 If \(\bL\) is a logic system and \(\Gamma , \Delta\) are sets of \(L\)-formulas, we will use the notation \(\Gamma \vdash_{\bL } \Delta\) to state that for every \(\psi \in \Delta , \Gamma \vdash_{\bL } \psi\). Thus \(\Gamma \vdash_{\bL } \Delta\) if and only if \(\Delta \subseteq C_{\bL }(\Gamma)\).