Empty domain

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 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, it follows immediately for the truth conditions for quantified sentences: 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 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, it follows immediately for the truth conditions for quantified sentences: