Proof Theory for a Recently-Proposed Logic of Adverbs

If Socrates is running slowly, he must be running. Likewise, if he’s not running, he must not be running slowly. Haze [2] proposes an extension of first-order logic inspired by this characteristic inferential behaviour of words like ‘slowly’: FOL-SA (first-order logic with scoped adverbs). Haze presents the logic model-theoretically using a hierarchy of models, where the level of a model corresponds to the number of nestings of adverb formulas within adverb formulas, and leaves open the investigation of its proof theory. In this paper we develop a semantic tableaux proof system for FOL-SA, prove its soundness and completeness, and outline some directions for further research.