ancestral (of a given relation R), the relation (also called the transitive closure of R) that relates one given individual to a second if and only if the first can be ‘reached’ from the second by repeated ‘applications’ of the given relation R. The ancestor relation is the ancestral of the parent relation since one person is an ancestor of a second if the first is a parent of the second or the first is a parent of a parent of the second or the first is a parent of a parent of a parent of the second, and so on. Frege discovered a simple method of giving a materially adequate and formally correct definition of the ancestral of a given relation in terms of the relation itself (plus logical concepts). This method is informally illustrated as follows: in order for one person A to be an ancestor of a second person B it is necessary and sufficient for A to have every property that belongs to every parent of B and that belongs to every parent of any person to whom it belongs. This and other similar methods made possible the reduction of all numerical concepts to those of zero and successor, which Frege then attempted to reduce to concepts of pure logic. Frege’s definition of the ancestral has become a paradigm in modern analytic philosophy as well as a historical benchmark of the watershed between traditional logic and modern logic. It demonstrates the exactness of modern logical analysis and, in comparison, the narrowness of traditional logic. See also FREGE, LOGICISM , RELATIO. J.Cor.