Antonyms of predicates on n-tuples of fuzzy sets. A characterization of involutions on [0,1]ⁿ

Obtaining the complement of a fuzzy set is usually done through a negation function. On the other hand, the antuonym of a predicate, which in classical logic could be considered to be the complement, is radically different from it in uncertain environments. To model it in fuzzy logic or in its extensions, it is common to use involutions, functions on the universe that satisfy some boundary, monotony and involution conditions. In particular, to obtain the antonym of a fuzzy predicate determined by n arguments, we will need an involution on [0,1]ⁿ. As involutions on [0,1]² were characterized in a previous work, in the present paper we firstly focus on involutions on [0,1]³, suggesting how involutions on [0,1]ⁿ could be. We then obtain the main result, the characterization of involutions in this set [0,1]ⁿ.