Abstract
In this work, we define a novel approach to the formalization of the unexpected hanging paradox, sometimes called the surprise examination paradox, mechanized in the Coq Proof Assistant. This paradox requires the definition of the notion of a surprise event, which, for the purposes of this paradox, is usually interpreted as the inability to predict what day a specific event takes place. Our use of constructive logic allows us to distinguish between possibility and certainty. We make the observation that an inevitable, but unexpected, event requires there being strictly more than one possible day on which it can occur, and define surprise accordingly. We formalize the paradox using this interpretation of surprise, then specify a family of propositions representing beliefs about whether a hanging occurs on a particular day, parametrized by the planned hanging day. We define what members of this family are in accordance with the paradox constraints, and demonstrate that this family is inhabited by classical propositions. We assert that this offers an unexpected, but satisfying resolution to the paradox, which agrees with our intuition, all without the need for self-referential predicates used in existing work. We compare our definition to a weaker interpretation of surprise, giving an analysis of how it interplays with the use of both classical and constructive logic, and could allow the prisoner to reach an apparently faulty conclusion. We note that this interpretation offers a satisfying solution to the “conditional” variation of the unexpected hanging paradox.
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Acknowledgements
I would like to thank my awesome graduate school supervisors, Dr. Amy Felty and Dr. Philip Scott, as well as numerous colleagues at IOG, for listening to me ramble on about this paradox. I would especially like to thank Dr. Pieter Hofstra, may he rest in peace, for making the bold move of asking for a resolution of this paradox as a (surprise) bonus question on a computability theory exam. I could not stop thinking about it ever since, until, hopefully, now.
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Vinogradova, P. (2023). Formalizing the Unexpected Hanging Paradox: A Classical Surprise. In: Herzig, A., Luo, J., Pardo, P. (eds) Logic and Argumentation. CLAR 2023. Lecture Notes in Computer Science(), vol 14156. Springer, Cham. https://doi.org/10.1007/978-3-031-40875-5_4
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