Editing First-order logic (section)

===Hilbert-style systems and natural deduction===
A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a ''logical axiom'', a hypothesis that has been assumed for the derivation at hand or follows from previous formulas via a rule of inference. The logical axioms consist of several [[axiom schema]]s of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only [[modus ponens]] and [[universal generalization]] as rules of inference.

Natural deduction systems resemble Hilbert-style systems in that a deduction is a finite list of formulas. However, natural deduction systems have no logical axioms; they compensate by adding additional rules of inference that can be used to manipulate the logical connectives in formulas in the proof.