# Ancestral relation

In mathematical logic, the ancestral relation (often shortened to ancestral) of an arbitrary binary relation R is defined below.

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 numbers (actually the finite cardinals). Hence the ancestral was a key part of his search for a logicist foundation of arithmetic.

## Definition

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

The property F is "R-hereditary" if, whenever x is F and xRy, y is also F:

$(Fx \and xRy) \to Fy.$

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

76: $\Vdash aR*b \leftrightarrow \forall F \forall x \forall y [((aRx \to Fx) \wedge (Fx \wedge xRy \to Fy)) \to Fb]$.

The ancestral is transitive:

98: $\vdash (aR*b \wedge bR*c) \to aR*c.$

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

115: $\Vdash I(R) \leftrightarrow \forall x \forall y \forall z [(xRy \wedge xRz) \to y=z],$

If R is functional, we say nowadays that the ancestral of R is connected:

133: $\vdash (I(R) \wedge aR*b \wedge aR*c) \to (bR*c \vee b=c \vee cR*b).$

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## Discussion

Principia Mathematica made repeated use of the ancestral, as does 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).

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