We present an untyped calculus of blame assignment for a higherorder contract system with two new operators: intersection and union. The specification of these operators is based on the corresponding type theoretic constructions. This connection makes intersection and union contracts their inevitable dynamic counterparts with a range of desirable properties and makes them suitable for subsequent integration in a gradual type system. A denotational specification provides the semantics of a

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... ct in terms of two sets: a set of terms satisfying the contract and a set of contexts respecting the contract. This kind of specification for contracts is novel and interesting in its own right. A nondeterministic operational semantics serves as the specification for contract monitoring and for proving its correctness. It is complemented by a deterministic semantics that is closer to an implementation and that is connected to the nondeterministic semantics by simulation. The calculus is the formal basis of TreatJS, a language embedded, higher-order contract system implemented for JavaScript.