The connective of necessity of modal logic \(S_ 5\) is metalogical (Q2266004)

A new one-place connective A is introduced into a language L for propositional calculus. A mapping * is defined from the extended language L(A) onto the sentences of L. * maps L-formulae to themselves and maps an L(A) formula of the form A(X) to a fixed classical thesis if it maps X to any classical thesis, to a fixed counterthesis otherwise. The set S is then defined to be the set of L(A) formulae any L(A) substitution instance of which is mapped by * to a classical tautology. It is shown that S is the propositional modal logic S5, and that the connective A obeys various rules which express metalogical properties of the underlying classical logic. An infinitary S5 is investigated, and a rule for A produced which is equivalent to Stone's Representation Theorem for Boolean algebras.