Editing First-order logic (section)

===Empty domains===
{{Main|Empty domain}}
The definition above requires that the domain of discourse of any interpretation must be nonempty. There are settings, such as [[inclusive logic]], where empty domains are permitted. Moreover, if a class of algebraic structures includes an empty structure (for example, there is an empty [[poset]]), that class can only be an elementary class in first-order logic if empty domains are permitted or the empty structure is removed from the class.

There are several difficulties with empty domains, however:
* Many common rules of inference are valid only when the domain of discourse is required to be nonempty. One example is the rule stating that <math>\varphi \lor \exists x \psi</math> implies <math>\exists x (\varphi \lor \psi)</math> when ''x'' is not a free variable in <math>\varphi</math>. This rule, which is used to put formulas into [[prenex normal form]], is sound in nonempty domains, but unsound if the empty domain is permitted.
* The definition of truth in an interpretation that uses a variable assignment function cannot work with empty domains, because there are no variable assignment functions whose range is empty. (Similarly, one cannot assign interpretations to constant symbols.) This truth definition requires that one must select a variable assignment function (μ above) before truth values for even atomic formulas can be defined. Then the truth value of a sentence is defined to be its truth value under any variable assignment, and it is proved that this truth value does not depend on which assignment is chosen. This technique does not work if there are no assignment functions at all; it must be changed to accommodate empty domains.
Thus, when the empty domain is permitted, it must often be treated as a special case. Most authors, however, simply exclude the empty domain by definition.