Type theory, in one sense, is the view that some category of abstract entities—sets, in the simplest example, but there are analogous views of properties, relations, concepts, and functions—come in a hierarchy of levels, with an entity of one level applying to (having as members, or having as instances, or…) entities only of a lower level. Such a view gives an intuitively comprehensible picture of the universe of abstracta and provides a principled way of avoiding Bertrand Arthur William Russell's Paradox and its analogues. In a second sense, the term refers to any of a wide range of formal axiomatic systems embodying some form of the view. The present entry gives a short history of the view and a brief survey of the systems.

The systems are generally formulated in many-sorted quantificational logic, with a separate alphabet of quantified variables ranging over each type...