Erratum to: On Definability in Dependence Logic

Abstract

A correction is needed for the singular case X = ∅ in Theorems 4.9, 4.10, 5.1, and 5.2. In Kontinen and Vaananen (2009), the open formulas of Dependence Logic (D) were studied. Formulas of dependence logic express properties of sets of assignments (teams). It was shown in Vaananen (2007) that every formula of dependence logic can be represented in an equivalent form in existential second-order logic ( 1 1) with an extra predicate, occurring only negatively, interpreting the team. In Theorems 4.9 and 4.10 of Kontinen and Vaananen (2009) it was claimed that also the converse holds, i.e., that for every vocabulary L and sentence φ ∈ 1 1[L ∪ {R}], in which R is k-ary (for some k ≥ 1) and occurs only negatively, there is a formula ψ(y1, . . . , yk) ∈ D[L] such that for all models A and teams X with domain {y1, . . . , yk} A | X ψ ⇔ (A, rel(X)) | φ, (1)