 Logicians today use a different notion of “order”. Today, first-order logic is a logic with only variables for individuals. Second order logic is a logic with variables for both individuals and properties of individuals. Third-order logic is a logic with variables for individuals, properties of individuals, and properties of properties of individuals. And so forth. By contrast, Church would call these logics, respectively, the logic of functions of the types \((\iota)/1\) and \((\iota,\ldots,\iota)/1\), the logic of functions of the types \(((\iota)/1)/1\) and \(((\iota,\ldots,\iota)/1,\ldots,(\iota,\ldots,\iota)/1)/1\), and the logic of functions of the types \((((\iota)/1)/1)/1\) etc. (i.e., the level-one functions of the functions of the preceding type). Given Church’s definitions, these are logics of first-, second- and third-order functions, respectively, thus coinciding with the modern terminology of “\(n\)th-order logic”.