Ancestral relation

Definition

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

A property P is called R-hereditary if, whenever x is P and xRy holds, then y is also P:

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

The ancestral is a transitive relation:

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

If R is functional, then the ancestral of R is what nowadays is called connected:

Relationship to transitive closure

The Ancestral relation R ∗ is equal to the transitive closure R + of R . Indeed, R ∗ is transitive (see 98 above), R ∗ contains R (indeed, if aRb then, of course, b has every R-hereditary property that all objects x such that aRx have, because b is one of them), and finally, R ∗ is contained in R + (indeed, assume a R ∗ b ; take the property F x to be a R + x ; then the two premises, ∀ x ( a R x → F x ) and ∀ x ∀ y ( F x ∧ x R y → F y ) , are obviously satisfied; therefore, F b , which means a R + b , by our choice of F ). See also Boolos's book below, page 8.

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. It is controversial whether second-order logic is really "logic" at all. Quine famously claimed that it was not, 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).