Death of paradox: The killer logic beneath the standards of proof

Notre Dame Law Review

Volumn 88, Issue 3, 2013, Pages 1061-1138

  Clermont, Kevin M ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84876272424     PISSN: 07453515     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (198)

1. See generally WILLIM TWINING, RETHINKING EVIDENCE 237-48 (2d ed. 2006)
2. Richard Lempert, The New Evidence Scholarship: Analyzing the Process of Proof, in PROBABILITY AND INFERENCE IN THE LAW OF EVIDENCE 61, 61 (Peter Tillers & Eric D. Green eds., 1988)
3. Roger C. Park & Michael J. Saks, Evidence Scholarship Reconsidered: Results of the Interdisciplinary Turn, 47 B.C. L. REV. 949, 949 (2006)
4. See, e.g., TERENCE ANDERSON ET AL., ANALYSIS OF EVIDENCE (2d ed. 2005)
5. See generally RICHARD EGGLESTON, EVIDENCE, PROOF AND PROBABILITY 3 (2d ed. 1983)
6. Eric D. Green, Foreword: Probability and Inference in the Law of Evidence, 66 B.U. L. REV. 377, 377 (1986)
7. Symposium, Decision and Inference in Litigation, 13 CARDOZO L. REV. 253, 253 (1991)
8. cf V.C. Ball, The Moment of Truth: Probability Theory and Standards of Proof, 14 VAND. L. REV. 807, 809-12 (1961)
9. Laurence H. Tribe, Trial by Mathematics: Precision and Ritual in the Legal Process, 84 HARV. L. REV. 1329, 1344-50 (1971)
10. See generally SHARON BERTSCH MCGRAYNE, THE THEORY THAT WOULD NOT DIE (2011)
11. Peter Tillers, Trial by Mathematics-Reconsidered, 10 LAW PROBABILITY & RISK 167, 169-170 (2011)
12. But see Roger C. Park et al., Bayes Wars Redivivus-An Exchange, 8 INT'L COMMENT. ON EVIDENCE iss. 1, art. 1 (2010)
13. See Lea Brilmayer, Second-Order Evidence and Bayesian Logic, 66 B.U. L. REV. 673, 688-91 (1986)
14. See, e.g., David A. Schum, Probability and the Processes of Discovey, Proof, and Choice, 66 B.U. L. REV. 825, 847-53, 865-69 (1986).
15. See THEODORE SIDER, LOGIC FOR PHILOSOPHY 72-73 (2010).
16. See Peter Suber, Non-Contradiction and Excluded Middle, http://www.earlham.edu/~peters/courses/logsys/pnc-pem.htm (last visited Jan. 15, 2013).
17. Bertrand Russell, Vagueness, 1 AUSTRALASIAN J. PSYCHOL. & PHIL. 84 (1923)
18. see Bertrand Russell, The Philosophy of Logical Atomism, in LOGIC AND KNOWLEDGE 175, 180 (Robert Charles Marsh ed., 1956)
19. see also, e.g., TIMOTHY WILLIAMSON, Preface to VAGUENESS, at xi (1996)
20. Hartry Field, No Fact of the Matter, 81 AUSTRALASIAN J. PHIL. 457 (2003)
21. Hartry Field, Indeterminacy, Degree of Belief; and Excluded Middle, 34 NOÛS 1, 20 (2000)
22. Hartry Field, Vagueness, Partial Belief; and Logic, in MEANINGS AND OTHER THINGS (G. Ostertag ed., forthcoming 2013), available at http://philosophy.fas. nyu.edu/docs/IO/1158/schiffer2004b.pdf
