Open and Closed Questions in Decision-making

By asking questions, an agent can modify the range of options from which a decision is made.[Zuojun Xiong and Jeremy Seligman. How questions guide choices: a preliminary logical investigation. In Dianhui Wang and Mark Reynolds, editors, The 24th Australasian Joint Conference on Artificial Intelligence, Lecture Notes in Artificial Intelligence. Springer-Verlag, 2011] introduced a logic for reasoning about this role of question-asking in the decision-making process. The base logic is a modal logic with an operator D interpreted by: Dϕ iff after any rational choice that the agent can make, ϕ holds. On top of this, we proposed an analysis of questions as dynamic operators [?Q] and [!Q] which alter the range of options available to the agent in various ways. In the present paper, we provide a complete axiomatisation for this dynamic logic, and extend the analysis to complex questions. A particular characterisation of the transitivity of the preference order in terms of invariance under changes of the order in which questions are asked. This is applied to a notorious case of transitivity failure: Condorcetʼs voting paradox.