Definitions

A quantifier controls names within a specified scope. The names it controls are called the bound variables of the quantifier, and we say "the variables are bound" by the quantifier within its scope.

lambda is a quantifier. Its parameters are the bound variables, and its scope is the body of the lambda expression.

Those variables that are not controlled by a quantifier are called free. Bound variables can be renamed by uniformly changing them in the bound variable list and throughout the scope of the controlling quantifier, provided that doing so doesn't change which lambda controls the uses of the name. Free variables can't be renamed in this way because there's no lambda that controls them.

Jim Miller