Full Idea
The contradiction in Frege's system is due to the presence of second-order quantification, ..and Frege's explanation of the second-order quantifier, unlike that which he provides for the first-order one, appears to be substitutional rather than objectual.
Gist of Idea
Contradiction arises from Frege's substitutional account of second-order quantification
Source
comment on Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893], §25) by Michael Dummett - Frege philosophy of mathematics Ch.17
Book Reference
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.217
A Reaction
In Idea 9871 Dummett adds the further point that Frege lacks a clear notion of the domain of quantification. At this stage I don't fully understand this idea, but it is clearly of significance, so I will return to it.
Related Idea
Idea 9871 Frege always, and fatally, neglected the domain of quantification [Dummett on Frege]