 In the Introduction to PM terminology is introduced for the two ways that variables may appear in formulas. The “apparent variables” are bound variables, whereas “real variables” are free variables. The proper interpretation of higher-order variables in PM is the subject of contemporary dispute among scholars of PM. Landini (1998) and Linsky (1999) offer two rival accounts. Landini holds that higher-order free variables should be interpreted as schematic letters, replaceable by formulas, and that the bound variables are to be interpreted “substitutionally”. The logic of the theory of types in PM can be seen as an extension of a theory of a standard first order logic developed in ∗10. Then the more distinctive notions of PM that depend on the theory of types can be explained. These include the Axiom of Reducibility, in ∗12 which underlies the so called ramification of the theory of types, the division into orders of predicates true of a single type of argument. The Axiom of Reducibility asserts that for an arbitrary function of any order there is an equivalent predicative function, that is, one true of exactly the same range of arguments. This will be explained below. Identity is defined, in ∗13, with a version of Leibniz’ notion of the identity of indiscernibles that is consistent with the theory of types. Replacing Leibniz’ notion that x and y are identical just in case they share the same properties, in PM, x and y are identical if and only if they share the same predicative functions.  Then using the notion of identity so defined, PM presents Russell’s theory of definite descriptions, precisely as it was defined in “On Denoting” (1905). This article will use the notation for “r-types” due to Alonzo Church 1976, which is explained in the accompanying article “The Notation of Principia Mathematica” in this Encyclopedia.