Modal Logic in Two Gestalts

We develop a translation-based view dual of modal logic as the study of intensional languages that are at the same time interesting expressive and decidable parts of standard logical systems. This tandem approach improves our understanding of modal logic – while at the same time, it extends the range of modal notions and techniques into broader areas of standard logic. 1 Translation as a Way of Life 1.1 Basic modal logic and the modal fragment of FOL Modal languages as used to-day can be considered a species of their own, inhabiting the realm of Intensional Logic. But they can also be translated into fragments of standard logical languages, mostly first-order, sometimes higher-order or infinitary. These translations reflect the truth conditions for modal operators in possible worlds models. The ur-example is the basic modal language of possibility and necessity, whose standard translation ST inspired Correspondence Theory (van Benthem 1976, 1985): an existential modality <>p goes to a bounded quantifier ∃y (Rxy & Py) stating that the current world x has a successor y in which p holds