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Thermometer ratings and Likert scales are ubiquitous in social psychology, political psychology, and political science, even though critics have cautioned that researchers take the scores too literally. A measurement procedure based on arbitrary assumptions risks the real danger of generating scientifically meaningless inferences. Adopting a decision theoretic point of view, we use the concept of semiorders to capture the idea that a person giving 2 candidates distinct scores might or might not actually prefer one to the other, depending on the size of her threshold of discrimination. Furthermore, one respondent giving a candidate a lower score than another respondent could nevertheless be the stronger supporter. We state formal assumptions about the nature of preferences and propose a novel probabilistic response mechanism by which respondents construct numerical scores heterogeneously when asked to represent their preferences in a numerical format. We provide a proof of concept using maximum likelihood tests of our models on public domain American National Election Study data.