In the literature there have been a few works [1–4] that have dealt with obtaining metrics from associative, commutative, and monotonically increasing fuzzy logic connectives such as t-norms, t-conorms, copulas, and quasi-copulas. Recently, it has been shown [9] that a distance function dI can also be obtained from fuzzy implications which do not satisfy any of the above properties. This work studies the above distance along two aspects. Firstly, we investigate those implications I that satisfy a particular form of transitivity, viz. the SLK transitivity, that is both necessary and sufficient for the proposed distance to be a metric. In the recent past, monodistances w.r.t. a ternary relation, called the betweenness relation, defined on a set, 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 monodistances on a given betweenness set (X, B ). By characterising betweenness relations that can be obtained from a bounded below poset, our second contribution in this work is in showing that a monodistance on such betweenness sets (X, B ) can be obtained through dI. Our work seems to suggest that fuzzy implications are rather a natural choice for constructing monodistances. © 2022, Springer Nature Switzerland AG.