 Since ‘\(x\)’ and ‘\(y\)’ are variables ranging over individuals, one can say that the function indicated by the transitive verb ‘likes’ yields the value T given the ordered pair \(\langle x,y \rangle\) as argument if and only if \(x\) likes \(y\). In this notational scheme, ‘\(y\)’ corresponds to the direct object (or internal argument), which combines with the verb to form a phrase; ‘\(x\)’ corresponds to the grammatical subject (or external argument) of the verb. If we think about ‘every boy sang’ analogously, ‘boy’ is the internal argument of ‘every’, since ‘every boy’ is a phrase. By contrast, ‘boy’ and ‘sang’ do not form a phrase in ‘every boy sang’. So let us introduce ‘\(X\)’ and ‘\(Y\)’ as second-order variables ranging over functions, from individuals to truth values, stipulating that the extension of such a function is the set of things that the function maps onto the truth value T. Then one can say that the function indicated by ‘every’ yields the value T given the ordered pair \(\langle X,Y \rangle\) as argument iff the extension of \(X\) includes the extension of \(Y\). Similarly, one can say that the function indicated by ‘some’ maps the ordered pair \(\langle X, Y \rangle\) onto T iff the extension of \(X\) intersects with the extension of \(Y\).