23. See generally J.C. BEALL & BAS C. VAN FRAASSEN, POSSIBILITIES AND PARADOX (2003)
24. See BART KOSKO, Fuzzy THINKING 94-97 (1993).
25. See R. M. Sainsbury & Timothy Williamson, Sorites, in A COMPANION TO THE PHILOSOPHY OF LANGUAGE 458, 475-77 (Bob Hale & Crispin Wright eds., 1997)
26. Nicholas J.J. Smith, Fuzzy Logic and Higher-Order Vagueness, in LoGICAL. MODELS OF REASONING WITH VAGUE INFORMATION 1, 1 (Petr Cintula et al. eds., forthcoming 2012), available at http://www-personal.usyd.edu.au/~njjsmith/ papers/SmithFuzLogHOVag.pdf
27. But see TIMOTHY A. O. ENDICOTT, VAGUENESS IN LAW 77-136 (2000)
28. See generally LIARS AND HEAPS (JC Beall ed., 2003)
29. Kevin M. Clermont, Foreword: Why Comparative Civil Procedure?, in Kuo-CHANG HUANG, INTRODUCING DISCOVERY INTO CIVIL LAW, at xviii n.50 (2003)
30. See generally Peter Walley, Statistical Reasoning with Imprecise Probabilities, 42 MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY (1991)
31. See James F. Brulé, Fuzzy Systems-A Tutorial, http://www. austinlinks.com/Fuzzy/tutorial.html (last visited Jan. 15, 2013)
32. See GEORGE BooLE, INVESTIGATION OF THE LAWS OF THOUGHT (London, Walton and Maberly 1854) (doing the early work)
33. cf JOHN MAYNARD KEYNES, A TREATISE ON PROBABILITY (1921).
34. L.A. Zadeh, Fuzzy Sets, 8 INFO. & CONTROL 338 (1965).
35. See Timothy J. Ross & W. Jerry Parkinson, Fuzzy Set Theoiy, Fuzzy Logic, and Fuzzy Systems, in FUZZY LOGIC AND PROBABILITY APPLICATIONS 29 (Timothy J. Ross et al. eds., 2002).
36. See Liu Sifeng, Jeffrey Forrest & Yang Yingjie, A Brief Introduction to Grey Systems Theory, in 2011 IEEE INTERNATIONAL CONFERENCE ON GREY SYSTEMS AND INTELLIGENT SERVICES 1, 6 (2011).
37. See, e.g., Kevin M. Clermont, Procedure's Magical Number Three: Psychological Bases for Standards of Decision, 72 CORNELL L. REV. 1115, 1122 n.36 (1987)
38. See Richard M. Royall, Statistical Evidence, 71 MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY, at 5-6, 28 (1997).
39. MIRCEA REGHIS & EUGENE ROVENTA, CLASSICAL AND Fuzzy CONCEPTS IN MATHEMATICAL LOGIC AND APPLICATIONS 354 (1998)
40. ee DAVID A. SCHUM, THE EVIDENTIAL FOUNDATIONS OF PROBABILISTIC REASONING 41, 201 (1994)
41. GLENN SHAFER, A MATHEMATICAL THEORY OF EVIDENCE (1976)
42. see also Glenn Shafer, Perspectives on the Theory and Practice of Belief Functions, 4 INT'L J. APPROXIMATE REASONING 323 (1990)
43. Lotfi A. Zadeh, Book Review, AI MAG., Fall 1984, at 81, 83
44. See Ron A. Shapira, Economic Analysis of the Law of Evidence: A Caveat, 19 CAR. DOZO L. REV. 1607, 1614 (1998)
45. See Glenn Shafer, The Constrnction of Probability Arguments, 66 B.U. L. REV. 799, 801-04 (1986).
46. cf. DAVID CHRISTENSEN, PUTTING LOGIC IN ITS PLACE 12-13, 69 (2004)
47. L. Jonathan Cohen, Should a Jury Say What It Believes or What It Accepts?, 13 CARDOZO L. REV. 465 (1991)
48. See Didier Dubois & Henri Prade, A Unified View of Uncertainty Theories (Mar. 7, 2012) (unpublished manuscript).
49. See IRVING M. COPI ET AL., INTRODUCTION TO LOGIC ch. 14 (14th ed. 2011).
50. Neil B. Cohen, Confidence in Probability: Burdens of Persuasion in a World of Imperfect Knowledge, 60 N.Y.U. L. REV. 385 (1985)
51. Compare Neil B. Cohen, Commentary, The Costs of Acceptability: Blue Buses, Agent Orange, and Aversion to Statistical Evidence, 66 B.U. L. REV. 563, 569 (1986)
52. Neil B. Cohen, Conceptualizing Proof and Calculating Probabilities: A Response to Professor Kaye, 73 CORNELL L. REV. 78, 91-93 (1987)
53. with D.H. Kaye, Commentary, Do We Need a Calculus of Weight to Understand Proof Beyond a Reasonable Doubt?, 66 B.U. L. REV. 657, 667 n.22 (1986)
54. D.H. Kaye, Apples and Oranges: Confidence Coefficients and the Burden of Persuasion, 73 CORNELL L. REV. 54, 54, 56-58 (1987)
55. A similarly suggestive attempt to explain standards of proof in terms of psychological confidence appeared in Christoph Engel, Preponderance of the Evidence Versus Intime Conviction: A Behavioral Perspective on a Conflict Between American and Continental European Law, 33 VT. L. REV. 435 (2009).
