Lower and Upper Bounds on the Vagueness Values of Logical Operations in Pliant Systems

This study focuses on the measurement of vagueness of continuous-valued logical variables and connectives. Although fuzziness measures are widely used to express the vagueness of fuzzy sets, these measures have not been extensively applied with the purpose of characterizing the vagueness of logical expressions using the vagueness of their arguments. We seek to find an answer to the question of how the vagueness value of the result of a continuous-valued conjunction (disjunction, respectively) and the vagueness values of the variables in this conjunction (disjunction, respectively) are related to each other. We answer this question for the special case where the logical operators are implemented in a specific logical system, called the Pliant system, which consists of a strict triangular norm, a strict triangular conorm and a negation operator, and all these operators are determined by a common generator function. We utilize a parametric, operator-dependent entropy to interpret the vagueness. We point out that the proposed vagueness entropy is closely related to the above-mentioned logical system, that is, the vagueness entropy and the system have a common generator function. This connection allows us to give lower and upper bounds for the vagueness value of the result of a continuous-valued conjunction (disjunction, respectively) when this operation is realized by a strict triangular norm (conorm, respectively) of a Pliant system.