What's an ancestral?

An "ancestral" is a kind of relation that is a sort of generalization of a more specific relation. It's easiest to explain this by means of an example. Consider the relation "being the parent of" This is a relation that holds between any person who has a child and that child, for example, Henry Fonda and Jane Fonda. Of course Henry Fonda was himself the child of another person -- maybe George Fonda. And George was the child of yet another person, Sarah Fonda, and so on. So we could, by going back through the generations, build a list of individuals, where each individual is the parent of the next individual on the list.
So consider the "Fonda List." Every person on that list is related to Jane in some way. Henry is her parent, George is her grandparent, Sarah is her great-grandparent, etc. But is there some way to talk about an arbitrary person on this list? There is! We can say that every person who is on this list is an *ancestor* of Jane's. Someone -- say Patrick -- is an ancestor of Jane's, if and only if Patrick is either a parent of Jane's, a parent of a parent of Jane's, a parent of a parent of a parent of Jane's, or...... That is, someone is an ancestor of Jane's just in case that person is somewhere on the Fonda List. The relation of "being an ancestor" is, in this way, a generalization of the "being a parent" relation. More precisely, the relation "being an ancestor" is the "transitive closure" of the relation of "being a parent of"
All ancestral relations have this character -- they are built out of reiterations of a specific relation that holds between two things that are, in some sense "adjacent" to each other. They are called "ancestrals" because the "is an ancestor of" relation gives the perfect model.
This notion is very useful in formal logic and mathematics. It was first defined formally by the logician and philosopher Gottlob Frege. If you'd like to see the formal details, I recommend this article in the Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/frege/

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