Three-way decision and triangular thinking

A fundamental notion of three-way decision is a triadic structure, namely, a triad of three elements (things) equipped with a structure. In this paper, I give a triadic explanation and formulation of three-way decision and examine the structures, patterns, and models of triangular thinking. A triangular representation of a triad uses the three corners (vertexes) of a triangle to represent the three elements of the triad and the three sides (edges) to represent the pairwise relations of elements. There are two important families of triangular thinking. One family focuses on the space of all points inside the triangle for discussion. The main idea is to represent and process any point in terms of the three elements or the three relations. The other family focuses on the interactions of the three elements and the three relations. According to the presence, absence, and the direction of an edge in the triangle, there are sixteen patterns of interaction for triangular thinking. With respect to the elements and the relations of elements, there are three basic modes of interaction, namely, element-element (i.e., vertex-vertex), relation-relation (i.e., edge-edge), and element-relation (i.e., vertex-edge) interactions. By combining the two notions, I discuss models of triangular thinking and review the uses of these models across various disciplines and fields. Finally, I discuss dynamic triangular thinking through the change and revision of triangles. It is evident that triangular thinking offers new insights into three-way decision and may lead to new theories, methods, and tools.