I defend an analog of probabilism that characterizes rationally coherent estimates for chances. Specifically, I demonstrate the following accuracy-dominance result for stochastic theories in the C * -algebraic framework: supposing an assignment of chance values is possible if and only if it is given by a pure state on a given algebra, your estimates for chances avoid accuracy-dominance if and only if they are given by a state on that algebra. When your estimates avoid accuracy-dominance (roughly: when you cannot guarantee that other estimates would be more accurate), I say that they are sufficiently coherent. In formal epistemology and quantum foundations, the notion of rational coherence that gets more attention requires that you never allow for a sure loss (or 'Dutch book') in a given sort of betting game; I call this notion full coherence. I characterize when these two notions of rational coherence align, and I show that there is a quantum state giving estimates that are sufficiently coherent, but not fully coherent.

It is straightforward to determine the size of the Earth and the distance to the Moon without making use of a telescope. The methods have been known since the 3rd century BC. However, few amateur or professional astronomers have worked this out from data they themselves have taken. Here we use a gnomon to determine the latitude and longitude of South Bend, Indiana, and College Station, Texas, and determine a value of the radius of the Earth of 6290 km, only 1.4 percent smaller than the true value. We use the method of Aristarchus and the size of the Earth's shadow during the lunar eclipse of 2011 June 15 to derive an estimate of the distance to the Moon (62.3 R ⊕ ), some 3.3 percent greater than the true mean value. We use measurements of the angular motion of the Moon against the background stars over the course of two nights, using a simple cross staff device, to estimate the Moon's distance at perigee and apogee. Finally, we use simultaneous CCD observations of asteroid 1996 HW1 obtained with small telescopes in Socorro, New Mexico, and Ojai, California, to derive a value of the Astronomical Unit of (1.59 ± 0.19) × 10 8 km, about 6 percent too large. The data and methods presented here can easily become part of a beginning astronomy lab class.

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Abstract Recently, there has been some discussion of how Dutch book arguments might be used to demonstrate the rational incoherence of certain hidden variable models of quantum theory (Feintzeig [2015]; Feintzeig and Fletcher [2017]; Wroński and Godziszewski [2017]). In this paper, we argue that the ‘form of inconsistency’ underlying this alleged irrationality is deeply and comprehensively related to the more familiar ‘inconsistency’ phenomenon of contextuality. Our main result is that the hierarchy of contextuality due to Abramsky and Brandenburger ([2011]) corresponds to a hierarchy of additivity/convexity-violations that yields formal Dutch books of different strengths. We then use this result to provide a partial assessment of whether these formal Dutch books can be given a convincing normative interpretation. 1Introduction2Contextuality and Inconsistency2.1Contextuality2.2Gluing inconsistency2.3Dutch bookability2.4Our motivation and results3Representing Empirical Models3.1Gluing inconsistency: empirical models3.2Weak probability spaces: representations of empirical models4The Hierarchies4.1Contextuality and additivity-violation4.2Dutch bookability and convexity-violation5Is Dutch Bookability Normative?