  Another common operation on justifications is sum: it has been introduced to make explicit the modal logic reasoning (Artemov 1995).  However, some meaningful justification logics like \({\mathsf{J}}^{-}\) (Artemov and Fitting 2019) do not use the operation sum.  With sum, any two justifications can safely be combined into something with broader scope.  If \(s : F\),  then whatever evidence \(t\) may be, the combined evidence  \(s\) + \(t\) remains a justification for \(F\). More  properly, the operation ‘+’ takes justifications  \(s\) and \(t\) and produces \(s\) + \(t\), which is a justification for everything justified by \(s\) or by  \(t\).