Novel basic operational laws for linguistic terms, hesitant fuzzy linguistic term sets and probabilistic linguistic term sets

In the process of decision making, people sometimes may feel more comfortable to express their preferences by linguistic terms instead of the quantitative form. However, as the basic premise of operations, the existing operational laws of linguistic terms and the extended linguistic term sets are very unreasonable. In order to overcome this issue, in this paper, we redefine some more logical operational laws for linguistic terms, hesitant fuzzy linguistic elements (HFLEs) and probabilistic linguistic term sets (PLTSs) based on two equivalent transformation functions. These novel operational laws can not only avoid the operation values exceeding the bounds of LTSs, but also keep the operation results more reasonable in decision making with linguistic information. Furthermore, the operational laws can keep the probability information complete when computing with PLTSs. Additionally, lots of properties of the operational laws are discussed, and some three-dimensional figures are drawn to show the regions of different operational laws of linguistic terms more vividly.