Definition
A predicate is the part of a sentence that says something about a subject — in classical grammar, the verb phrase that follows the subject; in modern logic, an expression that takes one or more arguments and yields a truth value. A one-place predicate ascribes a property to a single object: "Socrates is mortal" applies the predicate "is mortal" to the argument Socrates. A two-place predicate expresses a relation between two objects: "Athens is east of Sparta" applies the predicate "is east of" to the pair. Predicates of higher arity express relations among three, four, or more arguments.
Logically, a predicate is a function from objects to truth values. Saturating it with the right number of arguments produces a complete statement that is either true or false; leaving the argument slots open produces an open formula that becomes meaningful only once it is quantified or instantiated.
Why it matters
How it works
In predicate logic, each predicate is associated with a fixed number of argument places and a denotation in the domain of discourse. A one-place predicate denotes a set — the set of objects that satisfy it. A two-place predicate denotes a set of ordered pairs — the pairs that stand in the named relation. Higher-arity predicates denote sets of longer tuples. This set-theoretic interpretation is what makes predicates analysable: instead of reasoning about words, the logician reasons about the structures they describe.
The expressive gain over propositional logic is dramatic. Propositional logic treats "all humans are mortal" and "Socrates is human" as two unrelated atomic statements and cannot draw the inference to "Socrates is mortal." Predicate logic dissects each sentence into predicates and arguments, applies the universal quantifier to one, instantiates it for Socrates, and chains the result through the second sentence to reach the conclusion. The internal structure that propositional logic discards is precisely the structure on which the inference depends. This is why every serious logic curriculum extends from propositional to predicate calculus — and why predicates, quantifiers, and variables are the central vocabulary of mathematical and philosophical proof.