Full Idea
The two frequent modern objects to logicism are that set theory is not part of logic, or that it is of no interest to 'reduce' a mathematical theory to another, more complex, one.
Gist of Idea
Set theory isn't part of logic, and why reduce to something more complex?
Source
Michael Dummett (Frege philosophy of mathematics [1991], Ch.18)
Book Reference
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.224
A Reaction
Dummett says these are irrelevant (see context). The first one seems a good objection. The second one less so, because whether something is 'complex' is a quite different issue from whether it is ontologically more fundamental.