Unremoved Continuity: A Resolution of the Ship of Theseus

The Ship of Theseus puzzle is often taken to show that our ordinary judgments about repair, replacement, and reassembly pull in incompatible directions. This paper argues that, once we are explicit about which judgments we want to retain, a single overlapbased principle yields a stable and precise package of answers. The central claim is that numerical identity over time for artefacts (and, more cautiously, organisms and persons) is fixed by unremoved continuity: there must be a finite chain of assembled states each sharing some unremoved part with its immediate predecessor. I formalise this as the Unremoved Continuity Principle (UCP), represent histories as a state graph whose vertices are assembled states and whose edges are admissible time-steps with nonzero unremoved overlap, and define identity as undirected path-connectedness in this graph. On that basis I prove several core results: gradual and even ω-length replacement preserves identity; identity cannot cross intervals in which there are only heaps (no assembled states); and in the standard Theseus case the maintained ship is identical to the original, whereas a later vessel rebuilt from stored parts is numerically distinct. Further meta-results show that these verdicts are invariant under changes of temporal granularity (mesh invariance), robust under reasonable precisifications of the “ship” predicate, and stable under part-splitting and fusion. The overall picture is endurantist in spirit but compatible with coarse-grained four-dimensionalism, and it is broadly friendly to animalist views of organisms and persons. The proposal does not claim to settle every puzzle about persistence, but it shows that if we want to preserve a familiar set of judgments about repair, reassembly, and annihilation, then an overlap-based, graph-theoretic criterion like UCP offers a principled way of doing so.