Tautologies and positive solvability of linear homogeneous systems (Q1192331)

Let \(A=[a_{ij}]\) be an \(m\times n\) real matrix, and let \[ L(a_{ij})=\begin{cases} p_ i, & a_{ij}>0,\\ \neg p_ i, & a_{ij}<0,\\ \text{true}, & a_{ij}=0,\end{cases} \] with propositional variables \(p_ 1,\dots,p_ m\). The main result: The system \(Ax=0\), \(x\geq 0\) has a nontrivial solution iff the formula \(L(A)\) is a tautology.