First class and higher-order

The terms first-class and higher-order can be confusing to people first encountering them in relation to programming languages (usually through exposure to some functional programming language). At first glance it can seem that they mean sort-of the same thing, but this isn’t really the case. Both refer to the ability to pass functions around like other values, but they relate to different aspects. The term higher-order is a statement about the interface of a function. It means that the function is able to accept other functions as arguments (or even the same function) and/or return functions as its value. It originates in logic, where a first-order logic is able to make statements only about objects in the domain of discourse (e.g., “all men are tall”) , whereas a second-order logic is also able to make statements about first-order relations over those objects (e.g., “all properties of men are also properties of women”). A higher-order logic is then able to make statements about any lower-order relation. On the other hand, the term first-class is a statement about how a language handles certain kinds of values. In particular, it usually means that a certain kind of value (such as functions) can be treated on an equal level with all other kinds of value (such as strings or integers), and can be passed to/returned from functions, stored in data structures, and so on. These are distinct concepts: a language might permit higher-order functions that are not first-class, and vice-versa.