 A logic system \(\bL = \langle L, \vdash_{\bL } \rangle\) straightforwardly provides us with an equivalent abstract logic \(\langle \bFm_L, C_{\vdash_{ \bL} } \rangle\) and an equivalent generalized matrix \(\langle \bFm_L,\tTH(\bL) \rangle\), where \(\tTH(\bL)\) is the set of \(C_{\vdash_{ \bL}}\)-closed sets of formulas (i.e., the \(\bL\)-theories). We will move freely from one to the other.