Large language model (LLM) evaluations often assume there is a single correct response -- a gold label -- for each item in the evaluation corpus. However, some tasks can be ambiguous -- i.e., they provide insufficient information to identify a unique interpretation -- or vague -- i.e., they do not clearly indicate where to draw the line when making a determination. Both ambiguity and vagueness can cause task indeterminacy -- the condition where some items in the evaluation corpus have more than one correct response. In this paper, we develop a framework for evaluating LLMs under task indeterminacy. Our framework disentangles the relationships between task specification, human ratings, and LLM responses in the LLM evaluation pipeline. Using our framework, we conduct a synthetic experiment showing that evaluations that use the "gold label" assumption underestimate the true performance. We also provide a method for estimating an error-adjusted performance interval given partial knowledge about indeterminate items in the evaluation corpus. We conclude by outlining implications of our work for the research community.