Let $\mathcal{L}$ be a sentential language. Consider the set $\Sigma$ of sentences. We want to show that
$\Sigma$ is inconsistent if and only if $\Sigma \vdash S \wedge \neg S$ for any simple symbol in $\mathcal{L}$.
The forward implication is clear. If $\Sigma$ is inconsistent, then for any sentence $\phi$, $\Sigma \vdash \phi$. And we have $\phi = S \wedge \neg S$ being a valid sentence.
I'm not sure why I'm having difficulties with the reverse.
My attempt:
Assume $\Sigma \vdash \phi = S \wedge \neg S$ for some symbol $S$ in $\mathcal{L}$. Then, by definition, $\phi$ must be: $1)$ a tautology (clearly false), $2)$, in $\Sigma$ to begin with, or $3)$ deducible within $\Sigma$.
We need to show that for any arbitrary sentence $\psi$, $\Sigma \vdash \psi$. However, I don't see how we can use $2)$ or $3)$ to help us with that.
