Predicate logic

Predicate logic is a sub-discipline of logic that had its roots in the last quarter of the nineteenth century, though it had to wait until the second decade of the twentieth century for a solid foundation. Like any other logic, it is concerned with the validity of arguments, though not of any kind: its interest lies in reasoning about what is universally true. As such, predicate logic is especially suited to reason about mathematical statements and can be considered a generalization of Aristotelian syllogisms. Predicate logic goes beyond syllogisms by introducing predicates with arbitrary numbers of arguments, and quantifiers that allow to refer either to all or to some of the elements in the universe that is under consideration. It has a proof theory, which consists of a set of rules that describe how to mechanically derive sentences from a given set of premises (and such derivations are called “proofs”), as well as a model theory that assigns meaning to the sentences with respect to structures so that a given sentence is either true or false in a given structure. Predicate logic is sound, meaning that every sentence that can be derived using the rules in its proof theory holds in every possible structure. More importantly, a fundamental result by Gödel in 1929 shows that predicate logic is also complete in the sense that if a sentence is true in every structure then it can be derived from the empty set of premises.