truth table were defined the other way around, then would fields not have to be UFDs?

Answer 1

Usually in classical first-order logic (contains propositional logic) $\rightarrow$ is defined as :

$$p\rightarrow q\equiv \lnot p\lor q.$$

Also $\lor$ can be defined as :

$$p\lor q\equiv \lnot(\lnot p\land \lnot q).$$

You will only have to use $\lnot,\land$ and then you can define everything. So to answer your question, definition of $\rightarrow$ sign is simply for human convenience. Actually you don’t even have to define it, and just use two signs. So if you don’t like the definition, you can choose other semantic meaning for $\rightarrow$, (but specify what you mean at the beginning of your article. And also as mentioned in the comment, it will be just equivalent to $\land$ operation so why would you use two symbols for one operation) or just don’t use $\rightarrow$.