The measurement of ranks and the laws of iterated contraction

Ranking theory delivers an account of iterated contraction; each ranking function induces a specific iterated contraction behavior. The paper shows how to reconstruct a ranking function from its iterated contraction behavior uniquely up to multiplicative constant and thus how to measure ranks on a ratio scale. Thereby, it also shows how to completely axiomatize that behavior. The complete set of laws of iterated contraction it specifies amend the laws hitherto discussed in the literature.

[1]  Peter Gärdenfors,et al.  Knowledge in Flux , 1988 .

[2]  C. V. Negoiţă,et al.  Fuzzy logic in knowledge engineering , 1986 .

[3]  Mary-Anne Williams,et al.  Frontiers in Belief Revision , 2001 .

[4]  Craig Boutilier,et al.  Revision Sequences and Nested Conditionals , 1993, IJCAI.

[5]  Eduardo L. Fermé,et al.  Revision by comparison , 2004, Artif. Intell..

[6]  Arie Tzvieli.  Possibility theory: An approach to computerized processing of uncertainty , 1990, J. Am. Soc. Inf. Sci..

[7]  Daniel Lehmann.  Belief Revision, Revised , 1995, IJCAI.

[8]  Dov M. Gabbay,et al.  Handbook of logic in artificial intelligence and logic programming (vol. 1) , 1993 .

[9]  W. Salmon,et al.  Knowledge in Flux , 1991 .

[10]  Peter Gärdenfors,et al.  Belief Revision , 1995 .

[11]  Peter Gärdenfors,et al.  On the logic of theory change: Partial meet contraction and revision functions , 1985, Journal of Symbolic Logic.

[12]  Satu Eloranta,et al.  On Iterated Belief Change , 1998, SCAI.

[13]  Joseph Y. Halpern.  Reasoning about uncertainty , 2003 .

[14]  Mehmet A. Orgun,et al.  Iterated Belief Contraction from First Principles , 2007, IJCAI.

[15]  H. Kyburg.  Iterated Belief Change Based on Epistemic Entrenchment* , 1994 .

[16]  Jürg Kohlas,et al.  Handbook of Defeasible Reasoning and Uncertainty Management Systems , 2000 .

[17]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[18]  Judea Pearl,et al.  On the Logic of Iterated Belief Revision , 1994, Artif. Intell..

[19]  F. Huber,et al.  Degrees of Belief , 2009 .

[20]  Judea Pearl,et al.  Qualitative Probabilities for Default Reasoning, Belief Revision, and Causal Modeling , 1996, Artif. Intell..

[21]  Jaap Van Brakel,et al.  Foundations of measurement , 1983 .

[22]  A. Bochman,et al.  Contraction of Epistemic States: A General Theory , 2001 .

[23]  Hans Rott,et al.  Change, choice and inference - a study of belief revision and nonmonotonic reasoning , 2001, Oxford logic guides.

[24]  D. Dubois,et al.  Belief Change Rules in Ordinal and Numerical Uncertainty Theories , 1998 .

[25]  R. Jeffrey.  The Logic of Decision , 1984 .

[26]  G. L. S. Shackle,et al.  Decision Order and Time in Human Affairs , 1962 .

[27]  SVEN OVE HANSSON,et al.  Reversing the Levi identity , 1993, J. Philos. Log..

[28]  Dov M. Gabbay,et al.  Handbook of logic in artificial intelligence and logic programming (Vol. 4): epistemic and temporal reasoning , 1995 .

[29]  Mary-Anne Williams,et al.  Iterated Theory Base Change: A Computational Model , 1995, IJCAI.

[30]  Wolfgang Spohn,et al.  Ranking Functions, AGM Style , 1999 .

[31]  Dov M. Gabbay,et al.  Handbook of defeasible reasoning and uncertainty management systems: volume 2: reasoning with actual and potential contradictions , 1998 .

[32]  Wolfgang Spohn,et al.  Ordinal Conditional Functions: A Dynamic Theory of Epistemic States , 1988 .

[33]  John Cantwell,et al.  On the logic of small changes in hypertheories , 2008 .

[34]  Hans Rott,et al.  Change, choice and inference , 2001 .

[35]  Jonathan Barzilai,et al.  On the foundations of measurement , 2001, 2001 IEEE International Conference on Systems, Man and Cybernetics. e-Systems and e-Man for Cybernetics in Cyberspace (Cat.No.01CH37236).

[36]  Franz Huber,et al.  Ranking functions and rankings on languages , 2006, Artif. Intell..

[37]  H. Rott.  Coherence and Conservatism in the Dynamics of Belief , 1999 .

[38]  Brian Skyrms,et al.  Causation in Decision, Belief Change, and Statistics, vol. II , 1988 .

[39]  Craig Boutilier,et al.  Iterated revision and minimal change of conditional beliefs , 1996, J. Philos. Log..

[40]  Jacek Malinowski,et al.  Towards Mathematical Philosophy , 2009, Trends in logic.

[41]  Wolfgang Spohn,et al.  A Survey of Ranking Theory , 2009 .

[42]  Sven Ove Hansson.  Mild Contraction. Evaluating Loss of Information Due to Loss of Belief , 2006, Stud Logica.

[43]  L. Cohen.  The Implications of Induction , 2019 .

[44]  Hans Rott,et al.  Shifting Priorities: Simple Representations for Twenty-Seven Iterated Theory Change Operators , 2009, Towards Mathematical Philosophy.