Conditional Expressivity and Collective Deontic Admissibility

This paper makes a twofold contribution to the study of expressivity. First, we introduce and study the novel concept of conditional expressivity. Taking a universal logic perspective, we characterize conditional expressivity both syntactically and semantically. We show that our concept of conditional expressivity is related to, but different from, the concept of explicit definability in Beth's definability theorem. Second, we use the concept to explore inferential relations between collective deontic admissibility statements for different groups. Negative results on conditional expressivity are stronger than standard (unconditional) inexpressivity results: we show that the well-known inexpressivity results from epistemic logic on distributed knowledge and on common knowledge only concern unconditional expressivity. By contrast, we prove negative results on conditional expressivity in the deontic logic of collective agency. In particular, we consider the full formal language of the deontic logic of collective agency, define a natural class of sublanguages of the full language, and prove that a collective deontic admissibility statement about a particular group is conditionally expressible in a sublanguage from the class if and only if that sublanguage includes a collective deontic admissibility statement about a supergroup of that group. Our negative results on conditional expressivity may serve as a proof of concept for future studies.