There are two ways of handling this technical issue. The first is to pass to a larger language in which each element of the domain is named by a constant symbol. The second is to add to the interpretation a function that assigns each variable to an element of the domain. Then the T-schema can quantify over variations of the original interpretation in which this variable assignment function is changed, instead of quantifying over substitution instances. Some authors also admit propositional variables in first-order logic, which must then also be interpreted. A propositional variable can stand on its own as an atomic formula. The interpretation of a propositional variable is one of the two truth values true and false. Because the first-order interpretations described here are defined in set theory, they do not associate each predicate symbol with a property (or relation), but rather with the extension of that property (or relation). In other words, these first-order interpretations are extensional not intensional. Example of a first-order interpretation An example of interpretation of the language L described above is as follows. Domain: A chess set Individual constants: a: The white King b: The black Queen c: The white King's pawn F(x): x is a piece G(x): x is a pawn H(x): x is black I(x): x is white J(x, y): x can capture y In the interpretation of L: the following are true sentences: F(a), G(c), H(b), I(a) J(b, c), the following are false sentences: J(a, c), G(a). Non-empty domain requirement As stated above, a first-order interpretation is usually required to specify a nonempty set as the domain of discourse. The reason for this requirement is to guarantee that equivalences such as where x is not a free variable of φ, are logically valid. This equivalence holds in every interpretation with a nonempty domain, but does not always hold when empty domains are permitted. For example, the equivalence fails in any structure with an empty domain. Thus the proof theory of first-order logic becomes more complicated when empty structures are permitted. However, the gain in allowing them is negligible, as both the intended interpretations and the interesting interpretations of the theories people study have non-empty domains. Empty relations do not cause any problem for first-order interpretations, because there is no similar notion of passing a relation symbol across a logical connective, enlarging its scope in the process. Thus it is acceptable