How does second-order logic go beyond first-order logic?

Second-order logic extends first-order logic by expanding the scope of what can be quantified. While first-order logic—often called the lower predicate calculus—restricts quantifiers to range only over individual objects within a universe of discourse, second-order logic permits quantification over properties, relations, and sets of individuals.12

This added expressive power allows for the formation of formulas that cannot be constructed in first-order systems. For example, in the second-order predicate calculus, one can express the proposition that every property has at least one instance, represented as $(\forall \varphi )(\exists x)\varphi x$. Furthermore, second-order logic allows for the definition of identity through the principle of the identity of indiscernibles, defining $x=y$ as the condition that every property possessed by $x$ is also possessed by $y$ and vice versa.2

The separation of these systems was formalized in the early 20th century, notably by David Hilbert and Wilhelm Ackermann, to distinguish between systems that quantify only over individuals and those that include higher-order entities.1