In
first-order logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
, 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 and
Schönfinkel Schönfinkel ( ''Sheynfinkel'', ''Šejnfinkeľ''):
* Moses (Ilyich) Schönfinkel, born ''Moisei (Moshe) Isai'evich Sheinfinkel'' (1889, Ekaterinoslav - 1942, Moscow)
** The Bernays–Schönfinkel class (also ''Bernays–Schönfinkel-Ramsey class' ...
(though possibly earlier) but oft-attributed to
Quine's 1951 ''Mathematical Logic''.
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
George Boole ( ; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. H ...
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 (mat ...
, it follows immediately for the truth conditions for quantified sentences:
*