Abstract
Sarah Moss’ thesis that we have probabilistic knowledge is from some perspectives unsurprising and from other perspectives hard to make sense of. The thesis is potentially transformative, but not yet elaborated in sufficient detail for epistemologists. This paper interprets Mossean probabilistic knowledge in a suitably-modified Kripke framework, thus filling in key details. It argues that probabilistic knowledge looks natural and plausible when so interpreted, and shows how the most pressing challenges to the thesis can be overcome. Most importantly, probabilistic knowledge can satisfy factivity in the framework, though we are not forced to accept a specific account of probabilistic “facts”. The framework also reflects Moss’ claim that old-fashioned propositional knowledge is just a limiting case of probabilistic knowledge, and all knowledge is fundamentally probabilistic. Finally, Moss endorses a failure of contraposition: for example, p implies probably p, but not probably p does not imply not p. The framework makes clear the sense in which the valid inferences regarding probably p are as Moss claims.
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Notes
This is perhaps the best suggestion, since these are meant to pick out the probabilities the agent should have, in an important sense. My own concern with this proposal is that it’s not clear whether unique evidential probabilities exist or can be identified for the full range of real-world credences that interest us. Until these evidential probabilities are characterized more clearly, the proposal does not tell us more than that credences are knowledge when they are the right ones.
A \(\sigma\)-algebra is basically a collection of sets that includes the empty set and all complements, unions and intersections of its members.
I will sometimes omit the brackets of singleton sets, for neatness.
Personal communication.
Thanks to several audience members for suggesting this.
It is true that some of Moss’ audience, particularly philosophers of language, may not take truth (in the ordinary sense) to be a requirement of assertion; see e.g. Yalcin (2011). The goal of this paper is to explicate probabilistic knowledge for epistemologists, though, and for this audience truth is paramount, rivaling knowledge as the standard for belief and so forth.
Since the focus here is on conceptual issues, an axiomatization is not provided.
A model with multiple agents would have additional accessibility relations, one for each agent. For a set of agents A, a model is defined as \(M=(\Omega ,(R^{a})_{a\in A},V^{P})\).
Usually the agent only entertains spaces such that all worlds in the space have positive probability. The requirement that only impossibilities receive probability 0 is known as regularity. This principle is intuitive but problematic in special types of cases; see Hájek (2012).
We could impose basic coherence requirements on the agent. For example, it seems reasonable to require that, at a particular possible world, the probability spaces that the agent entertains have a common set of (“live”) possible worlds, those the agent cannot rule out. Absent regularity, this does not unduly restrict the probabilistic content of the spaces.
In the multi-agent case, there is a knowledge operator \(K_{a}\) for each agent a, and the relevant relation is \(R^{a}\).
Although this paper focuses on first-order knowledge, we can consider how the knowledge operator should iterate. It looks more complicated than usual, but we can appeal to the usual intuitions. Consider when it should be the case that \(M,\,\psi \,\vDash\, KK\psi '\). The first requirement is factivity, which in the case of second-order knowledge means that there is in fact first-order knowledge: so, \(M,\,\psi \,\vDash\, K\psi '\), which we have already interpreted. The second requirement is that the epistemic state imply that there is first-order knowledge. Again, substitute into the original definition: for all \(s\,\in \psi\), for all \(\omega\, \in \Omega _{s}\), if \(\omega Rs'\) then \(M,s'\,\vDash\, K\psi '\). Further iterations proceed in the same way. Essentially, each additional iteration of knowledge of some \(\psi '\) means taking an additional step in the probability spaces—looking at the spaces accessible from the live worlds at the present space—and checking that the new space still belongs to \(\psi '\). While this definition of higher-order knowledge is less obvious than its non-probabilistic counterpart, it is the most natural candidate because it retains the recursive character of the familiar iteration procedure.
On the natural reading of ‘nominally probabilistic,’ in order to model multi-agent or higher-order knowledge, we must also check that each s places weight only on worlds which agree about who knows what.
Thanks to Thomas Krödel for pressing these issues.
Formally, take the model \(M_{1}'\) that is just like \(M_{1}\) except that now \(\omega _{1},\omega _{2}Rs_{.9}\) where \(s_{.9}\) is the probability space such that \(\mu _{s_{.9}}(\omega _{1})=.9\). Take the reference point \(s_{.6}\) such that \(\mu _{s_{.6}}(\omega _{1})=.6\). \(M_{1}',s_{.6}\,\nvDash\, Ks_{.9}\) , but \(M_{1}',s_{.6}\,\vDash\, K\mathrm {prob}(p)\), since \(M_{1}',s_{.6}\,\vDash\, \mathrm {prob}(p)\)\(s_{.9}\in \mathrm {prob}(p)\).
This example is related to the argument put forward by Yalcin (2012) as a counterexample to Modus Tollens.
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Acknowledgements
This work was carried out as part of the project “Knowledge and Decision,” funded by the Deutsche Forschungsgemeinschaft (Project Number 315078566). Discussions during the project’s symposium on Sarah Moss’ book were instrumental in showing the need for such a paper. The author thanks especially Roman Heil, Jakob Koscholke, Thomas Krödel, Moritz Schulz, Sergiu Spatan and Jacques Vollet for helpful discussions of the manuscript, and audiences at the Charles University in Prague and the Mind, World and Action summer school in Dubrovnik for feedback on presentations of the material. She also thanks an anonymous referee of this journal for helpful suggestions.
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Rich, P. The logic of probabilistic knowledge. Philos Stud 177, 1703–1725 (2020). https://doi.org/10.1007/s11098-019-01281-5
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DOI: https://doi.org/10.1007/s11098-019-01281-5