 A somewhat surprising feature of the product of modal logics is that some new interactions between modalities arise. These new valid formulas are a sort of bridge principles (recall  section 1).  Using the standard notation \(\lozenge_1 \varphi\) for \(\neg \square_1 \neg \varphi\) (and analogously for \(\lozenge_2\)) for the possibility operator, the following bridge principles are always valid in the product logic: