Reasoning about permitted announcements ( extended

Four players are playing the french game ”la belote”. All the 32 cards are dealt, each player starts the game with eight cards and she knows them, and all of this is common knowledge. We can formalize this situation using epistemic logic (see [8] for more details): the set of propositional atoms of our language is {(V C)i|V ∈ {7, 8, 9, 10, J,Q,K,A}, C ∈ {♣,♥,♦,♠}, i ∈ {1, 2, 3, 4}}. The model of ”la belote” has four epistemic (equivalence) relations, a world corresponds to a deal, and we say that M, s |= (10♠)3 if player 3 has the ten of spades in the deal s. Then we can express sentences like ”player 2 knows that player 3 has the queen of hearts” (K2(Q♥)3). What is happening when the game starts? Player one playing the 7 of diamonds can be seen as the public announcement that player one has the 7 of diamonds. But she may be able to do such an announcement, because she actually has the 7 of diamonds, without having the permission to do it, because the rules of the game may forbid it to her. Which are the announcements that are permitted in this game, depending on the actual situation? This paper will try to answer this question. Our goal is to formalize the concept of ”having the permission to say”. The way we choose is to adapt the semantics considered in [11] in the particular case where actions are replaced by public announcements. Thus, for α in van der Meyden’s ♦(α,φ) we take an announcement ψ! such that ♦(ψ!, φ) now means ‘it is permitted to announce ψ, after which φ is true’. The primitive of our language is in fact somewhat different, as it also takes sequences of announcements into account, but we will be able to define such ♦(ψ!, φ) as a special case (see section 4.1). The logic of public announcements (PAL) proposed by Plaza in [9], which is an extension of epistemic logic, permits to express the way agents update their knowledge after public announcements of true propositions. We can for example write in this language 〈ψ〉φ which means that after the truthful public