Talk:Rational consequence relation

RMO generalizes CMO ?
It seems to me that rule RMO $$\frac{p \vdash r \quad\quad p \not\vdash \lnot q}{p \land q \vdash r}$$ is a generalization of rule CMO $$\frac{p \vdash r \quad\quad p \vdash q}{p \land q \vdash r}$$. For, if $$p \vdash q$$ holds, then, assuming consistency, $$p \not\vdash \lnot q$$ also holds. Therefore, I wonder why both rules are required. - Jochen Burghardt (talk) 15:57, 14 October 2023 (UTC)