Abstract
This chapter surveys recent work on probabilistic extensions of epistemic and dynamic-epistemic logics (the latter include the basic system of public announcement logic as well as the full product update logic). It emphasizes the importance of higher-order information as a distinguishing feature of these logics. This becomes particularly clear in the dynamic setting: although there exists a clear relationship between usual Bayesian conditionalization and public announcement, the probabilistic effects of the latter are in general more difficult to describe, because of the subtleties involved in higher-order information. Finally, the chapter discusses some applications of probabilistic dynamic epistemic logic, such as the Lockean thesis in formal epistemology and Aumann’s agreement theorem in game theory.
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Notes
- 1.
A notable exception is ‘Miller’s principle’, which states that \(P_1(\varphi \,|\,P_2(\varphi )=b)=b\). The probability functions \(P_1\) and \(P_2\) can have various interpretations, such as the probabilities of two agents, subjective probability (credence) and objective probability (chance), or the probabilities of one agent at different moments in time—in the last two cases, the principle is also called the ‘principal principle’ or the ‘principle of reflection’, respectively. This principle has been widely discussed in Bayesian epistemology and philosophy of science [29, 32, 38, 40, 41].
- 2.
See [33] for a further discussion of these and other properties, and their correspondence to knowledge/probability interaction principles.
- 3.
The agents’ probabilities are thus explicitly represented in the logic’s object language \(\fancyscript{L}\). Other proposals provide a probabilistic semantics for an object language that is itself fully classical (i.e. that does not explicitly represent probabilities). See [26] for a recent overview of the various ways of combining logic and probability.
- 4.
Romeijn [45] provides an analysis that stays closer in spirit to probability theory proper. He argues that the public announcement of \(\varphi \) induces a shift in the interpretation of \(\psi \) (in our terminology: from \(\psi \) to \([!\varphi ]\psi \), i.e. from \({{\mathrm{[\![\!}}}\psi {{\mathrm{\!]\!]}}}^\mathbb {M}\) to \({{\mathrm{[\![\!}}}\psi {{\mathrm{\!]\!]}}}^{\mathbb {M}|\varphi }\)), and shows that such meaning shifts can be modeled using Dempster-Shafer belief functions. Crucially, however, this proposal is able to deal with the case of \(\psi \) expressing second-order information (e.g. when it is of the form \(P_i(p) = b\)), but not with the case of higher-order information in general (e.g. when \(\psi \) is of the form \(P_j(P_i(p) = b) = a\), or involves even more deeply nested probabilities) [45, p. 603].
- 5.
Similarly, the success postulate for belief expansion in the (traditional) AGM framework [1, 30] states that after expanding one’s belief set with a new piece of information \(\varphi \), the updated (expanded) belief set should always contain this new information. Also here the explanation is that AGM is only concerned with first-order information. (Note that we talk about the success postulate for belief expansion, rather than belief revision, because the former seems to be the best analogue of public announcement in the AGM framework.)
- 6.
The occurrence of higher-order information is a necessary condition for this phenomenon, but not a sufficient one: there exist formulas \(\varphi \) that involve higher-order information, but still \(\models [!\varphi ]P_i(\varphi ) = 1\) (or epistemically: \(\models [!\varphi ]K_i\varphi \)).
- 7.
Occurrence probabilities are often assumed to be objective frequencies. This is reflected in the formal setup: the function \(\mathsf {pre}\) is not agent-dependent.
- 8.
Note that \(E\) and \(\varPhi \) are components of the probabilistic update model \(\mathbb {E}\) named by \(\mathsf {E}\); furthermore, the values \(k_{i,e,\varphi ,f}\) are fully determined by the model \(\mathbb {E}\) and event \(e\) named by \(\mathsf {E}\) and \(\mathsf {e}\), respectively (consider their definition in Fig. 13.3). Hence any \(i\)-probability formula involving \(\sigma \) is fully determined by \(\mathbb {E},e\), and can be interpreted at any probabilistic Kripke model \(\mathbb {M}\) and state \(w\).
- 9.
In practice, this distinction will not always be clear-cut, of course.
- 10.
As usual, \(R_i[w]\) denotes the set \(\{v \in W \, | \, (w,v) \in R_i\}.\)
- 11.
The logic in [19] does have a probabilistic component, but this logic is fully static.
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Acknowledgments
The authors wish to thank Alexandru Baltag, Johan van Benthem and Sonja Smets for their valuable feedback on earlier versions of this chapter. Lorenz Demey is financially supported by a PhD fellowship of the Research Foundation–Flanders (FWO).
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Demey, L., Kooi, B. (2014). Logic and Probabilistic Update. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_13
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