Solving modal equivalences

Nonmonotonic logics such as reflective logic, default logic, and autoepistemic logic, are usually defined in terms of set-theoretic fixedpoint equations defined over deductively closed sets of sentences of first order logic. Such systems may also be represented as necessary equivalences in a modal logic stronger than S5 with the added advantages that such representations may be generalized to allow quantified variables crossing modal scopes whereby both the Barcan formula and its converse hold. Herein, we discuss the problem of solving such necessary equivalences. Methods for solving equations and subsidiary methods for deducing what is logically possible are discussed. Solutions obtained by this method are then compared to related results obtained in the literature by circumscription in 2/sup nd/ order logic since the disjunction of all the solutions of a necessary equivalences containing just normal defaults is equivalent to circumscription in 2/sup nd/ order logic.