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":

:$(Fx\; and\; xRy)\; o\; Fy.$

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

The ancestral is transitive:

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

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

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

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

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

Discussion

"Principia Mathematica" made repeated use of the ancestral, as does Quine's (1951) "Mathematical Logic".

ee also

*" Begriffsschrift"
*Gottlob Frege

References

*George Boolos, 1998. "Logic, Logic, and Logic". Harvard Univ. Press.
*Ivor Grattan-Guinness, 2000. "In Search of Mathematical Roots". Princeton Univ. Press.

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.