Logical questions behind the lottery and preface paradoxes: lossy rules for uncertain inference

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.

[1]  D. Makinson.  The paradox of the preface , 1965 .

[2]  O. G. Morgan.  Conjunctivitis , 1931, Proceedings of the Royal Society of Medicine.

[3]  Christopher. Thornton,et al.  Coherent Systems , 2007 .

[4]  Leonard Goddard,et al.  Laws of thought , 1959 .

[5]  J. M. Larrazabal,et al.  Reasoning about change , 1991 .

[6]  E. W. Adams,et al.  The logic of conditionals , 1975 .

[7]  David Makinson,et al.  Bridges from classical to nonmonotonic logic , 2005, Texts in computing.

[8]  Ernest W. Adams,et al.  Four probability-preserving properties of inferences , 1996, J. Philos. Log..

[9]  Rohit Parikh,et al.  PROBABILISTIC CONDITIONALS ARE ALMOST MONOTONIC , 2008, The Review of Symbolic Logic.

[10]  Henry Ely Kyburg,et al.  Probability and Inductive Logic , 1970 .

[11]  James Hawthorne,et al.  The Quantitative/Qualitative Watershed for Rules of Uncertain Inference , 2007, Stud Logica.

[12]  Jeff B. Paris,et al.  O IS NOT ENOUGH , 2009, The Review of Symbolic Logic.

[13]  Marshall Swain,et al.  Induction Acceptance and Rational Belief , 1970 .

[14]  Judea Pearl,et al.  Probabilistic Semantics for Nonmonotonic Reasoning: A Survey , 1989, KR.

[15]  Henry E. Kyburg,et al.  Probability and the logic of rational belief , 1970 .

[16]  Henry E. Kyburg,et al.  The Rule of Adjunction and Reasonable Inference , 1997 .

[17]  Ernest W. Adams,et al.  A primer of probability logic , 1996 .

[18]  Simon Parsons,et al.  Propagating Probabilities in System P , 1998, FLAIRS.

[19]  E. W. Adams,et al.  Probability and the Logic of Conditionals , 1966 .

[20]  James Hawthorne,et al.  On the logic of nonmonotonic conditionals and conditional probabilities , 1996, J. Philos. Log..

[21]  F. Ramsey.  The Foundations of Mathematics and Other Logical Essays , 2001 .

[22]  David Makinson,et al.  Conditional Probability in the Light of Qualitative Belief Change , 2011, J. Philos. Log..

[23]  J. Hintikka,et al.  Aspects of Inductive Logic. , 1968 .

[24]  Henry E. Kyburg,et al.  Probability and Randomness. , 1970 .

[25]  G. Wheeler.  A Review of the Lottery Paradox , 2007 .

[26]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems , 1988 .

[27]  George Boole,et al.  The Laws of Thought , 2003 .

[28]  Sarit Kraus,et al.  Nonmonotonic Reasoning, Preferential Models and Cumulative Logics , 1990, Artif. Intell..