Full Idea
If the goal of logical study is to present a canon of inference, a calculus which codifies correct inference patterns, then second-order logic is a non-starter.
Gist of Idea
If the aim of logic is to codify inferences, second-order logic is useless
Source
Stewart Shapiro (Higher-Order Logic [2001], 2.4)
Book Reference
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.51
A Reaction
This seems to be because it is not 'complete'. However, moves like plural quantification seem aimed at capturing ordinary language inferences, so the difficulty is only that there isn't a precise 'calculus'.