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.