A Journey into Vagueness

Part 1 of the Global Example describes a situation that cannot be suitably represented by classical Boolean logic. Classical logic provides for two truth values: {true, false}. However, whether or not this paper has been rendered properly has three classes: {true, mostly, false}. Moreover, the distinction between each of the three classes is not well-defined. The problem contains borderline cases, or situations in which there are elements for which it is unclear to what class they belong. In addition to the propriety with which the paper is rendered, the colors “black” and “white” also have borderline cases. Exactly what constitutes the color “white”? Does a pale off-white count? Propositions or predicates for which borderline cases exist are said to be vague [1]. For the purposes of this discussion, a distinction generally is not made between propositions and predicates. Either or both of these logical elements have the capability of being vague. Furthermore, the treatment of vagueness here focuses on properties of objects; thus, the focus will be on predicates that operate on objects. However, for given logical rules and techniques for dealing with vagueness, a proposition may be substituted for a predicate by “wrapping” it in a predicate that returns the truth value of the proposition.