We present an implemented method of parsing with Combinatory Categorial Grammar (CCG) that for the first time derives the exceptional scope behavior of indefinites in a principled and plausibly practical way. The account implements Charlow’s (2014) monadic approach to dynamic semantics, in which indefinites’ exceptional scope taking follows from the way the side effect of introducing a discourse referent survives the process of delimiting the scope of true quantifiers in a continuized grammar. To efficiently parse with this system, we extend Barker and Shan’s (2014) method of parsing with continuized grammars to only invoke monadic lifting and lowering where necessary, and define novel normal form constraints on lifting and lowering to avoid spurious ambiguities. We also integrate Steedman’s (2000) CCG for deriving basic predicate-argument structure and enrich it with a method of lexicalizing scope island constraints. We argue that the resulting system improves upon Steedman’s CCG in terms of theoretical perspicuity and empirical coverage while retaining many of its attractive computational properties.