In this talk, I will discuss an ongoing line of research in the relational (non topological) semantics of non-distributive logics (aka LE-logics, i.e. those logics canonically associated with varieties of normal lattice expansions). The developments we consider are technically rooted in dual characterization results and insights from unified correspondence theory. However, these developments also have broader, conceptual ramifications for the intuitive meaning of non-distributive logics. Specifically, I will discuss two types of relational semantics for non-distributive logics: one based on polarities, or formal contexts from FCA, and another based on reflexive graphs. I will argue that the polarity-based semantics supports the recognition of non-distributive logics as the logics of categories or concepts, and the graph-based semantics supports their recognition as hyperconstructive logics of evidential truth.