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Abstract: Symmetry considerations are sometimes used to motivate “structuralist” views in metaphysics: e.g. the view that motion is fundamentally relative (not absolute), that quantities like mass are fundamentally relational (not intrinsic), and others besides. These structuralist views all entail a restricted possibility space: e.g. that there is no distinction between worlds that agree on all facts about relative motions, that there is no distinction between worlds that agree on all facts about mass relations, and so on. This restricted possibility space is often considered a virtue, but some have argued that it is a vice. For example, it has been argued (in separate works by Earman, Pooley, Baker, and others) that it leads to physical theories that are pathologically indeterministic and non-local. In response, I think the structuralist should distinguish between different senses of “possible”. Structuralist views do indeed imply a restricted possibility space, but not in the same sense of “possibility” in which determinism and non-locality are properly defined. If that’s right, then structuralist views do not yield pathological physical theories after all. The challenge is to clearly articulate these different senses of “possibility”, and I will make a start at doing so.