56. I tried to demonstrate his theory's prescriptive and descriptive failings in Kevin M. Clermont, Standards of Proof Revisited, 33 VT. L. REV. 469 (2009).
57. See Kevin M. Clermont & Emily Sherwin, A Comparative View of Standards of Proof, 50 AM. J. COMP. L. 243, 256-57 (2002).
58. John Kaplan, Decision Theory and the Factfinding Process, 20 STAN. L. REV. 1065 (1968)
59. Michael O. Finkelstein & William B. Fairley, A Bayesian Approach to Identification Evidence, 83 HARV. L. REV. 489 (1970).
60. See generally Rolf Haenni, Non-Additive Degrees of Belief in DEGREES OF BELIEF 121 (Franz Huber & Christoph Schmidt-Petri eds., 2009)
61. See David A. Schum, A Science of Evidence: Contributions from Law and Probability, 8 LAW PROBABILITY & RISK 197, 203-04 (2009) (crediting DAVID OLDROYD, THE ARCH OF KNOWLEDGE (1986), for this image).
62. See Martin F. Kaplan, Cognitive Processes in the Individual Juror, in THE PSYCHOLOGY OF THE COURTROOM 197 (Norbert L. Kerr & Robert M. Bray eds., 1982)
63. see also Jennifer Groscup & Jennifer Tallon, Theoretical Models of Jury Decision-Making, in JURY PSYCHOLOGY 41 (Joel D. Lieberman & Daniel A. Krauss eds., 2009)
64. Nancy Pennington & Reid Hastie, Juror Decision-Making Models: The Generalization Gap, 89 PSYCHOL. BULL. 246 (1981) (evaluating various models).
65. See generally DANIEL KANNEMAN, THINKING, FAST AND SLOW (2011).
66. See J. Alexander Tanford & Sarah Tanford, Better Trials Through Science: A Defense of Psychologist-Lauryer Collaboration, 66 N.C. L. REV. 741, 748-59 (1988).
67. See Anna Ronkainen, Dual-Process Cognition and Legal Reasoning, if 1 ARGUMENTATION 2011, at 1, 1 (Michal Araszkiewicz et al. eds., 2011), available at http://ssrn.com/abstract=2004336
68. See Peter Tillers, Introduction: A Personal Perspective on "Artficial Intelligence and Judicial Proof," 22 CARDOZO L. REV. 1365 (2001)
69. Compare Paul Bergman & Al Moore, Mistrial by Likelihood Ratio: Bayesian Analysis Meets the F-Word, 13 CARDOZO L. REV. 589, 590 (1991) (attacking)
70. with D.H. Kaye, Commentary, Credal Probability, 13 CARDOZO L. REV. 647 (1991) (defending).
71. See Leonard R. Jaffee, Of Probativity and Probability: Statistics, Scientific Evidence, and the Calculus of Chance at Trial, 46 U. PITT. L. REV. 925, 980-85 (1985).
72. See Lea Brilmayer & Lewis Kornhauser, Review: Quantitative Methods and Legal Decisions, 46 U. CHI. L. REV. 116, 135-48 (1978).
73. Compare SAMUEL KOTZ & DONNA F. STROUP, EDUCATED GUESSING (1983)
74. with Craig R. Callen, Notes on a Grand Illusion: Some Limits on the Use of Bayesian Theory in Evidence Law, 57 IND. L.J. 1 (1982)
75. Vulém Novák, Modeling with Words, SCHOLARPEDIA, http://www.scholarpedia.org/article/Modeling-with-words (last visited Jan. 15, 2013).
76. See Alf C. Zimmer, Verbal vs. Numerical Processing of Subjective Probabilities, in DECISION MAKING UNDER UNCERTAINTY 159, 180 (Roland W. Scholz ed., 1983).
