 The proof that every logic system has a reduced matrix semantics (i.e., a matrix semantics consisting of reduced matrices) is as follows. Let \(\bL\) be a logic system. Consider the matrices \(\langle \bFm_L, T \rangle\) over the formula algebra, where \(T\) is a theory of \(\bL\). These matrices are known as the Lindenbaum matrices of \(\bL\). It is not difficult to see that the class of those matrices is a matrix semantics for \(\bL\). Since a matrix and its reduction are models of the same logics, the reductions of the Lindenbaum matrices of \(\bL\) constitute a matrix semantics for \(\bL\) too, and indeed a reduced one. Moreover, any class of reduced matrix models of \(\bL\) that includes the reduced Lindenbaum matrices of \(\bL\) is automatically a complete matrix semantics for \(\bL\). In particular, the class of all reduced matrix models of \(\bL\) is a complete matrix semantics for \(\bL\). We denote this class by \(\bRMatr(\bL)\).