 The strict implication of the language of modal logic is defined using the \(\Box\) operator and the material implication \(\rightarrow\). We will use \(\Rightarrow\) for the strict implication. Its definition is \(\phi \Rightarrow \psi := \Box(\phi \rightarrow \psi)\). The language of the logic \(\bSilK\), that we call the strict implication fragment of the local modal logic \(\blK\), is the language \(L = \{\wedge , \vee , \bot , \top , \Rightarrow \}\). We can translate the formulas of \(L\) to formulas of the modal language by systematically replacing in an \(L\)-formula \(\phi\) every subformula of the form \(\psi \Rightarrow \delta\) by \(\Box(\psi \rightarrow \delta)\) and repeating the process until no appearance of \(\Rightarrow\) is left. Let us denote by \(\phi^*\) the translation of \(\phi\) and by \(\Gamma^*\) the set of the translations of the formulas in \(\Gamma\). Then the definition of the consequence relation of \(\bSilK\) is: