Symmetry, Invariance, and Imprecise Probability

IF 1.8 1区 哲学 0 PHILOSOPHY
MIND Pub Date : 2024-10-16 DOI:10.1093/mind/fzae048
Zachary Goodsell, Jacob M Nebel
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引用次数: 0

Abstract

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.
对称性、不变性和不精确概率
我们很容易想到,从球体表面随机选择一个点的过程在概率上是对称的,即球体上任何两个相差一圈的区域都同样有可能包含所选的点。Isaacs、Hájek 和 Hawthorne(2022 年)从这种对称性原理和度量衡的数学悖论出发,论证了不精确机会的存在和不精确信任的合理性。Williamson(2007)从相关的对称性原理论证了概率规律性的失败。我们认为这些论证都失败了,因为它们依赖于关于概率的辅助假设,而这些假设从一开始就与对称性不一致。此外,我们还认为,鉴于这种不一致性,对称性应该被摒弃,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则认为,任何两个区域的概率比较都不会因球体的旋转而改变。这一原则为不精确概率提供了更有说服力的论据。然而,我们表明,不变性与关于什么是概率的世俗判断是不相容的。最终,我们发现有理由怀疑对称性和不变性等原理是否适用于不可测量的区域。
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来源期刊
MIND
MIND PHILOSOPHY-
CiteScore
3.10
自引率
5.60%
发文量
47
期刊介绍: Mind has long been a leading journal in philosophy. For well over 100 years it has presented the best of cutting edge thought from epistemology, metaphysics, philosophy of language, philosophy of logic, and philosophy of mind. Mind continues its tradition of excellence today. Mind has always enjoyed a strong reputation for the high standards established by its editors and receives around 350 submissions each year. The editor seeks advice from a large number of expert referees, including members of the network of Associate Editors and his international advisers. Mind is published quarterly.
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