Logical questions behind the lottery and preface paradoxes

lossy rules for uncertain inference

David Makinson

pp. 511-529

in: Horacio Arló-Costa, Gregory Wheeler (eds), Commemorating the work of Henry E. Kyburg Jr., Synthese 186 (2), 2012.

Abstract

We reflect on lessons that the lottery and preface paradoxes provide for the logic of uncertain inference. One of these lessons is the unreliability of the rule of conjunction of conclusions in such contexts, whether the inferences are probabilistic or qualitative; this leads us to an examination of consequence relations without that rule, the study of other rules that may nevertheless be satisfied in its absence, and a partial rehabilitation of conjunction as a ‘lossy’ rule. A second lesson is the possibility of rational inconsistent belief; this leads us to formulate criteria for deciding when an inconsistent set of beliefs may reasonably be retained.