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It doesn't have to: logics which don't are called paraconsistent.

The most important paraconsistent logic is relevance logic, which repudiates the K axiom: $$\alpha \rightarrow (\beta \rightarrow \alpha)$$ and replaces it by axioms that do not allow there to be unused assumptions. This is equivalent to saying weakening, the principle that if $\Gamma \vdash \alpha$ then $\Gamma'\vdash \alpha$ for $\Gamma\subset\Gamma'$. This blocks derivations such as Weltschmertz's, which appeals to the K axiom once, Asaf's which uses it twice; Francesco appeals to monotonicity in his proof, which is another name for weakening.

It's not difficult to see that this also blocks proofs of everything from a contradictory pair of propositions in a logic satisfying compactness, since one can prove inductively about such proof systems that if $\alpha\rightarrow\beta$, then all positive atoms in $\beta$ must occur either negatively in $\beta$ or positively in $\alpha$. So if our contradictory pair (over an assumption) takes the form $\alpha\rightarrow\beta$ and $\alpha\rightarrow

eg\beta$, we need to prove for any $\gamma$ that $\alpha\rightarrow\gamma$. But if we choose $\gamma$ to be any positive atom not occuring in $\alpha$, our inductive proof tells us this cannot be done. We need compactness here, to be ensure that the basis for all contradictory pairs can be expressed by a finitary formula.