HAJNAL ANDRÉKA, ISTVÁN NÉMETI and JOHAN VAN BENTHEM MODAL LANGUAGES AND BOUNDED FRAGMENTS OF PREDICATE LOGIC 1. MODAL LOGIC AND CLASSICAL LOGIC

Modal Logic is traditionally concerned with the intensional operators “possibly” and “necessary”, whose intuitive correspondence with the standard quantifiers “there exists” and “for all” comes out clearly in the usual Kripke semantics. This observation underlies the well-known translation from propositional modal logic with operators ♦ and , possibly indexed, into the first-order language over possible worlds models (van Benthem 1976, 1984). In this way, modal formalisms correspond to fragments of a full first-order (or sometimes higher-order) language over these models, which are both expressively perspicuous and deductively tractable. In this paper, by the “modal fragment” of predicate logic we understand the set of all first-order formulas obtainable as translations of basic (poly-)modal formulas. As the modal fragment is merely a notational variant of the basic modal language, we will often refer to the two interchangeably. Basic modal logic shares several nice properties with full predicate logic, namely, finite axiomatizability, Craig Interpolation and Beth Definability, as well as model-theoretic preservation results such as the Loś–Tarski Theorem characterizing those formulas that are preserved under taking submodels. In addition, basic modal logic has some nice properties not shared with predicate logic as a whole: e.g., its axiomatization does not need side conditions on free or bound variables – and most evidently: basic modal logic is decidable. We shall concentrate on this list in what follows, in the hope that it forms a representative sample. Our aim is to find natural fragments of predicate logic extending the modal one which inherit the above-mentioned nice properties. This quest has two virtues. It forces us to understand why basic modal logic has these nice properties. And it points the way to new insights concerning predicate logic itself. Note that this study takes place over the universe of all models, without special restrictions on accessibility relations. This is the domain of the “minimal modal logic”, which