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The central claim of the Parfitian psychological approach to personal identity is that the fact about personal identity is underpinned by a “non-branching” psychological continuity relation. Hence, for the advocates of the Parfitian view, it is important to understand what it is for a relation to take or not take a branching form. Nonetheless, very few attempts have been made in the literature of personal identity to define the “non-branching clause.” This paper undertakes this task. Drawing upon a recent debate between Anthony Brueckner and Harold Noonan on the issue, I present three candidates for the non-branching clause.