Two conceptions of absolute generality

What is absolutely unrestricted quantification? We distinguish two theoretical roles and identify two conceptions of absolute generality: maximally strong generality and maximally inclusive generality. We also distinguish two corresponding kinds of absolute domain. A maximally strong domain contains every potential counterexample to a generalisation. A maximally inclusive domain is such that no domain extends it. We argue that both conceptions of absolute generality are legitimate and investigate the relations between them. Although these conceptions coincide in standard settings, we show how they diverge under more complex assumptions about the structure of meaningful predication, such as cumulative type theory. We conclude by arguing that maximally strong generality is the more theoretically valuable conception.