Many-valued logic in multistate and vague stochastic systems

The state of the art in coherent structure theory is driven by two assertions, both of which are limiting: (1) all units of a system can exist in one of two states, failed or functioning; and (2) at any point in time, each unit can exist in only one of the above states. In actuality, units can exist in more than two states, and it is possible that a unit can simultaneously exist in more than one state. This latter feature is a consequence of the view that it may not be possible to precisely define the subsets of a set of states; such subsets are called vague. The first limitation has been addressed via work labeled 'multistate systems'; however, this work has not capitalized on the mathematics of many-valued propositions in logic. Here, we invoke its truth tables to define the structure function of multistate systems and then harness our results in the context of vagueness. A key contribution of this paper is to argue that many-valued logic is a common platform for studying both multistate and vague systems but, to do so, it is necessary to lean on several principles of statistical inference. © 2008 The Authors. Journal compilation © 2008 International Statistical Institute.