  The Correspondence Theorem gives fresh insight into epistemic modal  logics. Most notably, it provides a new semantics for the major modal logics. In addition to the traditional Kripke-style  ‘universal’ reading of \(\Box F\) as F holds  in all possible situations, there is now a rigorous  ‘existential’ semantics for \(\Box F\) that can be read as there is a witness (proof, justification) for F.