Circumscriptions

Circumscriptions, the plural of circumscription, is a term used in several distinct fields to describe the act or result of circumscribing or enclosing a figure, as well as specialized theoretical frameworks in logic and anthropology.

In geometry, circumscriptions describe the relationship between figures that enclose others. A polygon circumscribed about a circle has each of its sides tangent to the circle, while the circle is inscribed in the polygon. A circle circumscribed about a polygon passes through all the polygon’s vertices, and the polygon is inscribed in the circle. These concepts are central to discussions of circumcircles, incircles, and the general study of how figures relate to each other through enclosing boundaries.

In logic and artificial intelligence, circumscription is a formalism for nonmonotonic reasoning introduced by John McCarthy. It minimizes the extension of certain predicates to model a closed-world assumption: what is not known to be true is assumed false unless it can be derived. This framework supports default reasoning and knowledge representation, with variants such as predicate circumscription and domain circumscription. While powerful, circumscription can be computationally demanding and sensitive to the choice of predicates being minimized.

In anthropology and archaeology, the circumscription theory, formulated by Robert Carneiro, explains the emergence of centralized political authority under conditions of circumscribed geography or resources. When populations are confined by environmental barriers or resource scarcity, increasing competition and warfare can drive the development of centralized leadership and state organization.

Circumscriptions thus refer to geometric relationships of enclosing figures, a formal reasoning method for AI and knowledge bases, and a theoretical model of state formation in human societies, depending on the disciplinary context.