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Per-PPs like "*per milliliter"* can appear in measure phrases (MPs) like "*0.9 grams per milliliter*." Such MPs raise questions about how measurement theory used in science relates to language. Measurement theory posits complex dimensions that are quotients of more basic dimensions, as well as corresponding complex measures, or quotient quantities. For example, given the dimensions weight (WT) and volume (VOL), measurement theory admits WT/ VOL, the dimension of density, as well as quotient quantities like 0.9g/ml. Do per-MPs denote these types of quantities? In pioneering work, Coppock (2021) hypothesizes that language indeed aligns with measurement theory in this way. Here, we challenge this view. While the truth conditions of sentences like "The oil weighs 0.9 grams per milliliter" can be rendered with reference to quotient quantities, we argue that such renditions do not reflect compositional structure. In the actual composition, we argue, per is covertly anaphoric. This allows for the same entity to be measured twice, in different basic dimensions, which can achieve the effect of measuring in a complex dimension, like density, without needing to interpret per-MPs as quotient quantities.
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