In recent years second-order logic has made something of a recovery, buoyed by George Boolos' interpretation of second-order quantification as plural quantification over the same domain of objects as first-order quantification (Boolos 1984). Boolos furthermore points to the claimed nonfirstorderizability of sentences such as "Some critics admire only each other" and "Some of Fianchetto's men went into the warehouse unaccompanied by anyone else" which he argues can only be expressed by the full force of second-order quantification. However, generalized quantification and partially ordered, or branching, quantification may suffice to express a certain class of purportedly nonfirstorderizable sentences as well and it does not appeal to second-order quantification.

https://en.wikipedia.org/wiki/Nonfirstorderizability