77. See Mandeep K Dhami, On Measuring Quantitative Interpretations of Reasonable Doubt, 14 J. EXPERIMENTAL PSYCHOL.: APPLIED 353, 362 (2008)
78. See Mandeep K. Dhami & Thomas S. Walisten, Interpersonal Comparison of Subjective Probabilities: Toward Translating Linguistic Probabilities, 33 MEMORY & COGNITION 1057 (2005).
79. See Petr Hajek, Fuzzy Logic, in THE STANFORD ENCYCLOPEDIA OF PHILOSOPHY (Edward N. Zalta ed., 2010), available at http://plato.stanford.edu/archives/ fall 2010/entries/logic-fuzzy
80. Jeffrey A. Barnett, Computational Methods for A Mathematical Theory of Evidence, in CLASSIC WORKS OF THE DEMPSTER-SHAFER THEORY OF BELIEF FUNCTIONS 197, 198-204 (Ronald R. Yager & Liping Liu eds., 2008).
81. See Kari Sentz & Scott Ferson, Combination of Evidence in Dempster-Shafer Theory 17-27 (Apr. 2002) (unpublished manuscript), available at http://www.sandia.gov/epistemic/Reports/SAND2002-0835.pdf
82. See Ronald R. Yager, New Paradigms for Reasoning with Uncertain Information, 13 CARDOZO L. REV. 1005, 1017-24 (1991)
83. See generally Mark Colyvan, Is Probability the Only Coherent Approach to Uncertainty?, 28 RISK ANALYSIS 645 (2008)
84. Bart Kosko, Fuzziness vs. Probability, 17 INT'L J. GEN. SYS. 211 (1990)
85. LEO KATZ, WHY THE LAW Is So PERVERSE 139-56 (2011)
86. Andrei Marmor, Varieties of Vagueness in the Law (USC Legal Studies Research Paper No. 12-8, Apr. 2012), available at http://ssrn.com/abstract= 2039076
87. Scott Soames, Vagueness in the Law, in THE ROUTLEDGE COMPANION TO PHILOSOPHY OF LAW 95 (Andrei Marmor ed., 2012)
88. See Michael T. Nguyen, Note, The Myth of "Lucky" Patent Verdicts: Improving the Quality of Appellate Review by Incorporating Fuzzy Logic in Jury Verdicts, 59 HASTINGS L.J. 1257, 1261 n.28 (2008)
89. See generally MACIEJ WYGRALAK, VAGUELY DEFINED OBJECRS (1996).
90. See Mark Jablonowski, An Ultra-fuzzy Model of Aggregate Growth in Catastrcphic Risk Potentials, 2008 ANN. MEETING N. AM. FUZZY INFO. PROCESSING SOC'Y (May 19-22, 2008)
91. See Jerry M. Mendel, Type-2 Fuzzy Sets and Systems: An Overview, IEEE COMPUTATIONAL INTELLIGENCE MAC., Feb. 2007, at 20, available at http://sipi.usc.edu/~mendel/publications/MENDEL%20CI%20Magazine%202007.pdf.
92. See Nilesh N. Karnik & Jerry M. Mendel, Operations on Type-2 Fuzzy Sets, 122 FUZZY SETS & SYS. 327 (2001)
93. see also Jerry M. Mendel, Type-2 Fuzzy Sets and Systems: How to Learn About Them, IEEE SMC ENEWSL. (Sys., Man & Cybernetics Soc'y, New York, N.Y.), June 2009, available at http://sipi.usc.edu/~inendel/publications/ MendelSMCeNewsletter6-09.pdf
94. See Ariel Porat & Eric A. Posner, Aggregation and Law, 122 YALE L.J. 2, 2 (2012)
95. Nicholas J.J. Smith, Degree of Belief Is Expected Truth Value, in CUTS AND CLOUDS 491, 491 (Richard Dietz & Sebastiano Moruzzi eds., 2010)
96. See Charles M. Yablon, On the Allocation of Burdens of Proof in Corporate Law: An Essay on Fairness and Fuzzy Sets, 13 CARDOZO L. REV. 497 (1991)
97. Bogdan R. Kosanovic, Fuzziness and Probability 2-3 (Feb. 8, 1995) (unpublished manuscript) (emphasis omitted), available at http://www.neuronet. pitt.edu/~bogdan/research/fuzzy/fvsp/fvsp.html.
98. Andrew Taylor, Congressional Offices Receive Mailed Threats, YAHOO! NEWS (Feb. 23, 2012, 1:47 AM), http://news.yahoo.com/congressional-offices-receive- mailed-threats-220538717.html.
