On a Parametric Measure of Vagueness

In the application of fuzzy sets, the greatest uncertainty appears when the membership grade in a fuzzy set is equal to the membership grade in the complement of this fuzzy set. That is, the membership grade is equal to the value of the fixed point (<inline-formula><tex-math notation="LaTeX">$\nu$</tex-math></inline-formula>) of the underlying complement (negation) operator. The fixed point of the standard Zadeh negation is 0.5. In fuzzy set theory, the complement of a fuzzy set can be defined using a strong complement (negation) operator that differs from the standard Zadeh negation. In this case, the fixed point of the strong complement (negation) operator is not necessarily 0.5. In this short paper, we present the concept of a parametric fuzziness measure called the <inline-formula><tex-math notation="LaTeX">$\nu$</tex-math></inline-formula>-maximal vagueness measure. This new measure may be regarded as a generalized fuzziness measure. Here, we present an operator system-dependent kernel, which we call the <inline-formula><tex-math notation="LaTeX">$\nu$</tex-math></inline-formula>-maximal vagueness entropy, and using this, we construct a <inline-formula><tex-math notation="LaTeX">$\nu$</tex-math></inline-formula>-maximal vagueness measure. The proposed vagueness entropy is based on a common generator function of a strict triangular norm and a strong negation operator. The <inline-formula><tex-math notation="LaTeX">$\nu$</tex-math></inline-formula>-maximal vagueness entropy is flexible and operator-dependent, and it can be readily adapted to a continuous-valued logical system. Furthermore, the parameter <inline-formula><tex-math notation="LaTeX">$\nu$</tex-math></inline-formula> has a clear semantic meaning.