On truth-gaps, bipolar belief and the assertability of vague propositions

This paper proposes an integrated approach to indeterminacy and epistemic uncertainty in order to model an intelligent agentʼs decision making about the assertability of vague statements. Initially, valuation pairs are introduced as a model of truth-gaps for propositional logic sentences. These take the form of lower and upper truth-valuations representing absolutely true and not absolutely false respectively. In particular, we consider valuation pairs based on supervaluationist principles and also on Kleeneʼs three-valued logic. The relationship between Kleene valuation pairs and supervaluation pairs is then explored in some detail with particular reference to a natural ordering on semantic precision. In the second part of the paper we extend this approach by proposing bipolar belief pairs as an integrated model combining epistemic uncertainty and indeterminacy. These comprise of lower and upper belief measures on propositional sentences, defined by a probability distribution on a finite set of possible valuation pairs. The properties of these measures are investigated together with their relationship to different types of uncertainty measure. Finally, we apply bipolar belief measures in a preliminary decision theoretic study so as to begin to understand how the use of vague expressions can help to mitigate the risk associated with making forecasts or promises. This then has potential applications to natural language generation systems.