Stopping to Reflect

Abstract

Our note is prompted by a recent article by Frank Arntzenius, “Some Problems for Conditionalization and Reflection,” which appeared in this Journal. Through a sequence of examples, that article purports to show limitations for a combination of two inductive principles that relate current and future rational degrees of belief: Temporal Conditionalization and Reflection: $${P}_{later}(A) = {P}_{later\}{\textbackslash}left(AB\right) = {P}_{now\}{\textbackslash}left(AB\right) $$$$ {P}_{now\}{\textbackslash}left(A{P}_{later}(A)=r\right)=r. $$ We will use the expression “Reflection holds with respect to the event A.” to apply to this equality for a specific event A.