If any class closed under isomorphism is definable in sort logic, then sort logic inherits the same structural abstraction that set-theoretic construction already provides, making the contrast with set theory superficial.
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sort logic(Extension of many-sorted first-order logic; Väänänen 2014)
A many-sorted second-order logic that allows quantification over new sorts via a new logical symbol ∃̃, enabling formulas to look outside the structure where the formula is being interpreted
structural abstraction(as used in logic and mathematics)
The process of focusing on the underlying patterns and relationships that matter in a system, while ignoring the specific details that don't affect how it works.