A higher order function is a function that takes another function as a parameter or returns a function as its result. Famous higher order functions include "map" and "filter", for example. Javascript, for example, uses higher order functions all the time. See [1].

The term "higher order logic" typically means program logic that uses higher order functions. An object-oriented programming style is inherently higher order because objects typically contain functions and are passed to methods.

The term "higher order logic" can also mean a system of logic that allows statements about logical statements. [2]

[1] https://en.wikipedia.org/wiki/Higher-order_function [2] https://en.wikipedia.org/wiki/Higher-order_logic