What can we achieve by arbitrary announcements?: A dynamic take on Fitch's knowability

  • Authors:
  • P. Balbiani;A. Baltag;H. van Ditmarsch;A. Herzig;T. Hoshi;T. de Lima

  • Affiliations:
  • IRIT, Toulousem, France;Oxford University, United Kingdom;University of Otago, New Zealand;IRIT, Toulouse, France;Stanford University;IRIT, Toulouse, France

  • Venue:
  • TARK '07 Proceedings of the 11th conference on Theoretical aspects of rationality and knowledge
  • Year:
  • 2007

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Abstract

Public announcement logic is an extension of multi-agent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: □ϕ expresses that ϕ is true after an arbitrary announcement ψ. As this includes the trivial announcement ⊤, one might as well say that □ϕ expresses what remains true after any announcement: it therefore corresponds to truth persistence after (definable) relativisation. The dual operation ⋄ϕ expresses that there is an announcement after which ϕ. This gives a perspective on Fitch's knowability issues: for which formulas ϕ does it hold that ϕ → ⋄Kϕ? We give various semantic results, and we show completeness for a Hilbert-style axiomatisation of this logic.