A set $\Gamma$ of propositional logic formulae over $(P, C)$ is said to be finitely satisfiable if and only if every finite subset $\Gamma_{0} \subset_{\text{fin}} \Gamma$ of $\Gamma$ is satisfiable.
Let $\Gamma$ be a set of propositional logic formulae over $(P, C)$. $\Gamma$ is said to be $P$-consistent if for every proposition $p \in P$, either $p \not\in \Gamma$ or $\neg p \not\in \Gamma$.
The above are definitions of finite satisfiabiity and P-consistency. The given question is to come up with counterexample for finite $\Gamma$ where $\Gamma$ is finitely satisfiable but not P-consistent.
According to the above definition, the only case where $\Gamma$ is not P-consistent is when there exists some $p \in P$ such that $p \in P$ and $\neg p \in P$, and that immediately implies that $\Gamma$ is not finitely satisfiable. How is it possible to consider the above counterexample? Am I missing something here?
Thanks!!