A logic allowing quantification over relations or subsets of the object domain as well as over its individual objects. The range of relation quantifiers depends on the chosen semantics.
Second-order quantifiers range over specified collections of relations, which can be proper subcollections of all external relations. With these collections as additional sorts, this is a form of first-order logic. Full-class conclusions cannot be inferred merely from truth with Henkin semantics.
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Second-order logic (SOL) is an extension of first-order logic (FOL) that allows quantification not only over individual variables (such as objects or elements of a domain) but also over predicates or sets of individuals. This additional expressive power makes second-order logic more powerful than first-order logic in certain ways, allowing for the formulation of more complex statements about mathematical structures and relationships.