Editing First-order logic (section)

===First-order logic without equality===
An alternate approach considers the equality relation to be a non-logical symbol. This convention is known as ''first-order logic without equality''. If an equality relation is included in the signature, the axioms of equality must now be added to the theories under consideration, if desired, instead of being considered rules of logic. The main difference between this method and first-order logic with equality is that an interpretation may now interpret two distinct individuals as "equal" (although, by Leibniz's law, these will satisfy exactly the same formulas under any interpretation). That is, the equality relation may now be interpreted by an arbitrary [[equivalence relation]] on the domain of discourse that is [[congruence relation|congruent]] with respect to the functions and relations of the interpretation.

When this second convention is followed, the term ''normal model'' is used to refer to an interpretation where no distinct individuals ''a'' and ''b'' satisfy ''a'' = ''b''. In first-order logic with equality, only normal models are considered, and so there is no term for a model other than a normal model. When first-order logic without equality is studied, it is necessary to amend the statements of results such as the [[Löwenheim–Skolem theorem]] so that only normal models are considered.

First-order logic without equality is often employed in the context of [[second-order arithmetic]] and other higher-order theories of arithmetic, where the equality relation between sets of natural numbers is usually omitted.