Decidability of ordered fragments of FOL via modal translation

We present a simplification and a modification of a method introduced by Herzig to prove the decidability of Quine’s ordered fragment of first-order logic. The method consists in an interpretation of quantifiers as modal operators. We show that our modification yields the decidability of two new ordered fragments of first-order logic, called the grooved fragment and the loosely grooved fragment, whose expressive power lies between Quine’s ordered fragment and the fluted fragment.