Full Idea
In second-order logic only the formation rules are completely formalizable, not the inference rules.
Gist of Idea
We can formalize second-order formation rules, but not inference rules
Source
Michael Potter (Set Theory and Its Philosophy [2004], 01.2)
Book Reference
Potter,Michael: 'Set Theory and Its Philosophy' [OUP 2004], p.13
A Reaction
He cites Gödel's First Incompleteness theorem for this.