Abstract
This paper introduces and studies a notion of cautious distributed belief. Different from the standard distributed belief, the cautious distributed belief of a group is inconsistent only when all group members are individually inconsistent. The paper presents basic results about cautious distributed belief, investigates whether it preserves properties of individual belief, and compares it with standard distributed belief. Although both notions are equivalent in the class of reflexive models, this is not the case in general. While we argue that an understanding of the concept of cautious distributed belief from first principles is important, cautious distributed belief can be expressed using standard distrbuted belief and we show that the propositional language extended only with cautious distributed belief is in fact strictly less expressive than the propositional language extended only with standard distributed belief. We, finally, identify a small extension of the language making the former as expressive as the latter.
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- 1.
A group has common knowledge of \(\varphi \) if and only if everybody in the group knows \(\varphi \), everybody in the group knows that everybody in the group knows \(\varphi \), and so on.
- 2.
Considering notions of maximal consistency is a standard approach in non-monotonic reasoning for resolving potential conflicts. Think, e.g., about the extensions of a theory in default logic [19], or the maximally admissible (i.e., preferred) sets of arguments in abstract argumentation theory [8]. The idea has been also used within epistemic logic (e.g., by [5] in the context of evidence-based beliefs) and also for distributed beliefs (by [13], in the context of explicit beliefs defined via belief bases).
- 3.
This corresponds to the skeptical reasoner in non-monotonic reasoning.
- 4.
This corresponds to the credulous reasoner in non-monotonic reasoning.
- 5.
In particular, individual belief operators \(\mathop {{B}_{a}}\) can be defined in terms of \(\mathop {D_{}}\), as \(\mathop {D_{\{a\}}}\varphi \) (abbreviated as \(\mathop {D_{a}}\varphi \)) holds in a world s if and only if \(\mathcal {M},s' \vDash \varphi \) for all \(s' \in C_{a}(s)\).
- 6.
Thus, e.g., it distributes over implications: \(\vDash \mathop {D^{\forall }_{G}}(\varphi \rightarrow \psi ) \rightarrow (\mathop {D^{\forall }_{G}}\varphi \rightarrow \mathop {D^{\forall }_{G}}\psi )\).
- 7.
Note: the individual relations are serial, transitive and Euclidean. While the paper uses the term “belief” in a rather loose way, these three properties are the ones most commonly associated with a belief operator.
- 8.
More precisely, a frame (a model without the valuation) has the given relational property if and only if the formula is valid in the frame (i.e., it is true in any world of the frame under any valuation).
- 9.
This includes distributed and common knowledge/belief, but leaves out the notion of somebody knows [3], which does not have a relational semantics.
- 10.
A group has general knowledge/belief of \(\varphi \) if and only if everybody in the group knows/believes \(\varphi \).
- 11.
A typical strategy for proving \(\mathcal {L}_{1} \preccurlyeq \mathcal {L}_{2} \) is to give a translation \( tr :\mathcal {L}_{1} \rightarrow \mathcal {L}_{2} \) such that for every \((\mathcal {M}, s)\) we have \(\mathcal {M},s \vDash \alpha _1\) iff \(\mathcal {M},s \vDash tr (\alpha _1)\). The crucial cases are those for the operators in \(\mathcal {L}_{1} \) that do not occur in \(\mathcal {L}_{2} \).
- 12.
A typical strategy for proving \(\mathcal {L}_{1} \not \preccurlyeq \mathcal {L}_{2} \) is to find two pointed models that satisfy exactly the same formulas in \(\mathcal {L}_{2} \), and yet can be distinguished by a formula in \(\mathcal {L}_{1} \).
- 13.
Note: this relies on the fact that G is finite (because A is finite).
- 14.
This assumes that only normal modalities are involved in the language.
- 15.
Note: a group that is inconsistent in a given world does not need to be inconsistent in \(\mathcal {L}_{\mathop {D^{\forall }_{}}} \)-bisimilar worlds (see the models in the proof of Fact 2). This is different from the collective bisimulation case, under which a group that is inconsistent in a given world must be also inconsistent at any collectively bisimilar one.
- 16.
A belief model \(\mathcal {M}\) is image-finite iff \(C_{a}(s)\) is finite for every \(s \in {\text {D}}(\mathcal {M})\) and every \(a \in A\) (equivalently, iff \(C_{G}(s)\) is finite for every \(s \in {\text {D}}(\mathcal {M})\) and every \(G \subseteq A\)).
- 17.
The argument is straightforward: for \(\mathop {D^{\exists }_{G}}\bot \) to hold in a given pointed model, one needs a maximally consistent subset of G that has distributed belief in \(\bot \). But no consistent set of agents believes \(\bot \) distributively.
- 18.
But note how, in particular, \(\vDash \mathop {{B}_{a}}\bot \leftrightarrow \lnot \mathop {D^{\exists }_{a}}\top \).
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Appendix: frame conditions
Appendix: frame conditions
Table 2 shows the 32 combinations of frame conditions, and the generic names of the corresponding logics. Combinations on the same line are equivalent
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Lindqvist, J., Velázquez-Quesada, F.R., Ågotnes, T. (2023). Cautious Distributed Belief. In: Areces, C., Costa, D. (eds) Dynamic Logic. New Trends and Applications. DaLí 2022. Lecture Notes in Computer Science, vol 13780. Springer, Cham. https://doi.org/10.1007/978-3-031-26622-5_7
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