![Square of opposition, set diagrams](https://upload.wikimedia.org/wikipedia/commons/5/51/Square_of_opposition%2C_set_diagrams.svg)
In
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
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 (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
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 Ire ...
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 formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
, it follows immediately for the truth conditions for quantified sentences:
*