“…To concretize the problem, suppose that the end result of understanding There are more apples than bananas is the LoT expression “A > B” and that of There is more sand than mud is “S > M.” Specifically, what's required is a specification of (i) how the single symbol “>” (Wellwood, 2019) can be interpreted by cognitive systems operating both over domains representing pluralities of objects and those representing stuff (Odic, Pietroski, Hunter, Lidz, & Halberda, 2012; see Rips & Hespos, 2015), and (ii) the logical relationships that an LoT expression enters into by virtue of being built around this symbol in a certain way (e.g., via proof-theoretic inference rules). We have suggested (Hunter & Wellwood, 2023) that for a nonlinguistic system to interpret a symbol such as “>” in the appropriate way is exactly for this interpretation to abide by some algebraic laws that are, in effect, specified by some inference rules governing expressions built out of “>.” For example, a rule licensing a logical inference from “A > B” and “B > C” to “A > C” essentially specifies that, in each content domain where “>” has an interpretation, that interpretation must correspond to a transitive binary relation.…”