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Abstract The preference relation is mathematically expressed as a binary“-relation”. For any two objects and , says that “ is at least as preferred as ”. This paper first reviews the research of preference relations in two directions, from qualitative to quantitative, and vice versa. On the basis of the review, we illustrate how to represent the“-relation” with a probability function in a quantitative way. In the other direction, the quantitative method in decision theory is explained in detail. After that, we introduce the pure qualitative way that logicians adopt to capture the quantitative interpretation of the“-relation”. We argue that there is a clear connection between these two approaches, but that what makes them differ from each other are their underlying methodologies. Keeping in mind the above-mentioned differences, we propose a novel approach called the “probabilistic preference comparing method”. It introduces a -function to record the number of occurrences of “-relation” and “≥-relation” between members in two separate sets. Then, the total probability of occurrences of both “-relation” and “≥-relation” is calculated in terms of the probability function. Consequentially, the “-relation” between these two sets can be defined. Clearly, this method is a combination of qualitative and quantitative approaches. It reflects the qualitative “-relation” between members, but also takes the probability of their occurrence into account. We illustrate how this new method can be used to explain some paradoxical scenarios. Finally, we propose a probabilistic preference model using the “probabilistic preference comparing method”,. By exploring its logical properties, we highlight the connections and clarify the differences between the new method and qualitative/quantitative ones. We believe that our attempt creates an example for further research in combing logic and probability in general.
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