If \(\bL = \langle L, \vdash_{\bL } \rangle\) and \(\bL' = \langle L', \vdash_{\bL' } \rangle\) are logic systems whose languages satisfy that  \(L'\subseteq L\) (hence all the \(L'\)-formulas are \(L\)-formulas) and   \[\Gamma \vdash_{\bL' } \phi \txtiff \Gamma \vdash_{\bL } \phi,\] for every set of \(L'\)-formulas \(\Gamma\) and every \(L'\)-formula \(\phi\) we say that \(\bL'\) is a fragment \(\bL\) (in fact, the \(\bL'\)-fragment) and   that \(\bL\) is an expansion of \(\bL'\).