99. See Richard Bellman & Magnus Giertz, On the Analytic Formalism of the Theory of Fuzzy Sets, 5 INFO. SCI. 149, 151-52 (1973)
100. See J. P. McBaine, Burden of Proof: Degrees of Belief, 32 CALIF. L. REV. 242 (1944)
101. C.M.A. McCauliff, Burdens of Proof: Degrees of Belief Quanta of Evidence, or Constitutional Guarantees?, 35 VAND. L. REV. 1293 (1982).
102. See WILLIAM N. DEMBER & JOEL S. WARM, PSYCHOLOGY OF PERCEPTION 113, 116-17 (2d ed. 1979).
103. See Terrence L. Fine, [The Axioms of Subjective Probability]: Comment, 1 STAT. SCI. 352, 353 (1986).
104. Yanlong Sun et al., Probabilistic Judgment on a Coarser Scale, 9 COGNITIVE SYS. RES. 161 (2008)
105. See Brian R. Gaines, Fuzzy and Probability Uncertainty Logics, 38 INFO. & CONTROL 154 (1978)
106. See Radim Bělohlávek et al., On the Capability of Fuzzy Set Theory to Represent Concepts, 31 INT'L J. GEN. SYS. 569, 575 (2002)
107. Ronald R. Yager, Connectives and Quantifiers in Fuzzy Sets, 40 Fuzzy SETS & SVS. 39. (1991).
108. Amos Tversky & Daniel Kahneman, Extensional Versus Intuitive Reasoning: The Conjunction Fallacy in Probability Judgment, in HEURISTICS AND BIASES 19, 20 (Thomas Gilovich et al. eds., 2002)
109. D. Dubois & H. Prade, A Review of Fuzzy Set Aggregation Connectives, 36 INFO. SCI. 85, 89-92 (1985)
110. The figure comes from 3 AVI SION, LOGICAL AND SPIRITUAL REFLECTIONS ch. 4 (2008), available at http://www.thelogician.net/6-reflect/6-Book-3/6c-chapter- 04.htm.
111. See H. G. WELLS, A MODERN UTOPIA 381 (1905)
112. See John Leubsdorf The Standard for Preliminary Injunctions, 91 HARV. L. REV. 525, 542 (1978)
113. Richard W. Wright, Proving Facts: Belief Versus Probability, in EUROPEAN TORT LAW 2008, at 79, 93 (Helmut Koziol & Barbara C. Steininger eds., 2009).
114. L. JONATHAN COHEN, THE PROBABLE AND THE PROVABLE (1977)
115. see also BERTRAND RUSSELL, HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS 359-61 (1948)
116. Susan Haack, The Embedded Epistemologist: Dispatches from the Legal Front, 25 RATIO JURIS 206, 217-18 (2012)
117. JASON ROSENHOUSE, THE MONRY HALL. PROBLEM 5 (2009).
118. See John Tierney, And Behind Door No. 1, a Fatal flaw, N.Y. TIMES, Apr. 8, 2008, at F1, available at http://www.nytimes.com/2008/04/08/science/08tier. html
119. See Alex Stein, Of Two Wrongs That Make a Right: Two Paradoxes of the Evidence Law and Their Combined Economic Justification, 79 TEX. L. RZV. 1199 (2001).
120. See Saul Levmore, Conjunction and Aggregation, 99 MICH. L. REV. 723, 734-45 (2001).
121. See Ronald J. Allen & Sarah A. Jehl, Burdens of Persuasion in Civil Cases: Algorithms v. Explanations, 2003 MICH. ST. L. REV. 893, 904-19
122. Paul H. Edelman, On Legal Interpretations of the Condorcet Jury Theorem, 31 J. LEGAL STUD. 327, 343-48 (2002).
123. The figure comes from TIMOTHY J. Ross, Fuzzy LOGIC WITH ENGINEERING APPLICATIONS 37 (3d ed. 2010).
124. Charles Nesson, The Evidence or the Event? On Judicial Proof and the Acceptability of Verdicts, 98 HARV. L. REV. 1357, 1385-88 (1985).
125. See generally J.S. COVINGTON, J.R., THE STRUCTURE OF LEGAL ARGUMENT AND PROOF 347-57 (2d ed. 2006)
126. See James A. Henderson, Jr. et al., Optimal Issue Separation in Modern Products Liability Litigation, 73 TEX. L. REV. 1653, 1655-59, 1667-75 (1995).
