I am given the following definition of L-formulas:
"Positive formulas are defined with the following properties:
(i) Every atomic formula is positive.
(ii) If $\phi,\psi$ are positive that $\phi\land\psi$ and $\phi\lor\psi$ are positive.
(iii) If $\phi$ is positive and $x$ is a variable, then $\forall x \phi$ and $\exists x\phi $ are positive."
How should I approach this question? My strategy now is to assume that $\Sigma$ is the set of positive formula and that it is inconsistent, i.e. $\Sigma \vdash \phi \land \lnot\phi$, and go case by case (the type of formula $\phi$ can be) to show that $\Sigma$ must contain non-positive formula. It's been difficult to write down the proof and I'm not sure if I'm right in arguing it this way. Is there another method I can try?