 It has been suggested that PFO be used to accommodate the post-Fregean logicists’ need for second-order quantification. Since the ancestral of a dyadic predicate can be defined using only monadic second-order quantification, PFO does indeed serve the logical needs of Boolos’s  sub-logicism.[13]  But since the neo-logicist definition of \(F\approx G\) uses dyadic second-order logic, PFO alone does not have sufficient expressive power to accommodate the needs of neo-logicism. The neo-logicist may attempt to solve this problem by regarding equinumerosity as a primitive logical quantifier or by simulating dyadic second-order quantification in some suitable extension of PFO, as discussed in  Section 2.2[14]  (see Boccuni 2013 for another option).