127. David A. Moran, Jury Uncertainty, Elemental Independence and the Conjunction Paradox: A Response to Allen and Jehl, 2003 MICH. ST. L. REV. 945, 946-47, 950.
128. See generally JEFFREY T. FREDERICK, THE PSYCHOLOGY OF THE AMERICAN JURY 296-99 (1987)
129. REID HASTIE ET AL., INSIDE THE JURY 22-23 (1983)
130. Paula L. Hannaford et al., The Timing of Opinion Formation by Jurors in Civil Cases: An Empirical Examination, 67 TENN. L. REV. 627, 629-32 (2000)
131. Jill E. Huntley & Mark Costanzo, Sexual Harassment Stories: Testing a Story-Mediated Model of Juror Decision-Making in Civil Litigation, 27 LAW & HUM. BEHAV. 29, 29 (2003)
132. Nancy Pennington & Reid Hastie, The Story Model for Juror Decision Making, in INSIDE THE JUROR 192 (Reid Hastie ed., 1993)
133. Compare Reid Hastie, What's the Story? Explanations and Narratives in Civil Jury Decisions, in CIVIL JURIES AND CIVIL JUSTICE 23, 31-32 (Brian H. Bornstein et al. eds., 2008)
134. with Michael S. Pardo, The Nature and Purpose of Evidence Theory, 66 VAND. L. REV. (forthcoming 2013), available at http://ssrn.com/abstract=2060340
135. KEVIN F. O'MALLEY ET AL., FEOERAL JURY PRACTICE AND INSTRUCTIONS: CIVIL § 104.01 (6th ed. 2011)
136. See Bruce Ching, Narrative Implications of Evidentiary Rules, 29 QUINNIPIAC L. REV. 971 (2011)
137. Doron Menashe & Mutal E. Shamash, The Narrative Fallacy, 3 INT'L COMMENT. ON EVIDENCE iss. 1, art. 3 (2006)
138. But cf. Elizabeth G. Thornburg, The Power and the Process: Instructions and the Civil Jury, 66 FORDHAM L. REV. 1837, 1857-63 (1998)
139. See Gregg C. Oden, Integration of Fuzzy Logical Information, 3 J. EXPERIMENTAL PSYCHOL.: HUM. PERCEPTION & PERFORMANCE 565, 568-572 (1977).
140. See Rami Zwick, David V. Budescu & Thomas S. Wallsten, An Empirical Study of the Interpretation of Linguistic Probabilities, in Fuzzy SETS IN PSYCHOLOGY 91, 114-16 (Tamás Zétényi ed., 1988)
141. see Massachusetts v. U.S. Dep't of Health & Human Servs., 682 F.3d 1 (1st Cir. 2012) (striking down DOMA on both equal protection and federalism grounds)
142. Mike Dorf, Is the First Circuit's Opinion in the DOMA Case Insufficiently "Fuzzy"?, DORF ON LAW (June 4, 2012, 12:30 AM), http://www.dorfonlaw. org.
143. Compare Alon Harel & Ariel Porat, Aggregating Probabilities Across Cases: Criminal Responsibility for Unspecified Offenses, 94 MINN. L. REV. 261, 261-62 (2009)
144. Frederick Schauer & Richard Zeckhauser, On the Degree of Confidence for Adverse Decisions, 25 J. LEGAL STUD. 27, 41-47 (1996)
145. See Kevin M. Clermont, Aggregation of Probabilities and Illogic, 47 GA. L. REV. 165 (2012).
146. A. P. Dempster, Upper and Lower Probabilities Induced by a Multivalued Mapping, 38 ANNALS MATHEMATICAL STAT. 325 (1967)
147. L.A. Zadeh, Fuzzy Sets as a Basis for a Theory of Possibility, 1 Fuzzy SETS & SYS. 3 (1978).
148. Kevin M. Clermont, Standards of Proof in Japan and the United States, 37 CORNELL. INT'L L.J. 263 (2004)
149. Richard W. Wright, Proving Causation: Probability Versus Belief in PERSPECTIVES ON CAUSATION 195 (Richard Goldberg ed., 2011).
