Second-Order Logic (SOL) extends first-order logic by allowing quantification not only over individual variables but also over predicates and relations. This means that in addition to saying "for all x" or "there exists an x," we can say "for all properties P" or "there exists a relation R." This added expressiveness enables more complex statements about mathematical structures and concepts. In SOL, we can express statements that are not possible in first-order logic, such as properties of sets or functions. For example, we can formalize concepts like completeness or compactness in mathematics more naturally. However, this increased power comes with greater complexity in terms of semantics and proof theory.