It is tempting to think that a process of choosing a point at random from the surface of a sphere can be probabilistically symmetric, in the sense that any two regions of the sphere which differ by a rotation are equally likely to include the chosen point. Isaacs, Hájek and Hawthorne (2022) argue from such symmetry principles and the mathematical paradoxes of measure to the existence of imprecise chances and the rationality of imprecise credences. Williamson (2007) has argued from a related symmetry principle to the failure of probabilistic regularity. We contend that these arguments fail, because they rely on auxiliary assumptions about probability which are inconsistent with symmetry to begin with. We argue, moreover, that symmetry should be rejected in light of this inconsistency, and because it has implausible decision-theoretic implications. The weaker principle of probabilistic invariance says that the probabilistic comparison of any two regions is unchanged by rotations of the sphere. This principle supports a more compelling argument for imprecise probability. We show, however, that invariance is incompatible with mundane judgements about what is probable. Ultimately, we find reason to be suspicious of the application of principles like symmetry and invariance to non-measurable regions.

中文翻译：

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对称性、不变性和不精确概率


人们很容易认为，从球体表面随机选择一个点的过程在概率上是对称的，因为球体中任何两个因旋转而不同的区域都同样有可能包含所选点。Isaacs、Hájek 和 Hawthorne （2022） 从这种对称原则和测量的数学悖论出发，论证了不精确机会的存在和不精确可信度的合理性。Williamson （2007） 从相关的对称性原则论证了概率规律性的失败。我们认为这些论点是失败的，因为它们依赖于关于概率的辅助假设，而这些假设一开始就与对称性不一致。此外，我们认为，鉴于这种不一致，对称性应该被拒绝，因为它具有难以置信的决策理论含义。较弱的概率不变性原理表示，任意两个区域的概率比较因球体的旋转而保持不变。该原则支持更令人信服的不精确概率论点。然而，我们表明，不变性与关于可能性的世俗判断是不相容的。最终，我们发现有理由对对称性和不变性等原则应用于不可测量的区域持怀疑态度。