## A Theory of Three-way Decisions

Three-way decisions are thinking in threes. In contrast to dichotomous thinking in terms of two options, three-way decisions introduce a third option. We move from true/false, black/white, yes/no etc. into true/unsure/false, black/grey/white, yes/maybe/no etc. The third option provides the necessary flexibility and universality of thinking in threes.

A theory of three-way decisions (3WD) may be interpreted within a trisecting-and-acting framework. With respect to trisecting, we divide a whole into three parts. With respect to acting, we design most effective strategies for processing the three parts. In a set-theoretic model of three-way decisions, we divide a universal set into three pair-wise disjoint regions known as a trisection or a weak tri-partition. We design most effective strategies for processing the three regions. The identification and explicit investigation of different strategies for different regions are a distinguishing feature of three-way decisions.

A specific model of three-way decisions may be formulated based on the notions of acceptance, rejection and noncommitment. It is an extension of the commonly used binary-decision model with an added third option. Three-way decisions play a key role in everyday decision-making and have been widely used in many fields and disciplines.

The notion of three-way decisions was originally introduced by the needs to explain the three regions of probabilistic rough sets. One can construct rules for acceptance from the positive region and rules for rejection from the negative region. When neither a decision of an acceptance nor a decision of rejection can be made, a third option of noncommitment is chosen. Recent studies show that rough set theory is only one of possible ways to construct three regions. A more general theory of three-way decisions has been proposed, embracing ideas from rough sets, interval sets, shadowed sets, three-way approximations of fuzzy sets, orthopairs, square of oppositions, and others.

### 中文论文 (Chinese Papers)

• 姚一豫, 于 洪, 三支决策概述, in: 于洪, 王国胤, 李天瑞, 梁吉业, 苗夺谦, 姚一豫, (编著) 《三支决策：复杂问题求解方法与实践》 科学出版社, 北京, pp. 1-19, 2015.
(Yao, Y.Y. and Yu, H., Introduction to three-way decisions, in: Yu, H., Wang, G.Y., Li, T.R., Liang, J.Y., Miao, D.Q., and Yao, Y.Y. (Eds.), Three-Way Decisions: Methods and Practices of Complex Problem Solving, Sience Press, Beijing, pp. 1-19, 2015. )
• 姚一豫, 三支决策研究的若干问题 , 刘盾，李天瑞，苗夺谦，王国胤，粱吉业 （编著). 《三支决策与粒计算》，科学出版社， 北京，1-13, 2013.
(Yao, Y.Y., Several issues in studies on three-way decisions (in Chinese), in: Dun, L., Li, T.R., Miao, D.Q., Wang, G.Y., Liang, J.Y. (Eds.), Three-way Decisions and Granular Computing, Science Press, Beijing, China, pp. 1-13, 2013.) ( Scanned version)
• 姚一豫, 三支决策 , 姚一豫, 三支决策, 贾修一, 商琳, 周献中, 梁吉业, 苗夺谦, 王国胤, 李天瑞, 张燕平 （编著）. 《三支决策理论与应用》, 南京大学出版社, 南京, 1-16, 2012.
(Yao, Y.Y., Three-way decisions (in Chinese), in: Jia, X.Y., Shang, L., Zhou, X.Z., Liang, J.Y., Miao, D.Q., Wang, G.Y., Li, T.R., Zhang, Y.P. (Eds.), Theory of Three-way Decisions and Application, Nanjing University Press, Nanjing, China, pp. 1-16, 2012.) ( Scanned copy)

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