First-order logic builds upon propositional logic by adding objects and relations between objects in a domain. It allows for more expressive power than propositional logic through the use of variables, quantifiers, and relations. Logical sentences in first-order logic make assertions about objects and their relations that are either true or false. Key aspects of first-order logic include atomic sentences using relations, complex sentences with connectives, quantification over variables, and nested quantifiers. Common mistakes to avoid include using the wrong main connective with quantifiers. Examples demonstrate using first-order logic to represent relationships in a kinship domain.