For the simple theory, which Frank Plumpton Ramsey (1903-1930) separated out from the ramified theory, properties of objects are of type one, properties of type one properties are of type two, and so on.
The ramified theory further classifies properties of each type into orders. A first-order type n+1 property is a property of things of type n. A second-order type n+1 property is still of things of type n, but it involves a reference to first-order type n+1 properties.
Red is a first-order type one property applying to objects. Applying to objects is a first-order type two property applying to type one properties. Having some first-order type one property is a second-order type one property; it still applies to objects, but makes reference to first-order properties. Thus each type has its own ramification (or branching) of orders. (But when accuracy is not needed, ‘type’ and ‘order’ are often used more loosely or even interchangeably.)
Because of certain unwanted technical results, the ramified theory led Bertrand Russell (1872-1970) to introduce the axiom of reducibility.
Also see: vicious circle principle, simple theory of types
F P Ramsey, Foundations of Mathematics (1931), ch. 1
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