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This paper argues that extant approaches to presupposition projection that either rely on strict linear order (Schlenker, 2009) or hierarchy (Romoli and Mandelkern, 2017) cannot provide a uniform account of data drawn from head-final languages. While building on Schlenker’s theory, this paper resolves the issues by restricting the calculation of local contexts to specific points in the parsing process. The consequence is that the theory makes a prediction robust to the head directionality parameter