A Formal Analysis of Hollis' Paradox

Thomas Ågotnes and Chiaki Sakama

in: Proceedings of the 9th International Conference on Logic, Rationality and Interaction (LORI IX), Lecture Notes in Computer Science 14329, Springer-Verlag, pages 306-321, 2023.

Abstract

In Hollis' paradox, A and B each chose a positive integer and whisper their number to C. C then informs them, jointly, that they have chosen different numbers and, moreover, that neither of them are able to work out who has the greatest number. A then reasons as follows: B cannot have 1, otherwise he would know that my number is greater, and by the same reasoning B knows that I don't have 1. But then B also cannot have 2, otherwise he would know that my number is greater (since he knows I don't have 1). This line of reasoning can be repeated indefinitely, effectively forming an inductive proof, ruling out any number -- an apparent paradox. In this paper we formalise Hollis' paradox using public announcement logic, and argue that the root cause of the paradox is the wrongful assumption that A and B assume that C's announcement necessarily is successful. This resolves the paradox without assuming that C can be untruthful, or that A and B are not perfect reasoners, like other solutions do. There are similarities to the surprise examination paradox. In addition to a semantic analysis in the tradition of epistemic logic, we provide a syntactic one, deriving conclusions from a set of premises describing the initial situation -- more in the spirit of the literature on Hollis' paradox. The latter allows us to pinpoint which assumptions are actually necessary for the conclusions resolving the paradox.

Full Paper (PDF 300K)

Slide