Conceptual scaling: from many-valued tables to formal contexts

A formal context is a binary table — an object either has an attribute or it doesn’t. Real datasets, of course, are rarely so accommodating. Ages are numbers. Colours are categories. Heights are floats. Conceptual scaling is the bridge between the many-valued world and the binary world that FCA needs.

A many-valued context is a quadruple $(G, M, W, I)$ where $G$ is a set of objects, $M$ a set of (many-valued) attributes, $W$ a set of possible values, and $I \subseteq G \times M \times W$ is a partial function: each pair $(g, m)$ either has no value, or exactly one value in $W$. A conceptual scale for an attribute $m$ is a formal context $\mathbb{S}_m = (W_m, M_m, I_m)$ whose objects are the possible values of $m$ and whose attributes encode the distinctions one wants to draw. Scaling the many-valued context against a family of scales — one per attribute — produces an ordinary (binary) formal context whose concept lattice can then be computed.

The choice of scale is not innocent. A nominal scale on a categorical attribute treats values as pairwise incomparable; an ordinal scale imposes a chain; an interordinal scale captures “at most $x$” and “at least $x$” simultaneously. The same numerical column can yield very different concept lattices depending on which scale one picks, and those differences correspond to different prior commitments about which distinctions matter.

This is an entry-point note; future case studies in this section will walk through specific scales on real data, including the use of ordinal motifs as analytical units for ordinal substructures. For a formal treatment, the canonical reference remains Ganter & Wille (1999).