#### Date of Original Version

9-2003

#### Type

Technical Report

#### Rights Management

All Rights Reserved

#### Abstract or Description

The theory of coherent structures is a key element of ``complex stochastic systems.'' Such systems are germane to the biomedical, engineering and statistical sciences. With regard to the biomedical sciences, the theory helps address issues such as the failure of paired organs. For the engineering sciences, it helps to address issues pertaining to the robustness of the electrical power grid and the efficient design of networks. Regarding statistical sciences, its underpinnings facilitate a deeper appreciation of the assumptions in graphical models, neural networks, and ``logic regression''.

The state of the art in coherent structure theory is driven by two assertions, whose principles were laid out over forty years ago. The first is that all units of a system can exist in one of two possible states: failed or functioning. The second is that, at any given point in time, each unit can exist in only one of the above states. Both assertions are limiting. Units do exist in more than two states - a notion acknowledged before; however, what has not been recognized is that it is possible to declare that a unit can *simultaneously* exist in more than one state. This feature is a consequence of the view that it is not always possible to precisely define the subsets of a set of possible states. Such subsets are known as *vague sets*.

Regarding the first limitation, work has been done under the general label of ``multi-state systems''; however, this work has not capitalized on the mathematics of many-valued propositions developed by logicians. Here, we invoke the truth tables of many-valued logic to define the structure function of multi-state systems and then harness these results in the context of vagueness. In essence, many-valued logic provides a common platform for studying both multi-state and vague systems.

Our development, which has some interplay with philosophy and logic is a contribution to the mathematics of complex stochastic systems whose impact on science, technology and medicine continues to grow.