Abstract
In this paper we describe the use of two mathematical constructs in developing objective measures of explainability. The first one is measure theory, which has a long and interesting history, and which establishes abstract principles for comparing the size of general sets. At least some of the underpinnings of this theory can equally well be applied to evaluate the degree of explainability of given explanations. However, we suggest that it is meaningless, or at least undesired, to construct objective measures that allow the comparison of any two given explanations. Explanations might be non compatible, in the sense that integrating such explanations results in decreasing rather than increasing explainability. In other words, explainability is best considered as a partial order relation. Notwithstanding the use of partial order relations and measure theory, it is unwise to unconditionally apply these mathematical concepts to the field of explainability. It is demonstrated that the law of diminishing returns from economics offers a neat way to make these concepts applicable to the domain of explainability. The legal field is used as an illustration of the presented ideas.
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Notes
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Ibid.
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Definition from Encyclopaedia Britannica.
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De Mulder, W. (2022). The Use of Partial Order Relations and Measure Theory in Developing Objective Measures of Explainability. In: Calvaresi, D., Najjar, A., Winikoff, M., Främling, K. (eds) Explainable and Transparent AI and Multi-Agent Systems. EXTRAAMAS 2022. Lecture Notes in Computer Science(), vol 13283. Springer, Cham. https://doi.org/10.1007/978-3-031-15565-9_11
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