Full Idea
Shapiro preferred second-order logic to set theory because second-order logic refers only to the relations and operations in a domain, and not to the other things that set-theory brings with it - other domains, higher-order relations, and so forth.
Gist of Idea
Second-order logic is better than set theory, since it only adds relations and operations, and nothing else
Source
report of Stewart Shapiro (Foundations without Foundationalism [1991]) by Shaughan Lavine - Understanding the Infinite VII.4
Book Reference
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.225