Back and Forth Between Modal Logic and Classical Logic

Modal Logic is traditionally concerned with the intensional operators \possibly" and \necessary", whose intuitive correspondence with the standard quantiiers \there ex-ists" and \for all" comes out clearly in the usual Kripke semantics. This observation underlies the well-known translation from modal logic into the rst-order language over possible worlds models 12, 13]. In this way, modal formalisms correspond to fragments of a full rst-order (or sometimes higher-order) language over these models , which are both expressively perspicuous and deductively tractable. In this paper, we shall enquire which features of`modal fragments' are responsible for these attractions. Throughout, we shall concentrate on the basic language of modal propositional logic, which still serves as thèpure paradigm' in a rapidly expanding eld of more expressive modal formalisms 56, 51]. What precisely arèmodal fragments' of classical rst-order logic? Perhaps the most innuential answer is that of Gabbay 25], which identiies them with so-called``nite-variable fragments', using only some xed nite number of variables (free or bound). This viewpoint has been endorsed by many authors (cf. 16]). Our paper presents a critical review of its supporting evidence, adding some new results about nite-variable fragments, including failures of the Lo s{Tarski preservation theorem. But there is also a second answer to our question, implicit in much of the literature, which emphasizes so-called`bounded quantiication'. As our positive contribution, we shall develop the latter perspective here, showing its utility as a guide towards generalization of modal notions and techniques to larger fragments of classical logics. In particular, we prove decidability for a largèbounded fragment' of predicate logic, and point out several applications. One can also combine the two views on modal logic, as will be illustrated. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantiiers, but quan-tiiers behave like modal operators. This observation inspires a generalized modal semantics for rst-order predicate logic using accessibility constraints on assignments (cf. 45, 47]) which moves the earlier quantiier restrictions into the semantics. This provides a fresh look at the landscape of possible predicate logics, including candi-685