 Even more modest is Boolos’s sub-logicism, which rejects the idea (endorsed by both logicists and neo-logicists) that there are logical objects, but insists that Frege’s definition of the ancestral of a relation can be used to show, as against Kant, that at least some non-trivial mathematics is analytic (Boolos 1985b). Recall that a relation \(R\) stands to its ancestral \(\Rarel\) as the relation is a parent of stands to is an ancestor of. (More precisely, \(\Rarel\) holds between two objects \(x\) and \(y\) just in case \(x\) and \(y\) are connected through a finite sequence of objects each of which bears \(R\) to its successor.) Frege gives a second-order definition of the ancestral relation \(\Rarel\) by laying down that \(x\) and \(y\) are related by \(\Rarel\) just in case \(y\) has every property that is had by \(x\)’s \(R\)-successors and inherited under the \(R\)-relation: