 Another type of unsoundness arguments attempt to show that there is some higher-order formula that is model-theoretically valid but is intuitively false in a structure whose domain is a proper class. (The “intended interpretation” of set theory, if it exists at all, might be one such structure, for it is certainly not a set; see the entry on  set theory.)  These arguments thus question the claim that each meaning assignment's validity-refuting power is modeled by some set-theoretic structure, a claim which is surely a corollary of the first implication in (5). (In McGee 1992 there is a good example; there is critical discussion in Gómez-Torrente 1998/9.) The most widespread view among set theorists seems to be that there are no formulae with that property in Fregean languages, but it's certainly not an absolutely firm belief of theirs. Note that these arguments offer a challenge only to the idea that universal validity (as defined in section 2.3) is adequately modeled by set-theoretic validity, not to the soundness of a characterization of logical truth in terms of universal validity itself, or in terms of a species of validity based on some notion of “meaning assignment” different from the usual notion of a set-theoretic structure. (The arguments we mentioned in the preceding paragraph and in 2.4.1 would have deeper implications if correct, for they easily imply challenges to all characterizations in terms of species of validity as well). In fact, worries of this kind have prompted the proposal of a different kind of notions of validity (for Fregean languages), in which set-theoretic structures are replaced with suitable values of higher-order variables in a higher-order language for set theory, e.g. with “plural interpretations” (see Boolos 1985, Rayo and Uzquiano 1999, Williamson 2003, Florio and Incurvati 2021; see also Florio and Linnebo 2021 and the entry on  plural quantification).  Both set-theoretic and proper class structures are modeled by such values, so these particular worries of unsoundness do not affect this kind of proposals.