Abstract
This paper studies Aumann’s agreeing to disagree theorem from the perspective of dynamic epistemic logic. This was first done by Dégremont and Roy (J Phil Log 41:735–764, 2012) in the qualitative framework of plausibility models. The current paper uses a probabilistic framework, and thus stays closer to Aumann’s original formulation. The paper first introduces enriched probabilistic Kripke frames and models, and various ways of updating them. This framework is then used to prove several agreement theorems, which are natural formalizations of Aumann’s original result. Furthermore, a sound and complete axiomatization of a dynamic agreement logic is provided, in which one of these agreement theorems can be derived syntactically. These technical results are used to show the importance of explicitly representing the dynamics behind the agreement theorem, and lead to a clarification of some conceptual issues surrounding the agreement theorem, in particular concerning the role of common knowledge. The formalization of the agreement theorem thus constitutes a concrete example of the so-called dynamic turn in logic.
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Notes
A detailed comparison between Dégremont and Roy’s qualitative approach and the present probabilistic approach falls outside the scope of this paper, but can be found in Demey (2010), where I argue that the probabilistic approach is to be preferred on both philosophical and technical grounds. Some aspects of the model theory of epistemic plausibility models are studied in Demey (2011).
A similar explanatory scenario is described more extensively by Bonanno and Nehring (1997).
This paper will thus not deal with any of the stronger and more general agreement theorems that exist in the game-theoretical literature (Bacharach 1985; Bonanno and Nehring 1997; Feinberg 2000), because the methodological claim about the advantages of explicitly representing the dynamics can already be made for Aumann’s original result. Obviously, it will be interesting to explore how the results obtained in this paper can be generalized to stronger agreement theorems.
Probabilistic dynamic epistemic logic is one particular combination of logic and probabilities. Demey et al. (2013) provide a comprehensive overview of the various ways of combining logic and probability theory.
In game-theoretical contexts, \(R_i\) is usually implicitly taken to be the universal relation \(W \times W\) (and is therefore often not explicitly mentioned at all), while the equivalence relation \(E_i\) is identified with the partition \(\Pi _i\) that it generates. Furthermore, if \(R_i = W\times W\), then condition (ii) of Definition 1 is vacuously satisfied, so that the prior \(\mu _i(w)\) is defined over the entire domain W.
At the end of this subsection, we will see that condition (i) corresponds to the formula \(p \rightarrow P_i(p) > 0\). Furthermore, the prudence principle from which condition (i) follows (for all w,v: if \((w,v) \in R_i\), then \(\mu _i(w)(v) > 0\)) corresponds to the formula \(\lnot K_i\lnot p \rightarrow P_i(p) > 0\). Note (trivially, perhaps) that the formula corresponding to condition (i) follows from the formula corresponding to the prudence principle by the formula \(p \rightarrow \lnot K_i\lnot p\), which is exactly the formula corresponding to the reflexivity of \(R_i\).
For any binary relation \(\mathcal R \subseteq W\times W\), I abbreviate \(\mathcal R [w] := \{v \in W \,|\, (w,v) \in \mathcal R \}\). Furthermore, I will write \(\mathcal R ^*\) for the reflexive transitive closure of \(\mathcal R \), and \(\mathcal R ^{+}\) for the transitive closure of \(\mathcal R \).
Hence there are two \(R_i\)’s: on the one hand, \(R_i\) is agent i’s epistemic accessibility relation in a probabilistic Kripke model \(\mathbb{M }\); on the other hand, \(R_i\) is a unary modal operator of the language \(\mathcal L \). The main reason for not using another letter for the post-experimental knowledge operator is to ensure uniformity of notation with van Benthem and Minică (2012). I trust that the meaning of \(R_i\) will always be clear from the context.
The formula \(P_i(\varphi )\ge 2P_i(\psi )\) is actually an abbreviation for \(P_i(\varphi ) + (-2)P_i(\psi ) \ge 0\). One easily sees that the format of i-probability formulas is sufficiently general to express any ‘equation’ concerning i’s probabilities; cf. Fagin and Halpern (1994).
I already discussed the analogy between carrying out an experiment and asking a question. Modeling the experiments as intersecting \(R_i\) with \(E_i\) is analogous to the ‘resolve’ action in the dynamic epistemic logic of questions (van Benthem and Minică (2012), Definition 6): carrying out an experiment means getting an answer to a question posed to nature.
