Abstract
In this paper, we introduce and formalize the concept of epistemic informativeness (EI) of statements: the set of new propositions that an agent comes to know from the truthful announcement of the statements. We formalize EI in multi-agent Public Announcement Logic and characterize it by proving that two basic statements are the same in EI iff the logical equivalence of the two is common knowledge after a certain announcement. As a corollary applied to identity statements, \(a=b\) and \(a=a\) are different in EI iff \(a=b\) is not common knowledge. This may shed new light on the differences in cognitive value of \(a=a\) and \(a=b\), even when they are both known to be true, as long as \(a=b\) is not commonly known to all.
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Notes
- 1.
It is definitely possible to consider other communication methods which have different effects on the knowledge states of agents. The choice here is simple enough to make our points clear.
- 2.
Cf. [7] for a detailed technical discussion of the Moore sentences in this setting.
- 3.
In the context of Frege’s puzzle, a similar informal ‘dynamic’ conception of informativeness was briefly mentioned by [3], but not formalized precisely.
- 4.
False statements can also be informative in general if we are talking about belief.
- 5.
See [14] for a discussion on common knowledge. Here common knowledge of \(\phi \) means all the agents know \(\phi \) and all the agents know that all the agents know \(\phi \), and so on, ad infinitum.
- 6.
Rendsvig [10] also used the update mechanism to define the equality of informativeness of identity statements, but in a very strict way: \(\phi \) and \(\psi \) are equally informative if the corresponding announcements can delete the same states in the given model. Here we give a natural definition of informativeness itself, which induces a weaker notion of informational equivalence.
- 7.
Actually, \([\phi \vee \psi ] C(\phi \leftrightarrow \psi )\) can be formulated as a weaker version of common knowledge of \(\phi \leftrightarrow \psi \), which is called relativized common knowledge (cf. [13]).
- 8.
For example, ‘\(2^{57885161}-1\) is a prime number’ may have the same EI as ‘2 is a prime number’, although intuitively these two sentences should induce different knowledge updates if not all the tautologies are common knowledge.
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Acknowledgments
We are grateful to Wen-fang Wang for his insightful comments on an earlier version of this paper. Yanjing Wang acknowledges support of the key project 12&ZD119 of the National Social Science Foundation of China.
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Wang, Y., Fan, J. (2015). Epistemic Informativeness. In: Ju, S., Liu, H., Ono, H. (eds) Modality, Semantics and Interpretations. Logic in Asia: Studia Logica Library. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47197-5_7
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