I'm having some difficulty proving $\overline{M}$
H1: $P \wedge Q$
H2: $P \rightarrow \overline{Q \wedge S}$
H3: $G \rightarrow S$
H4: $M \wedge P \rightarrow G$
So far, I've only done a few steps, but am stuck. I'm trying to apply the rules of inference, but I'm not totally sure where to go.
H1: $P \wedge Q$
S1: $P$ : Simplification of H1
H2: $P \rightarrow \overline{Q \wedge S}$
S2: $P \rightarrow \overline{Q} \vee \overline{S}$ : De Morgan's Law of H2
H3: $G \rightarrow S$
S3: $P \rightarrow G$ by resolution of S2 and H3 (is this possible?)
H4: $M \wedge P \rightarrow G$
S5: ?
I'm not sure how to express this in terms of the formal logic rules you're allowed to use, but the reasoning is clear.
We know from $H_1$ that $P$ is true, so $H_2$ tells us that at least one of $Q$ or $S$ must be false. But we also know from $H_1$ that $Q$ is true, so $S$ must be false.
$H_3$ tells us that $G \rightarrow S$, but we already know that $S$ is false, so $G$ must also be false.
Similarly, $H_4$ tells us that $M \land P \rightarrow G$, but we already know that $G$ is false, so $M \land P$ also must be false. We also know from $H_1$ that $P$ is true, so that means $M$ must be false, so $\overline{M}$ must be true.