If one assumes that \(R_i = W\times W\) (recall Footnote 6), then \(R_i^e = E_i\), i.e. agent i’s knowledge after the experiments consists entirely of what she has learned from carrying out her experiment. Furthermore, it then holds that \(\mu _i^e(w)(v) = \mu _i(w)(v | \Pi _i(w))\), where \(\Pi _i(w)\) is the cell of the partition \(\Pi _i\) that contains w (obviously, since \(\Pi _i\) is generated by the equivalence relation \(E_i\), it holds that \(\Pi _i(w) = E_i[w]\)).
Note that \(\mu _i^e\) is well-defined (no dangerous division by 0): since \(E_i\) is an equivalence relation, it holds that \(w \in E_i[w]\), so by condition (i) in Definition 1 it follows that \(\mu _i(w)(E_i[w])\ge \mu _i(w)(w)>0\).
Note that \(\mu _i^\varphi \) is well-defined (no dangerous division by 0): \(\mu _i^\varphi (w)\) is only defined for states \(w \in W^\varphi = [[\varphi ]]^\mathbb{M }\), so by condition (i) of Definition 1 it follows that \(\mu _i(w)([[\varphi ]]^\mathbb{M })\ge \mu _i(w)(w)>0\).
Note that I have tacitly moved outside the official object language here, because the formula \(P_1(\varphi )=a \wedge P_2(\varphi )=b\) involves real numbers which might not be rational (\(a,b \in \mathbb{R }-\mathbb{Q }\)), whereas the official object language only contains rational numbers. Technically speaking, this can be ‘repaired’ (cf. Demey 2010), and it does not matter from a modeling perspective, so I will not dwell on it further.
Recall that probabilistic Kripke models are assumed to be finite in this paper; cf. Definitions 1 and 2. If infinite models are allowed as well, then Lemma 3 no longer holds. However, because the submodels of \(\mathbb{M }\) (ordered by the submodel relation) form a chain-complete poset and \(f_{w,\varphi }\) is a deflationary map on this poset, the Bourbaki–Witt theorem (1949) still guarantees that \(f_{w,\varphi }\) has a fixed point; however, it might take transfinitely many steps to reach this fixed point. From an application-oriented perspective, such transfinite dialogues make little sense, and I will therefore not pursue this topic any further.
This strategy is no longer available if we restrict to binary experiments (as will be done in Subsect. 5.1).
Here’s another way of putting the problem. Both experimentation and public announcement of a formula \(\varphi \) change the probabilistic component of a Kripke model via Bayesian conditionalization: \(\mu _i^e(w)(v) = \mu _i(w)(v\,|\,E_i[w])\) and \(\mu _i^\varphi (w)(v) = \mu _i(w)(v\,|\,[[\varphi ]]^\mathbb{M })\). In the case of public announcement, this fact can also be expressed in the object language by means of the following formula (which was already mentioned before):
$$\begin{aligned} \varphi \longrightarrow \big ([!\varphi ]P_i(\psi )=k \leftrightarrow P_i([!\varphi ]\psi \,|\,\varphi )=k\big ). \end{aligned}$$In the case of experimentation, however, the fact that the agents’ probabilities get updated by means of Bayesian conditionalization cannot be expressed in the object language (because the set \(E_i[w]\) might be undefinable).
From a technical perspective, this solution is analogous to the construction of general frames in modal logic (Blackburn et al. 2001). An entirely different solution, based on hybrid logic, is explored in detail in Demey (2010). There it is also argued that the ‘binary experiments’-solution is preferable on technical as well as methodological grounds.
Given this definability result, it might be asked why \(R_i\) is still introduced as a primitive operator. The reason for doing this is that this operator is only definable if we make use of the special proposition letters \(\alpha _i\); I remind the reader that these were only introduced at the beginning Sect. 5, when we shifted from a semantic to a syntactic pespective.
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Acknowledgments
Earlier versions of this paper were presented at local seminars in Tilburg (TiLPS), Amsterdam (ILLC) and Leuven (CLAW). I would like to thank the audiences of these talks for their helpful remarks and suggestions. Furthermore, I would like to thank Johan van Benthem, Cédric Dégremont, Dick de Jongh, Eric Pacuit, and two anonymous referees for their feedback on earlier versions of this paper. The early stages of this research were funded by the University of Leuven’s Formal Epistemology Project and the Huygens Scholarship Programme (NUFFIC); during the final stages I held a PhD fellowship from the Research Foundation—Flanders (FWO).
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Demey, L. Agreeing to disagree in probabilistic dynamic epistemic logic. Synthese 191, 409–438 (2014). https://doi.org/10.1007/s11229-013-0280-6
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DOI: https://doi.org/10.1007/s11229-013-0280-6