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	\ո		In [[mathematical logic]], the '''ancestral relation''' (often shortened to '''ancestral''') of an arbitrary [[binary relation]] ''R'' is defined below.
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			The ancestral makes its first appearance in [[Frege]]'s ''[[Begriffsschrift]]''. Frege later employed it in his ''Grundgesetze'' as part of his definition of the [[natural number]]s (actually the finite [[cardinal number\|cardinals]]). Hence the ancestral was a key part of his search for a [[logicism\|logicist]] foundation of arithmetic.

			==Definition==
			The numbered propositions below are taken from his ''[[Begriffsschrift]]'' and recast in contemporary notation.

			A [[property (philosophy)\|property]] ''F'' is called "''R''-'''hereditary'''" if, whenever ''x'' is ''F'' and ''xRy'' holds, then ''y'' is also ''F'':

			:(''Fx'' ∧ ''xRy'') → ''Fy''.

			Frege then defined ''b'' to be an ''R''-'''ancestor''' of ''a'', written ''aR<sup>*</sup>b'', if ''b'' has every ''R''-hereditary property that all objects ''x'' such that ''aRx'' have:

			76: <math>\Vdash</math> ''aR<sup>*</sup>b'' ↔ ∀''F'' ∀''x'' ∀''y'' [((''aRx'' → ''Fx'') ∧ (''Fx'' ∧ ''xRy'' → ''Fy'')) → ''Fb''].

			The ancestral is [[transitive relation\|transitive]]:

			98: ⊢ (''aR<sup></sup>b'' ∧ ''bR<sup></sup>c'') → ''aR<sup>*</sup>c''.

			Let the notation ''I''(''R'') denote that ''R'' is [[function (mathematics)\|functional]] (Frege calls such relations "many-one"):

			115: <math>\Vdash</math> ''I''(''R'') ↔ ∀''x'' ∀''y'' ∀''z'' [(''xRy'' ∧ ''xRz'') → ''y''=''z''],

			If ''R'' is [[function (mathematics)\|functional]], then the ancestral of ''R'' is what nowadays is called '''[[connectedness\|connected]]'''{{clarify\|reason=The article referred to apparently doesn't handle relations. The notion of a connected graph defined there is different from the notion defined in prop.133.\|date=September 2013}}:

			133: ⊢ (''I''(''R'') ∧ ''aR<sup></sup>b'' ∧ ''aR<sup></sup>c'') → (''bR<sup></sup>c'' ∨ ''b''=''c'' ∨ ''cR<sup></sup>b'').

			==Discussion==
			''[[Principia Mathematica]]'' made repeated use of the ancestral, as does [[W. V. Quine\|Quine's]] (1951) ''Mathematical Logic''.

			However, it is worth noting that the ancestral relation cannot be defined in [[first-order logic]], and following the resolution of [[Russell's paradox]] both Frege and Quine largely considered the use of [[second-order logic]] a questionable approach. In particular, Quine did not consider second-order logic to be "logic" at all, despite his reliance upon it for his 1951 book (which largely retells Principia in abbreviated form, for which second-order logic is required to fit its theorems).

			==See also==
			*''[[ Begriffsschrift]]''
			*[[Gottlob Frege]]
			[[Transitive closure]] - The ancestral of a relation is not to be confused with its [[Reflexive transitive closure]], although the notation ''R<sup></sup>'' is used for both.

			==References==
			*[[George Boolos]], 1998. ''Logic, Logic, and Logic''. Harvard Univ. Press.
			*[[Ivor Grattan-Guinness]], 2000. ''In Search of Mathematical Roots''. Princeton Univ. Press.
			*[[Willard Van Orman Quine]], 1951 (1940). ''Mathematical Logic''. Harvard Univ. Press. ISBN 0-674-55451-5.

			==External links==
			*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/frege-logic/ Frege's Logic, Theorem, and Foundations for Arithmetic]" -- by [[Edward N. Zalta]]. Section 4.2.

			[[Category:Mathematical relations]]

			[[ja:概念記法]]

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