 The ITL translation of “a man” has type   \[(\{0, \ldots ,i\} \rightarrow[e]^{i+1} \rightarrow [e]^j \rightarrow t) \rightarrow [e]^i \rightarrow [e]^j \rightarrow t.\]    In \(P\lvert c\rvert (c\mcaret x)c'\), the \(P\) variable marks the slot for the VP interpretation; \(\lvert c\rvert\) gives the length of the input context to \(P\); it picks up the value \(i\), which is the position of the next available slot when the context is extended. This slot is filled by an object \(x\) denoting a man. Note that \(c\mcaret x[\lvert c\rvert ] = c\mcaret x[i] = x\), so the index \(i\) serves to pick out that man from the context.