The language of classical first-order logic
First-order logic (sometimes called predicate logic) includes all of the connectives of propositional logic. Unlike propositional logic, however, first-order logic analyzes simple sentences into terms and predicates. We use uppercase roman letters as predicates, lowercase roman letters a through t as (individual) constants, and lowercase roman letters u through z as (individual) variables. Predicates, constants, and variables may be augmented with subscripts if necessary, thus guaranteeing an infinite supply of each.
Constants function like names in English, and variables function like pronouns. Together constants and variables count as terms. Predicates have arities, where an arity is the number of terms to which a predicate applies. In English, for example, the arity of the predicate runs in John runs is 1 – it combines with a single term, John in this case – while the arity of the predicate loves in John loves Sue is 2 – it combines with two terms. Atomic formulas are formed by writing predicates in initial position followed by an appropriate number of terms (determined by the predicate's arity). John runs and John loves Sue might thus be symbolized as Rj and Ljs.
There are two standard quantifiers in first-order logic, the universal and the existential quantifiers. We'll use ∀ as the universal quantifier symbol and ∃ as the existential quantifier symbol.