Quantified modal logics: one approach to rule (almost) them all!
The interaction between standard modal operators and the quantifiers of first-order logic introduces various possibilities and challenges. Using the semantics of possible worlds and accessibility relations, one must decide whether the domains of quantification remain fixed, grow, shrink, or vary as one moves between worlds. Additionally, a choice must be made regarding whether terms have consistent denotations across all worlds (rigid designators) or not (non-rigid designators).\N\NIn this article, Orlandelli proposes a general approach to quantified modal logics capable of simulating most other frameworks. The approach is grounded in a language that employs operators indexed by terms, enabling the expression of \textit{de re} modalities and providing precise control over the interaction between modalities, first-order logic mechanisms, and non-rigid designators. The semantics are built upon a primitive counterpart relation that applies to \(n\)-tuples of objects across possible worlds. This is a primitive two-place relation between \(n+1\)-tuples composed of one world and \(n\) objects inhabiting that world which allows an object to be represented by one, many, or no objects in an accessible world, addressing some limitations of standard individual counterpart theories.\N\NFinally, Orlandelli utilizes cut-free labeled sequent calculi to offer a proof-theoretic characterization of the quantified extensions of first-order definable propositional modal logics. This approach also provides a means to complete many axiomatically incomplete quantified modal logics.
scientific article; zbMATH DE number 7914752