In relational databases the class of weak functional dependencies is finitely axiomatisable and the associated implication problem is coNP-complete in general. Our first main result extends this axiomatisation to databases in which complex elements can be derived from atomic ones by finitely many nestings of record, list and disjoint union constructors. In particular, we construct two nested tuples that can serve as a counterexample relation for the implication of weak functional dependencies. We further apply this construction to show an equivalence to truth assignments that serve as counterexamples for the implication of propositional clauses. Hence, we characterise the implication of weak functional dependencies in complex-value databases in completely logical terms. Consequently, state-of-the-art SAT solvers can be applied to reason about weak functional dependencies in relational and complex-value databases.