Builds on top of propositional logic, adding notably existential quantification.
Existence and uniqueness results are fundamental in mathematics because we often define objects by their properties, and then start calling them "the object", which is fantastically convenient.
But calling something "the object" only makes sense if there exists exactly one, and only one, object that satisfies the properties.
One particular context where these come up very explicitly is in solutions to differential equations, e.g. existence and uniqueness of solutions of partial differential equations.
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First-order logic (FOL), also known as predicate logic or first-order predicate logic, is a formal system used in mathematical logic, philosophy, linguistics, and computer science to express statements about objects and their relationships. It expands upon propositional logic by introducing quantifiers and predicates, allowing for a more expressive representation of logical statements.