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We present a model to tackle a fundamental but understudied problem in computational argumentation: proposition extraction. Propositions are the basic units of an argument and the primary building blocks of most argument mining systems. However, they are usually substituted by argumentative discourse units obtained via surface-level text segmentation, which may yield text segments that lack semantic information necessary for subsequent argument mining processes. In contrast, our cascade model aims to extract complete propositions by handling anaphora resolution, text segmentation, reported speech, questions, imperatives, missing subjects, and revision. We formulate each task as a computational problem and test various models using a corpus of the 2016 U.S. presidential debates. We show promising performance for some tasks and discuss main challenges in proposition extraction.