 It has been proved that Carnap’s notion of maximal validity makes completeness impossible for quantified S5, i.e., that there are L-truths that are not theorems of Carnap’s MFC. Let \(A\) be a non-modal sentence of MFC. By convention (1), \(\Box A\) is true in MFC if and only if \(A\) is L-true in MFC. But \(A\) is also a sentence of FC, thus if L-true in MFC it is also L-true in FC, since the state descriptions (models) of modal functional logic are the same as those of functional logic (1946: 54). This means that the state descriptions hold the triple role of (i) first-order models of FC thereby defining first-order validity, (ii) worlds for MFC thereby defining truth for \(\Box A\) sentences of MFC, and (iii) models of MFC thereby defining validity for MFC. The core of the incompleteness argument consists in the fact that the non-validity of a first-order sentence \(A\) can be represented in the modal language, as \(\neg \Box A\), but all models agree on the valuation of modal sentences, making \(\neg \Box A\) valid. Roughly, and setting aside complications created by the fact that Carnap’s semantics has only denumerable domains, if \(A\) is a first-order non-valid sentence of FC, \(A\) is true in some but not all the models or state-descriptions. Given Carnap’s conventions, it follows that \(\neg \Box A\) is true in MFC. But then \(\neg \Box A\) is L-true in MFC, i.e., in MFC \(\mvDash \neg \Box A\). Given that the non-valid first-order sentences are not recursively enumerable, neither are the validities for the modal system MFC. But the class of theorems of MFC is recursively enumerable. Hence, MFC is incomplete vis-à-vis Carnap’s maximal validity. Cocchiarella (1975b) attributes the result to Richard Montague and Donald Kalish. See also Lindström 2001: 209 and Kaplan 1986: 275–276.