The question as to whether the propositional logic of Heyting, which was a formalization of Brouwer's intuitionistic logic, is finitely many valued or not, was open for a while (the question was asked by Hahn). Kurt Gödel (1932) introduced an infinite decreasing chain of intermediate logics, which are known nowadays as Gödel logics, for showing that the intuitionistic logic is not finitely (man...

As a further extension of the fuzzy set and intuitive set, interval-valued (IIFS) is more effective tool to deal with uncertain problems. However, classical rough based on equivalence relation, which do not apply IIFS. In this paper, we combine IIFS ordered information system obtain (IIFOIS). On basis, three types multiple granulation models dominance relation are established effectively overco...

The Atanassov’s intuitionistic fuzzy (IF) set theory has become a popular topic of investigation in the fuzzy set community. However, there is less investigation on the representation of level sets and extension principles for interval-valued intuitionistic fuzzy (IVIF) sets as well as algebraic operations. In this paper, firstly the representation theorem of IVIF sets is proposed by using the ...

The notion of uncertainty in expert systems is dealing with vague data, incomplete information, and imprecise knowledge. Different uncertainty types which are imprecision, vagueness, ambiguity, and inconsistence need different handling models. Uncertain knowledge representation and analysis is an essential issue. Classical probability, Bayes theory, Dempster–Shafer theory, certainty factor and ...

The linear programming technique for multidimensional analysis of preference (LINMAP) is one of the well-known methods for multiple attribute group decision making (MAGDM). In this paper, we extend the LINMAP method to solve MAGDM problems in interval-valued intuitionistic fuzzy environment in which all the preference information provided by the decisionmakers is presented as interval-valued in...

One of the tools for building new fixed-point results is use symmetry in distance functions. The symmetric property metrics particularly useful constructing contractive inequalities analyzing different models practical consequences. A lot important invariant point crisp mappings have been improved by using metrics. However, more than a handful theorems spaces are yet to be investigated fuzzy ve...

In some multi-attribute decision making (MADM) problems, various relationships among the decision attributes should be considered. This paper investigates the prioritization relationship of attributes in MADM with trapezoidal intuitionistic fuzzy numbers (TrIFNs). TrIFNs are a special intuitionistic fuzzy set on a real number set and have the better capability to model ill-known quantities. Fir...