The semantic characterization of modal logical connectives (Q1092890)

Let the propositional modal connective \(c(p_ 1,...,p_ n)\) be defined in Kripke models for S4 by a first order formula in terms of the accessibility relation R, the root world 0 and unary predicates \(P_ i(x)\) for values of \(p_ i\) in a world x. Then connective c is equivalent to a propositional formula in standard language \(\neg\), \&, \(\square\) iff it is stable under elementary equivalence (of first order models), p-morphisms and restrictions to worlds accessible from any fixed world.