$a \lor b$ is equivalent to $\neg a \implies b$. If I'm trying to prove "i am human" $\lor$ "i am in Mars", it seems to me that it should make a difference which of the two propositions you pick as $a$ to prove the other one.
If I were to prove $a \lor b$, since it is true that "i am human" then i can say that "i am human" $\lor$ "i am in Mars" is true. I wouldn't symmetrically be able to take the same approach in trying to prove the disjunction by proving "i am in Mars". How then, can we symmetrically apply the same approach in proving $\neg a \implies b$? Shouldn't it make a difference which of the two propositions we take as $a$ and assume the negation of?
Something feels off and I'm trying my best to put it in words. Sorry if I'm failing to do so. Is this the difference between Classical and Intuitionistic Logic?