Fuzzy Reasoning Symmetric Quintuple-Implication Method for Mixed Information and Its Application

Rule-based reasoning with different kinds of uncertain information has been identified in numerous applications within the real world. Any reasoning method must be able to coherently obtain the inference result by composing the given if–then rule with the assertion of the given input. The symmetric quintuple-implication principle was established by introducing symmetry into the five implication operators included. For example, the first, third and fifth implication operators exhibit symmetric properties, i.e., the three implication operators are taken as the same kind of operator and the second and fourth implication operators satisfy symmetry, that is, the two implication operators take the same kind of operator. So, the reasoning method induced by this principle possesses significant advantages in terms of its logical foundation and reductivity. This paper derives and studies reasoning methods for the mixture of fuzzy information and intuitionistic fuzzy information based on the symmetric quintuple-implication principle where all five implication operators satisfy symmetry (also called the quintuple-implication principle). Such reasoning methods are based on the ideas that the input and the given if–then rule can be combined for calculation only when their information representations exhibit consistency. An inconsistent given if–then rule with two different representations should be regarded as the composition of two different consistent given if–then rules with their own unique representations. This paper then elaborates on the methods by employing the possibility and necessity operators and the quintuple-implication principle from the perspective of whether the representation of rule antecedent and rule consequent is consistent or not. The reductivity of all the proposed reasoning methods is also analyzed in detail. This paper mainly contributes to the development of a novel mixed information reasoning framework, along with the introduction of the quintuple-implication principle into reasoning with mixed information. The proposed methods have also been applied to pattern recognition, and several experiments demonstrate that the mixed information reasoning methods based on the quintuple-implication principle are superior to the corresponding methods based on the triple I principle.