 Focusing on propositional functions, Whitehead and Russell distinguish between expressions with a free variable (such as “\(x\) is hurt”) and names of functions (such as “\(\hat{x}\) is hurt”) (PM, 14–15). The propositions which result from the formula by assigning allowable values to the free variable “\(x\)” are said to be the “ambiguous values” of the function. Expressions using the circumflex notation, such as \(\phi \hat{x}\) only occur in the introductory material in the technical sections of PM and not in the technical sections themselves (with the exception of the sections on the theory of classes), prompting some scholars to say that such expressions do not really occur in the formal system of PM. This issue is distinct from that surrounding the interpretation of such symbols. Are they “term-forming operators” which turn an open formula into a name for a function, or simply a syntactic device, a placeholder, for indicating the variable for which a substitution can be made in an open formula? If they are to be treated as term-forming operators, the modern notation for \(\phi \hat{x}\) would be \(\lambda x\phi x\). The \(\lambda\)-notation has the advantage of clearly revealing that the variable \(x\) is bound by the term-forming operator \(\lambda\), which takes a predicate \(\phi\) and yields a term \(\lambda x\phi x\) (which in some logics is a singular term that can occur in the subject position of a sentence, while in other logics is a complex predicative expression). Unlike \(\lambda\)-notation, the PM notation using the circumflex cannot indicate scope. The function expression “\(\phi(\hat{x},\hat{y}\))” is ambiguous between \(\lambda x\lambda y\phi xy\) and \(\lambda y\lambda x\phi xy\), without some further convention. Indeed, Whitehead and Russell specified this convention for relations in extension (on p. 200 in the introductory material of ∗21, in terms of the order of the variables), but the ambiguity is brought out most clearly by using \(\lambda\) notation: the first denotes the relation of being an \(x\) and \(y\) such that \(\phi xy\) and the second denotes the converse relation of being a \(y\) and \(x\) such that \(\phi xy\).