Polythetic Classes. In contrast to monothetic classes, polythetic classes do not contain cases that are identical on all variables, but rather group cases by overall greatest similarity. For example, Case I and Case 2 might be similar on variables I and 2, while Case I and Case 3 are similar on variables I and 3. Thus Cases 1,2, and 3 are not identical on variables I, 2, and 3, but share an overall similarity greater than the similarity with cases in other classes. That is, they are more similar to the cases in their class than to the cases in other classes. As defined by Beckner (1959, p. 22) a monothetic class requires all cases to have a set of properties that are both necessary and sufficient. In contrast, a polythetic class is defined differently. We can define a polythetic class in terms of a set G of properties fl ,f2 . . . ,fn such that:
1. Each one possesses a large (but unspecified) number of the properties in G
2. Each f in G is possessed by large numbers of these individuals
3. No fin G is possessed by every individualin the aggregate. (Beckner, 1959, p.22)
A class is called polythetic if just the first two conditions are fulfilled, and fully polythetic if all three are fulfilled. That is, a class is fully polythetic if the objects in it do not share even a single characteristic in common.
