In mathematical logic, the ancestral relation (often shortened to ancestral) of a binary relation R is its transitive closure, however defined in a different way, see below.

Ancestral relations make their first appearance in Frege's Begriffsschrift. Frege later employed them in his Grundgesetze as part of his definition of the finite cardinals. Hence the ancestral was a key part of his search for a logicist foundation of arithmetic.

The numbered propositions below are taken from his Begriffsschrift and recast in contemporary notation.

A property P is called R-hereditary if, whenever x is P and xRy holds, then y is also P:

An individual b is said to be an R-ancestor of a, written aR^*b, if b has every R-hereditary property that all objects x such that aRx have:

The ancestral is a transitive relation:

Let the notation I(R) denote that R is functional (Frege calls such relations "many-one"):

If R is functional, then the ancestral of R is what nowadays is called connected^{[clarification needed]}: