text_combined: kevin van horn bqkxbksvanhorncom gzrdaeecsberkeleyedu wrote consider following line reasoning let p proposition ronald born new york p infer q ronald born united states q infer r possible ronald born new jersey hand p infer possible ronald born new jersey arrived contradiction wrong paradox use probability theory extended logic analyze situation let x represent background information understanding spatial relationships us geography lets analyze 1 p infer q background information x tells us p q true symbols pp q x 1 hence pq p x 1 2 q infer r possible ronald born new jersey pr1 q x 0 r1 ronald born new jersey 3 hand p infer possible ronald born new jersey pr1 p x 0 4 arrived contradiction bayesians rule probability theory allows one conclude pr1 p x 0 pr1 q x 0 pq p x 1 appear using form modus ponens constructing paradox modus ponens doesnt generalize probabilities closest get pr1 p x since p q known true pr1 q p x product rule pq p x pr1 q p x since p q known true pr1 q p x get paradox would conclude pr1 q p x 0 pr1 q x 0 probability theory quite sensibly allow easy find cases new information p case turn possibility impossibility sense happening proposed paradox important context dropped happen probability theory product rule ensures contextual information retained kevin van horn wed 06 aug 2008 083732 0600 uai paradox