Abstract
We give informal definitions of the concepts of direct and indirect common belief, illustrating them by an example. We then provide an analysis of these concepts within public announcement logic (PAL). The conceptual distinction between direct common belief and indirect common belief is important because it identifies two social phenomena which rely on different forms of agents’ cognitive capabilities. It is moreover relevant for speech act theory.
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Notes
- 1.
In this chapter we consider the sentences “Giovanni and Maria mutually believe that ϕ”, “there is mutual belief between Giovanni and Maria that ϕ” and “there is common belief between Giovanni and Maria that ϕ” to be synonymous.
- 2.
Aumann formalizes common knowledge which contrarily to belief, has the property of being truthful.
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.
We use the terms “fact” and “proposition” indistinguishably.
- 9.
In the present analysis, the term ‘group’ just denotes a set of individuals (alias agents). We use the symbols i,j,k,… to refer to individuals and G,H,J,… to refer to groups.
- 10.
- 11.
Otherwise, agent i would believe both that p is true and that he does not believe that p, which makes i’s beliefs contradictory as soon as we accept the principle of positive introspection, in formulas: \( {{\mathbf{B}}_{\mathrm{i}}}\upvarphi \to {{\mathbf{B}}_{\mathrm{i}}}{{\mathbf{B}}_{\mathrm{i}}}\upvarphi \).
- 12.
We are here assuming that every agent in G trusts his perception (so that he automatically believes what he perceives) and that this is mutual belief among the agents in G.
- 13.
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- 15.
A relation B i is transitive if (w,v) ∈ B i and (v,u) ∈ B i implies (v,u) ∈ B i. B i is Euclidean if (w,v) ∈ B i and (w,u) ∈ B i implies (v,u) ∈ B i.
- 16.
A relation B i is serial if for every w ∈ W, there exists v such that (w,v) ∈ B i.
- 17.
- 18.
For simplicity, we assume that Maria and Giovanni only consider Italian citizenship and French citizenship.
- 19.
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Lorini, E., Herzig, A. (2014). Direct and Indirect Common Belief. In: Konzelmann Ziv, A., Schmid, H. (eds) Institutions, Emotions, and Group Agents. Studies in the Philosophy of Sociality, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6934-2_22
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