 A set of \(L\)-formulas \(\Gamma\) is called a theory of a logic system \(\bL\), or \(\bL\)-theory, if it is closed under the relation \(\vdash_{\bL}\), that is, if whenever \(\Gamma \vdash_{\bL } \phi\) it also holds that \(\phi \in \Gamma\). In other words, the theories of \(\bL\) are the closed sets of the consequence operation \(C_{\vdash_{ \bL}}\) on the set of \(L\)-formulas. In order to simplify the notation we denote this consequence operation by \(C_{\bL}\). A formula \(\phi\) is a theorem (or validity) of \(\bL\) if \(\varnothing \vdash_{\bL } \phi\). Then \(C_{\bL }(\varnothing)\) is the set of theorems of \(\bL\) and is the least theory of \(\bL\). The set of all theories of \(\bL\) will be denoted by \(\tTH(\bL)\).