Simon Schmitt

This paper develops a Kripkean truth theory for higher-order languages that is compatible with absolute generality. Existing accounts either omit truth-theoretical elements (Rayo and Uzquiano in Notre Dame J Formal Logic 40(3):315–325, 1999) or focus only on first-order languages (Rossi in Notre Dame J Formal Logic 64(1):95–127, 2023). To fill this gap, I generalize existing literature to cover unrestricted higher-order languages and develop an absolutist-friendly interpretation of higher-order expressions. The theory models key features of natural language, including categorical reference to mathematical structures, a type-free truth predicate, and generality absolutism.

Can we quantify over absolutely every set? Absolutists typically affirm, while relativists typically deny, the possibility of unrestricted quantification (in set theory). In the first part of this article, I develop a novel and intermediate philosophical position in the absolutism versus relativism debate in set theory. In a nutshell, the idea is that problematic sentences related to paradoxes cannot be interpreted with unrestricted quantifier domains, while prima facie absolutist sentences (e.g., “no set is contained in the empty set”) are unproblematic in this respect and can be interpreted over a domain containing all sets. In the second part of the paper, I develop a semantic theory that can implement the intermediate position. The resulting framework allows us to distinguish between inherently absolutist and inherently relativist sentences of the language of set theory.