If $\Sigma$ is finitely satisfiable, can both $\Sigma \cup \{ \phi\}$ and $\Sigma \cup \{ \neg \phi\}$ be finitely satisfiable?

Question

In propositional logic, if $\Sigma$ is finitely satisfiable, can both $\Sigma \cup \{ \phi\}$ and $\Sigma \cup \{ \neg \phi\}$ be finitely satisfiable?

I proved that at least one of $\Sigma \cup \{ \phi\}$ and $\Sigma \cup \{ \neg \phi\}$ must be finitely satisfiable, but could both be?

Answer 1

Of course both are satisfiable:

Recall that a set of sentences $S$ is complete if, for every sentence $\varphi$, either $S \vdash \varphi$ or $S \vdash \neg \varphi$.

All we need to show is that not every finite satisfiable set is complete. That is, as observed in the comments, it suffices give a counterexample of an incomplete finite satisfiable set. And for any signature you have, there are certainly many of them:

• $\emptyset$ is (vacuously) finite satisfiable, but incomplete.
• $\{\alpha\}$ is finite satisfiable, but incomplete.
• $\{\alpha, \alpha\rightarrow \alpha\}$ is finite satisfiable, but incomplete and so on

For any arbitrary sentences $\alpha, \beta$ of your language.