Proof in sequent calculus

Question

Is it actually possible to prove this in sequent calculus?

if $\Gamma \vdash \lnot A$ , then $\Gamma \vdash A ⊃ B$

where $\Gamma$ is a set of formulas and A and B are formulas.

Answer 1

Details will depend on the system in play. But you should be able e.g. to argue from (i) $F \vdash \neg A$ to (ii) $F, A \vdash \bot$ to (iii) $F, A \vdash B$ to (iv) $F \vdash A \to B$.

Answer 2

$(A \rightarrow B)$ is a logical consequence of $\lnot A$, so if your sequent calculus is complete, it is certainly possible to find such a proof.