 The implication from left to right says that \(\bL\) is sound relative to \(\bM\), and the other implication says that it is complete. In other words, \(\bM\) is a matrix semantics for \(\bL\) if and only if every matrix in \(\bM\) is a model of \(\bL\) and moreover for every \(\Gamma\) and \(\phi\) such that \(\Gamma \not\vdash_{\bL } \phi\) there is a model \(\langle \bA, \tD \rangle\) of \(\bL\) in \(\bM\) that witnesses the fact, namely there is a valuation on the model that sends the formulas in \(\Gamma\) to designated elements and \(\phi\) to a non-designated one.