The Ditmarsch Tale of Wonders - The Dynamics of Lying
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We propose a dynamic logic of lying, wherein a 'lie that phi' (where phi is a formula in the logic) is an action in the sense of dynamic modal logic, that is interpreted as a state transformer relative to the formula phi. The states that are being transformed are pointed Kripke models encoding the uncertainty of agents about their beliefs. Lies can be about factual propositions but also about modal formulas, such as the beliefs of other agents or the belief consequences of the lies of other agents. We distinguish (i) an outside observer who is lying to an agent that is modelled in the system, from (ii) one agent who is lying to another agent, and where both are modelled in the system. For either case, we further distinguish (iii) the agent who believes everything that it is told (even at the price of inconsistency), from (iv) the agent who only believes what it is told if that is consistent with its current beliefs, and from (v) the agent who believes everything that it is told by consistently revising its current beliefs. The logics have complete axiomatizations, which can most elegantly be shown by way of their embedding in what is known as action model logic or the extension of that logic to belief revision.
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