Monothetic

A monothetic class is defined in terms of characteristics that are both necessary and sufficient in order to identify members of that class. This way of defining a class is also termed the Aristotelian definition of a class.

A polythetic class is defined in terms of a broad set of criteria that are neither necessary nor sufficient. Each member of the category must possess a certain minimal number of defining characteristics, but none of the features has to be found in each member of the category. This way of defining classes is associated with Wittgenstein's concept of "family resemblances."

Monothetic type is a type in which all members are identical on all characteristics. Polythetic type is a type in which all members are similar, but not identical.

The distinction between monothetic and polythetic classification is discussed by Rijsbergen:

“An early statement of the distinction between monothetic and polythetic is given by Beckner (1959, p. 22): 'A class is ordinarily defined by reference to a set of properties which are both necessary and sufficient (by stipulation) for membership in the class. It is possible, however, to define a group K in terms of a set G of properties f1, f2, . . . , fn in a different manner. Suppose we have an aggregate of individuals (we shall not yet call them a class) such that

The first sentence of Beckner's statement refers to the classical Aristotelian definition of a class, which is now termed monothetic. The second part defines polythetic.” (Rijsbergen, 1979).

“To illustrate the basic distinction consider the following example (Figure 3.1) of 8 individuals (1-8) and 8 properties (A-H). The possession of a property is indicated by a plus sign. The individuals 1-4 constitute a polythetic group each individual possessing three out of four of the properties A,B,C,D. The other 4 individuals can be split into two monothetic classes {5,6} and {7,8}. The distinction between monothetic and polythetic is a particularly easy one to make providing the properties are of a simple kind, e.g. binary-state attributes. When the properties are more complex the definitions are rather more difficult to apply, and in any case are rather arbitrary.

Polythetic classification has been seriously criticized by, for example, Sutcliffe (1993, 1994, 1996). According to this criticism denies the theory of "polythetic classes" such as family, cluster and category the need for definition: there is said to be no property common to "members" of any such polythetic class. However, to be consistent with that denial, one cannot then use terms such as "cluster" or "category" as if they were classes in the sense of "monothetic class". Lacking that sense, terms such as "class" (family, category, cluster) "kind", and "membership" then become meaningless because in discourse one can make no categorical distinctions between things. Hence there can be no classification, and hence there can be no polythetic classification. This shows, according to Sutcliffe, that the theory of polytypy is logically incoherent.

Beckner, M. (1959). The Biological Way of Thought, Columbia University Press, New York.

Sutcliffe, J. P. (1993). Concept, class, and category in the tradition of Aristotle. IN: van Mechelen, I.; Hampton, J.; Michalski, R. S. & Theuns, P. (eds.). Categories and Concepts. London: Academic Press, pp. 35-65.

Sutcliffe, J. P. (1994). On the logical necessity and priority of a monothetic conception of class, and on the consequent inadequacy of polythetic accounts of category and categorization. IN: Diday, E.; Lechevallier, Y.; Schrader, M.; Bertrand, P. & Burtchy, B. (Eds.): New Approaches in Classification and Data Analysis. Berlin: Springer-Verlag, 55-63.