Set-theoretic models of three-way decision

The theory of three-way decision is about a philosophy of thinking in threes, a methodology of working with threes, and a mechanism of processing in threes. We approach a whole through three parts, in terms of three units, or from three perspectives. A trisecting-acting-outcome (TAO) model of three-way decision involves trisecting a whole into three parts and acting on the three parts, in order to produce an optimal outcome. In this paper, we further explore the TAO model in a set-theoretic setting and make three new contributions. The first contribution is an examination of three-way decision with nonstandard sets for representing concepts under the two kinds of objective/ontic and subjective/epistemic uncertainty. The second contribution is an introduction of an evaluation-based framework of three-way decision. We present a classification of trisections and investigate the notion of an evaluation space. The third contribution is, within the proposed framework, a systematical study of three-way decision with rough sets, interval sets, fuzzy sets, shadowed sets, rough fuzzy sets, interval fuzzy sets (or equivalently, vague sets, interval-valued fuzzy sets, intuitionistic fuzzy sets), and soft sets.