 If \(\bL_1 = \langle L,\vdash_{\bL_1 } \rangle\) is a logic system with \(\vdash_{\bL_1}\) defined by a proof-system and \(\bL_2 = \langle L, \vdash_{\bL_2 } \rangle\) is a logic system over the same language with \(\vdash_{\bL_2}\) defined semantically, we say that the proof-system used to define \(\vdash_{\bL_1}\) is sound for the semantics used to define \(\vdash_{\bL_2}\) if \(\vdash_{\bL_1}\) is included in \(\vdash_{\bL_2}\), namely if \(\Gamma \vdash_{\bL_1 } \phi\) implies \(\Gamma \vdash_{\bL_2 } \phi\). If the other inclusion holds the proof-system is said to be complete with respect to the semantics that defines \(\vdash_{\bL_2}\), that is, when \(\Gamma \vdash_{\bL_2 } \phi\) implies \(\Gamma \vdash_{\bL_1 } \phi\).