From Classical to Intuitionistic Probability
From Classical to Intuitionistic Probability
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从经典概率到直觉概率
DOI：
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发表时间：
2003
期刊：
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通讯作者：
B. Weatherson
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作者：
B. Weatherson
我们推广了概率演算的Kolmogorov公理，以得到定义任何给定逻辑的一类概率函数的条件，这些条件与经典逻辑的特殊情况下的标准概率函数一致，但允许相对于其他逻辑考虑其他类型的“本质上Kolmogorov”的概率函数。我们广泛地认为，贝叶斯方法规定，从给定逻辑的角度来看，理性信念程度是那些可以由适用于该逻辑的类别的概率函数表示的程度。古典贝叶斯主义将逻辑固定为经典逻辑，只是这种一般方法的一个版本。另一种，我们称之为直觉贝叶斯主义，选择直觉逻辑作为首选逻辑，并选择相关的概率函数类作为认知状态(信任度的合理分配)候选表示的正确类。我们认为，对经典贝叶斯主义的各种反对意见最好是转向直觉主义贝叶斯主义--其中概率函数是相对于直觉逻辑取的--而不是通过采用根本上非科尔莫哥罗主义的(例如，非加法的)概率函数的概念(或替代)，尽管后者在提出这些反对意见的人中很受欢迎。直觉主义贝叶斯主义的兴趣进一步增强，因为荷兰书中的一篇论证证明，当适当考虑到只有在下注的结果已知时才下注的事实时，选择直觉主义概率函数作为理性下注行为的指南是合理的。
We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of “essentially Kolmogorovian” probability functions relative to other logics. We take a broad view of the Bayesian approach as dictating inter alia that from the perspective of a given logic, rational degrees of belief are those representable by probability functions from the class appropriate to that logic. Classical Bayesianism, which fixes the logic as classical logic, is only one version of this general approach. Another, which we call Intuitionistic Bayesianism, selects intuitionistic logic as the preferred logic and the associated class of probability functions as the right class of candidate representions of epistemic states (rational allocations of degrees of belief). Various objections to classical Bayesianism are, we argue, best met by passing to intuitionistic Bayesianism – in which the probability functions are taken relative to intuitionistic logic – rather than by adopting a radically non-Kolmogorovian, e.g. non-additive, conception of (or substitute for) probability functions, in spite of the popularity of the latter response amongst those who have raised these objections. The interest of intuitionistic Bayesianism is further enhanced by the availability of a Dutch Book argument justifying the selection of intuitionistic probability functions as guides to rational betting behaviour when due consideration is paid to the fact that bets are settled only when/if the outcome betted on becomes known.