🤖 AI Summary
This study addresses the challenge of effectively transferring soundness and completeness from highly expressive modal logics to weaker linguistic systems. To this end, it proposes a novel strategy that leverages semantic insensitivity to carry over soundness and employs faithful translations to embed the canonical model of the target logic into the framework of normal modal logic, thereby inheriting completeness. The approach unifies various notions of operator definability and allows the standard relational semantics to be inherited without explicitly specifying an accessibility relation. Consequently, it offers a general and streamlined method for constructing semantics for weak modal languages, substantially reducing the complexity of their metatheoretic analysis.
📝 Abstract
We develop general strategies for transferring soundness and completeness from more expressive modal languages to less expressive ones, unifying several existing notions of operator definability along the way. For soundness, we exploit semantic insensitivity: if a less expressive language is insensitive to a frame operation, soundness extends to the operation's closure of the original frame class. For completeness, restricting to relational semantics and languages with a single operator, we present strategies for relating the target logic's canonical model to that of a normal modal logic via a truth-preserving translation. Three of those dispense entirely with specifying an accessibility condition for the target logic, inheriting it from a normal modal logic instead.