In
first-order logic the empty 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
Bernays Bernays is a surname. Notable people with the surname include:
* Adolphus Bernays (1795–1864), professor of German in London; brother of Isaac Bernays and father of:
** Lewis Adolphus Bernays (1831–1908), public servant and agricultural write ...
and
Schönfinkel (though possibly earlier) but oft-attributed to
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
George Boole in the late 19th century but this is debatable. In modern
model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
, it follows immediately for the truth conditions for quantified sentences:
*