  There is a lack of formalism for some key foundational concepts in systems
engineering. One of the most recently acknowledged deficits is the inadequacy
of systems engineering practices for engineering intelligent systems. In our
previous works, we proposed that closed systems precepts could be used to
accomplish a required paradigm shift for the systems engineering of intelligent
systems. However, to enable such a shift, formal foundations for closed systems
precepts that expand the theory of systems engineering are needed. The concept
of closure is a critical concept in the formalism underlying closed systems
precepts. In this paper, we provide formal, systems- and information-theoretic
definitions of closure to identify and distinguish different types of closed
systems. Then, we assert a mathematical framework to evaluate the subjective
formation of the boundaries and constraints of such systems. Finally, we argue
that engineering an intelligent system can benefit from appropriate closed and
open systems paradigms on multiple levels of abstraction of the system. In the
main, this framework will provide the necessary fundamentals to aid in systems
engineering of intelligent systems.
