Several works have proposed the construction of distance functions using fuzzy logic connectives to proffer further applications of the corresponding connectives. In these works, the authors define a distance function using t-norms, t-conorms, copulas, or quasi-copulas, all of which are either associative, commutative or monotonic fuzzy logic connectives. In this work, we define a distance function, denoted dI, from a non-associative, non-commutative, and mixed-monotonic fuzzy logic connective, viz., a fuzzy implication I, and study the above distance function along two aspects. Firstly, we investigate the necessary and sufficient conditions for dI to be a metric, wherein the role played by a transitivity type functional inequality involving the considered fuzzy implication and the Łukasiewicz t-conorm is highlighted. In the recent past, monometrics w.r.t. a ternary relation, called the betweenness relation, have garnered a lot of attention for their important role in decision-making and penalty-based data aggregation. One of the major challenges herein is that of obtaining monometrics on a given betweenness set. Our second contribution in this work is in establishing the existence of pseudo-monometrics using dI, from whence it appears that fuzzy implications are a natural choice for obtaining pseudo-monometrics on a given betweenness set. © 2022 Elsevier B.V.