On truth-gaps, truth-gluts, and bipolar propositions

• A truth model of Belnap valuations for bipolar propositions is proposed.
• Two truth normalization to reduce inconsistent cases are proposed.
• A calculus of bipolar belief measures for vague propositions is proposed.
• The differences with Atanassov's IFS and interval fuzzy sets are discussed.

This paper assumes that each proposition can be defined by a positive criterion together with a negative criterion. Then, within the framework of propositional logic, this paper proposes Belnap valuations as a truth model for bipolar propositions. The truth value of bipolar proposition is meant to be one of four values: absolutely true, absolutely false, borderline, and inconsistent. The borderline and inconsistent cases represent the truth-gap and truth-glut of a bipolar proposition, respectively. In order to reduce inconsistency two truth normalization methods, strong truth normalization and Kleene truth normalization, are proposed to approximate the original Belnap valuations. By integrating uncertainty of the true interpretations of bipolar propositions this paper introduces the following bipolar belief measures: positive and negative belief measures, lower and upper belief measures, and Kleene lower and upper belief measures. A mass function characterization of these bipolar belief measures are further explored. The relationship with Atanassov's intuitionistic fuzzy sets and interval fuzzy sets is also discussed in detail.