Another Paradox of Standard Logic

Prelinearity isn't the only counterintuitive consequence of propositional logic. Another is called explosiveness or ex falso quodlibet, which is Latin for "from the false, anything follows". That is, from a contradiction, you can deduce any proposition; (P & ~P) implies Q or, in another form, (P implies (~P implies Q)). These statements are logically true because of the principle of non-contradiction (which states that there are no true contradictions) combined with the formal semantics of implication; the formal semantics of implication guarantees that if the antecedent of an implication is false, the implication is true. (The antecedent of an implication is the proposition that comes before "implies"; the proposition after "implies" is called the consequent.) My intuitions regarding this are that, if you presuppose a true contradiction (in order to exploit explosiveness), then you are undermining the very principle which licenses concluding anything from a contradiction; consequently, when confronted with a contradiction, one ought not to conclude anything they like, but instead consider that at least one of their premises must be false.