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{{Short description|Concept in first-order logic}}
[[File:Square of opposition, set diagrams.svg|thumb|In modern logic only the contradictories in the [[Philosophy:Square of opposition|square of opposition]] apply, because domains may be empty.<br><br>(Black areas are empty,<br>red areas are nonempty.)]]
In [[First-order logic|first-order logic]], the '''empty [[Philosophy:Domain of discourse|domain]]''' is the empty set having no members. In traditional and classical logic domains are restrictedly non-empty in order that certain theorems be valid. Interpretations with an empty domain are shown to be a trivial case by a convention originating at least in 1927 with [[Biography:Paul Bernays|Bernays]] and Schönfinkel (though possibly earlier) but oft-attributed to [[Biography:Willard Van Orman Quine|Quine]] 1951. The convention is to assign any formula beginning with a universal quantifier the value ''truth'' while any formula beginning with an existential quantifier is assigned the value ''falsehood''. This follows from the idea that existentially quantified statements have existential import (i.e. they imply the existence of something) while universally quantified statements do not. This interpretation reportedly stems from [[Biography:George Boole|George Boole]] in the late 19th century but this is debatable. In modern [[Model theory|model theory]], it follows immediately for the truth conditions for quantified sentences:

*<math>A\models\exists x\phi(x) \text{ iff there is an } a\in A\text{ such that }A\models\phi[a]</math>
*<math>A\models\forall x\phi(x) \text{ iff every } a\in A\text{ is such that }A\models\phi[a]</math>

In other words, an existential quantification of the open formula &phi; is true in a model iff there is some element in the domain (of the model) that satisfies the formula; i.e. iff that element has the property denoted by the open formula. A universal quantification of an open formula &phi; is true in a model iff every element in the domain satisfies that formula. (Note that in the metalanguage, "everything that is such that X is such that Y" is interpreted as a [[Universal generalization|universal generalization]] of the [[Material conditional|material conditional]] "if anything is such that X then it is such that Y". Also, the quantifiers are given their usual objectual readings, so that a positive existential statement has existential import, while a universal one does not.) An analogous case concerns the empty conjunction and the empty disjunction. The semantic clauses for, respectively, conjunctions and disjunctions are given by

*<math>A\models \phi_1\land\dots\land\phi_n \iff \forall\phi_i (1\leq i\leq n), A\models \phi_i</math>
*<math>A\models \phi_1\lor\dots\lor\phi_n \iff \exists\phi_i (1\leq i\leq n), A\models \phi_i</math>.

It is easy to see that the empty conjunction is trivially true, and the empty disjunction trivially false.

Logics whose theorems are valid in every, including the empty, domain were first considered by Jaskowski 1934, Mostowski 1951, Hailperin 1953, Quine 1954, Leonard 1956, and Hintikka 1959. While Quine called such logics "inclusive" logic they are now referred to as [[Philosophy:Free logic|free logic]].

== See also ==
* [[Philosophy:Logical cube|Logical cube]]
* [[Philosophy:Logical hexagon|Logical hexagon]]
* Octagon of Prophecies
* [[Philosophy:Square of opposition|Square of opposition]]
* [[Triangle of opposition]]
* Table of logic symbols

[[Category:Predicate logic]]


{{Sourceattribution|Empty domain}}