Talk:Ancestral relation
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R-ancestor
editIt would perhaps be helpful for some readers if the article could find a way to explain how it comes to be that the terms R-hereditary and R-ancestor both use a genealogical metaphor, but apply it in opposite directions.
It seems clear enough to say that a property P is inherited in a relation R, just as heirlooms are inherited from parent to child and from ancestor to descendant. And this metaphor is Frege's: where this article might say "Property P is R-hereditary", Frege would say (see section 24 of the Begriffsschrift) "Die Eigenschaft P vererbt sich in der R-Reihe" (property P is inherited in the R-series").
It's also clear enough (after a little explanation), to say that if aRb means b is the parent of a, then aR*b means that b is the ancestor of a.
But notice that the genealogical relation at the basis of the metaphorical terminology has been turned around: the explanation of R-hereditary requires that aRb mean a is the parent of b, and the explanation of R-ancestor requires it to mean that a is the child of b. How can Russell and Whitehead, or Quine, or Boolos, have thought that that was a good idea? From here it looks like a choice of terminology contemptuously designed to fight with any reader's intuition, but presumably they had some rationale for it. What was that rationale?
That choice (for what it's worth) was not Frege's: he defines no term that serves even approximately the same function as R-ancestor. In his words (sec. 26 of Begriffsschrift), the sentence "a is the R-ancestor of b" would be rendered "b folgt in der R-Reihe auf a", oder: "a geht in der R-Reihe dem b vorher" ("b follows a in the series R", or "a precedes b in the series R").
I tried to add a sentence explaining that the term R-ancestor is derived from taking the transitive closure of the parent relation as an example, but gave up when I could not find a clean formulation, or one that would not draw 'original research' flags. Perhaps someone else can do better. C.M.Sperberg-McQueen (talk) 01:24, 4 May 2023 (UTC)