In deductive systems in which the truth of an atomic formula $\mathrm{Ra}$ does not entail that $a$ exists, e.g., free logics, a distinguished unary predicate $E$ corresponding to a construction ‘$x$ exists’. Existence predicates are also used to model variable domain modal logics with a single domain by representing the domain of a world $w$ as the interpretation of the existence predicate at $w$. In many contexts, existence predicates are definable so that $Ea$ is equivalent to the formula $\exists x\left(x=a\right)$.