Playing cards with Hintikka An introduction to dynamic epistemic logic

This contribution is a gentle introduction to so-called dynamic epistemic logics, that can describe how agents change their knowledge and beliefs. We start with a concise introduction to epistemic logic, through the example of one, two and finally three players holding cards; and, mainly for the purpose of motivating the dynamics, we also very summarily introduce the concepts of general and common knowledge. We then pay ample attention to the logic of public announcements, wherein agents change their knowledge as the result of public announcements. One crucial topic in that setting is that of unsuccessful updates: formulas that become false when announced. The Moore-sentences that were already extensively discussed at the conception of epistemic logic in Hintikka’s ‘Knowledge and Belief ’ (1962) give rise to such unsuccessful updates. After that, we present a few examples of more complex epistemic updates. Professor Jaakko Hintikka kindly gave permission to use his name in the title. He also observed that “My late wife Merrill was one of the best female blackjack players in the world and a championship level bridge player. Hence twenty years ago you would have been well advised to specify which Hintikka you refer to in your title!” Section 5 is partly based on a chapter of [vDvdHK06], and Section 6 is partly based on a section of [vDK05]. We thank an anonymous referee for very detailed helpful comments. H. P. van Ditmarsch, W. van der Hoek and B. P. Kooi, “Playing Cards with Hintikka”, Australasian Journal of Logic (3) 2005, 108–134 http://www.philosophy.unimelb.edu.au/ajl/2005 109 Our closing observations are on recent developments that link the ‘standard’ topic of (theory) belief revision, as in ‘On the Logic of Theory Change: partial meet contraction and revision functions’, by Alchourron et al. (1985), to the dynamic epistemic logics introduced here. 1  Imagine three players Anne, Bill, and Cath, each holding one card from a stack of three (known) cards clubs, hearts, and spades, such that they know their own card but do not know which other card is held by which other player. Assume that the actual deal is that Anne holds clubs, Bill holds hearts and Cath holds spades. Now Anne announces that she does not have hearts. What was known before this announcement, and how does this knowledge change as a result of that action? Before, Cath did not know that Anne holds clubs, but afterwards she knows that Anne holds clubs. This is because Cath can reason as follows: “I have spades, so Anne must have clubs or hearts. If she says that she does not have hearts, she must therefore have clubs.” Bill knows that Cath now knows Anne’s card, even though he does not know himself what Anne’s card is. Both before and after, players know which card they hold in their hands. Note that the only change that appears to have taken place is epistemic change, and that no factual change has taken place, such as cards changing hands. How do we describe such an information update in an epistemic setting? We can imagine various other actions that affect the knowledge of the players, for example, the action where Anne shows her clubs card to Bill, in such a way that Cath sees that Anne is doing that, but without seeing the actual card. How does that affect the knowledge of the players about each other? After that action, Cath still does not know whether Anne holds clubs or hearts. But Cath now knows that Bill knows Anne’s card. This contribution is a gentle introduction to so-called dynamic epistemic logics, that can describe how agents change their knowledge and beliefs. We start with a concise introduction to epistemic logic, through the example of one, two and finally three players holding cards; and, mainly for the purpose of motivating the dynamics, we also very summarily introduce the concepts of general and common knowledge. We then pay ample attention to the logic of public announcements, wherein agents change their knowledge as the result of, indeed, public announcements. One crucial topic in that setting is that of unsuccessful updates: formulas that become false when announced. The Moore-sentences that were already extensively discussed at the conception of epistemic logic in [Hin62] give rise to such unsuccessful updates. After that, we present a few examples of more complex epistemic updates. Our closing observations are on recent developments that link the ‘standard’ topic of (theory) belief revision [AGM85] to the dynamic epistemic logics introduced here. H. P. van Ditmarsch, W. van der Hoek and B. P. Kooi, “Playing Cards with Hintikka”, Australasian Journal of Logic (3) 2005, 108–134 http://www.philosophy.unimelb.edu.au/ajl/2005 110 2   We introduce epistemic logic by a simple example, even simpler than the one in the introduction. Suppose there is only one player: Anne. Anne draws one card from a stack of three different cards clubs, hearts, and spades. Suppose she draws the clubs card—but she does not look at her card yet; and that one of the remaining cards is put back into the stack holder, suppose that is the hearts card; and that the remaining card is put (face down) on the table. That must therefore be the spades card! Anne now looks at her card. What does Anne know? We would like to be able to evaluate system descriptions such as: • Anne holds the clubs card. • Anne knows that she holds the clubs card. • Anne does not know that the hearts card is on the table. • Anne can imagine that the hearts card is not on the table. • Anne knows that the hearts card or the spades card is in the stack holder. • Anne knows her own card. • The card on the table is not held by Anne. • Anne knows that she holds one card. Facts about the state of the world are in this case facts about card ownership. We describe such facts by atoms such as Clubsa standing for ‘the clubs card is held by Anne’, and similarly Clubsh for ‘the clubs card is in the stack holder’, and Clubst for ‘the clubs card is on the table’, etc. The standard propositional connectives are ∧ for ‘and’, ∨ for ‘or’, ¬ for ‘not’, → for ‘implies’, and ↔ for ‘if and only if ’. A formula of the form Kφ expresses that ‘Anne knows thatφ’, and a formula of the form K̂φ (K̂ is the dual of K) expresses that ‘Anne can imagine that φ’. The informal descriptions above become • Anne holds the clubs card: Clubsa • Anne knows that she holds the clubs card: KClubsa • Anne does not know that the hearts card is on the table: ¬KHeartst • Anne can imagine that the hearts card is not on the table: K̂¬Heartst • Anne knows that the hearts card or the spades card is in the stack holder: K(Heartsh ∨ Spadesh) • Anne knows her own card: KClubsa ∨ KHeartsa ∨ KSpadesa H. P. van Ditmarsch, W. van der Hoek and B. P. Kooi, “Playing Cards with Hintikka”, Australasian Journal of Logic (3) 2005, 108–134 http://www.philosophy.unimelb.edu.au/ajl/2005 111