Propositional logic is the logic of statements that can be true or false, or take some value in a boolean algebra. The logic of most mathematical arguments involves more than just this: it involves mathematical objects from one or other domain, such as the set of natural numbers, real numbers, complex numbers, etc. If we introduce such objects into our formal system for proof we get what is known as first-order logic, or predicate logic.
As for any of our other logics, first-order logic would not be so interesting if it was just a system for writing and mechanically checking formal proofs for one particular domain of mathematical work. But fortunately it can be interpreted in a rather general class of mathematical structures and the theory of these structures is a sort of generalised algebraic theory that applies equally well to groups, rings, fields, and many other familiar structures, so first-order logic can be applied to a wide range of mathematical subject areas.
There are Completeness and Soundness Theorems for first-order logic too. In a similar way to the Completeness and Soundness Theorems we have already seen, they can be read as stating the correctness and adequacy of our logical system, or as much more interesting constructive statements that enable new structures to be created and analysed.
We will start here by discussing the idea of first-order language, and the sorts of things that can (and cannot) be expressed in first-order logic. Later on, we will give some rules for a proof system for this logic.