What would happen if we just made vacuous truths false instead? It's well known that vacuous truths are a concept, i.e. an implication being true even if the premise is false. What would be the problem with simply redefining this to be evaluated to false? Would we still be able to make systems work with this definition or would it lead to a problem somewhere? Why must it be the case that false -> false is true and false -> true is true?

Notice that 3=5 is false. but if 3=5 we can prove 8=8 which is true.

$$ 3=5$$

therefore $$ 5=3$$

Add both sides, $$8=8$$

We can also prove that $$ 8=10$$ which is false.

$$ 3=5$$

Add $5$ to both sides, we get $$8=10$$

The point is that if we assume a false assumption, then we can claim whatever we like.

That means " False $\implies$ False " is true.

And " False $\implies$ True " is true.