This chapter outlines what the authors believe a theoretically grounded cognitive neuroscience of semantics should look like. It focuses on combinatory semantics, i.e. the composition operations that serve to build complex meanings from smaller parts. The chapter examines the formal syntax and semantics of the generative tradition (e.g., Heim and Kratzer 1998) as the cognitive model that guides this research and defines the operations whose neurobiology is to be investigated. It shows that linguistic theory offers a tremendous amount of detail, which may seem daunting to try to relate to neurobiological data. Theories of formal semantics have a crisp definition of semantic well-formedness: an expression is semantically well formed if the rules of composition yield a meaning for it. The chapter examines the anterior midline field (AMF) in a violation paradigm, to better connect the findings to event-related potentials (ERP) research which has been dominated by this type of design.