Drawing Sound Conclusions from Unsound Premises

Given sets $\Phi_1={\phi_{11},…,\phi_{1u(1)}}, …,\Phi_{z}={\phi_{z1},…,\phi_{zu(z)}}$ofbooleanformulas,aformula$\omega$followsfromtheconjunction$\bigwedge\Phi_i= \bigwedge \phi_{ij}$iff$\neg \omega\wedge \bigwedge_{i=1}^z \Phi_i$ is unsatisfiable. Now assume that, given integers $0\le e_i<u(i)$, we must check if $\neg \omega\wedge \bigwedge_{i=1}^z \Phi’_i$remainsunsatisfiable,where$\Phi’i\subseteq \Phi_i$isobtainedbydeleting$,,e{i}$ arbitrarily chosen formulas of ${\mathrm{\Phi }}_i$, for each $i=1,\mathrm{.}\mathrm{.}\mathrm{.},z\mathrm{.}$ Intuitively, does $\omega$ {\it stably} follow, after removing $e_i$ random formulas from each ${\mathrm{\Phi }}_i$? We construct a quadratic reduction of this problem to the consequence problem in infinite-valued \luk\ logic \L${}_{\mathrm{\infty }}$. In this way we obtain a self-contained proof that the \L${}_{\mathrm{\infty }}$-consequence problem is coNP-complete.