 The set of formulas valid on a class of frames \(\mathsf{F}\) is called the logic of \(\mathsf{F}\). Denote this logic that is, the set \(\{\varphi\in\mathcal{L}_{K}\colon\mathsf{F}\vDash\varphi\}\) by \(\Lambda_{\mathsf{F}}\). This is a semantic approach to defining logics, each just a set of formulas. One may also define logics proof-theoretically by defining a logic as the set of formulas provable in some system. With logics as just sets of formulas, soundness and completeness results may then be expressed using set inclusion. To exemplify, let \(\mathsf{A}\) be a set of axioms and write \(\mathsf{A}\vdash\varphi\) when \(\varphi\) is provable from \(\mathsf{A}\) using some given set of deduction rules. Let the resulting logic the set of theorems be denoted \(\Lambda_{\mathsf{A}}\). It is the set of formulas from \(\mathcal{L}_{K}\) provable from \(\mathsf{A}\), i.e., the set \(\{\varphi\in\mathcal{L}_{K}\colon\mathsf{A}\vdash\varphi\}\). The logic \(\Lambda_{\mathsf{A}}\) is sound with respect to \(\mathsf{F}\) iff \(\Lambda_{\mathsf{A}}\subseteq\Lambda_{\mathsf{F}}\) and complete with respect to \(\mathsf{F}\) iff  \(\Lambda_{\mathsf{F}}\subseteq\Lambda_{\mathsf{A}}\).[3]