A discussion of the introduction and elimination forms for higher-order logic is beyond the scope of this article.

It is possible to be in between first-order and higher-order logics.

The Lawvere programme was to write higher-order logic in terms of category theory.

There are two kinds of interpretations commonly employed for higher-order logic.

It is possible to produce sound deductive systems for higher-order logics, but no such system can be complete.

This is the first step towards creating a higher-order logic.

Other higher-order logics allow quantification over even higher types than second-order logic permits.

From 1996 the scope broadened to cover all theorem proving in higher-order logics.

Henkin's proof for higher-order logic uses a variant of the standard semantics.

Second-order logic is in turn extended by higher-order logic and type theory.