150. Adrian A.S. Zuckerman, Law, Fact or Justice?, 66 B.U. L. REV. 487, 508 (1986)
151. See, e.g., Richard Lempert, The New Evidence Scholarship: Analyzing the Process of Proof 66 B.U. L. REV. 439, 462-67 & n.60 (1986)
152. See 9 JOHN H. WIGMOI, EVIDENCE § 2487 (James H. Chadbourn rev., 1981)
153. cf John T. McNaughton, Burden of Production of Evidence: A Function of a Burden of Persuasion, 68 HARV. L. REV. 1382 (1955)
154. See John Fancy Thorne III, Comment, Mathematics, Fuzzy Negligence, and the Logic of Res Ipsa Loquitur, 75 Nw. U. L. REV. 147 (1980)
155. See Stimpson v. Hunter, 125 N.E. 155, 157 (Mass. 1919)
156. See Cruzan v. N.Y. Cent. & Hudson River R.R. Co., 116 N.E. 879, 880 (Mass. 1917)
157. See Dyer v. MacDougall, 201 F.2d 265, 269 (2d Cir. 1952)
158. See Guenther v. Armstrong Rubber Co., 406 F.2d 1315, 1318 (3d Cir. 1969)
159. See RICHARD H. FIELD, BENJAMIN KAPLAN & KEVIN M. CLERMONT, MATERIALS FOR A BASIC COURSE IN CIVIL PROCEDURE 1352-56 (10th ed. 2010)
160. See Baxter v. Palmigiano, 425 U.S. 308, 316-20 (1976)
161. See Robert S. Summers, Evaluating and Improving Legal Processes-A Plea for "Process Values, "60 CORNELL L. REV. 1, 4 (1974)
162. See Anne W. Martin & David A. Schum, Quantifying Burdens of Proof A Likelihood Ratio Approach, 27 JURIMETRICS J. 383, 390-93 (1987)
163. See David Kaye, The Limits of the Preponderance of the Evidence Standard: Justifiably Naked Statistical Evidence and Multiple Causation, 1982 AM. B. FOUND. RES. J. 487
164. See Peter Tillers & Jonathan Gottfried, Case Comment-United States v. Cope-land, 369 F. Supp. 2d 275 (E.D.N. Y. 2005)
165. A Collateral Attack on the Legal Maxim That Proof Beyond a Reasonable Doubt Is Unquantifiable?, 5 LAW PROBABILITY & RISK 135, 142 (2006)
166. See, e.g., Brown v. Bowen, 847 F.2d 342, 345 (7th Cir. 1988)
167. See United States ex rel. Bilyew v. Franzen, 686 F.2d 1238, 1248 (7th Cir. 1982)
168. See Eyal Zamir & Ilana Ritov, Loss Aversion, Omission Bias, and the Burden of Proof in Civil Litigation, 41 J. LEGAL STUD. 165, 197 n.23 (2012).
169. See Amos Tversky & Daniel Kahneman, Judgment Under Uncertainty: Heuristics and Biases, 185 SCIENCE (n.s.) 1124, 1128 (1974).
170. See Santosky v. Kramer, 455 U.S. 745, 757 (1982)
171. see also Edward K. Cheng, Reconceptualizing the Burden of Proof 122 YALE L.J. (forthcoming 2013) (manuscript at 3-4, 6), available at http://ssrn.com/abstract =2087254
172. See Milward v. Acuity Specialty Prods. Grp., Inc., 639 F.3d 11, 18 (1st Cir. 2011)
173. Westberry v. Gislaved Gummi AB, 178 F.3d 257 (4th Cir. 1999)
174. See Ronald J. Allen & Michael S. Pardo, Juridical Proof and the Best Explanation, 27 LAW & PHIL. 223, 226 (2008)
175. However, Larry Laudan, Strange Bedfellows: Inference to the Best Explanation and the Criminal Standard of Proof, 11 INT'L J. EVIDENCE & PROOF 292 (2007)
176. See Richard D. Friedman, "E" Is for Eclectic: Multiple Perspectives on Evidence, 87 VA. L. REV. 2029, 2046-47 (2001)
177. But cf Ronald J. Allen, The Nature of Juridical Proof, 13 CARDOZO L. REV. 373, 413 (1991)
178. Ronald J. Allen & Brian Leiter, Naturalized Epistemology and the Law of Evidence, 87 VA. L. REV. 1491, 1528 (2001)
179. See Ronald J. Allen, Standards of Proof and the Limits of Legal Analysis 14 (May 25-26, 2011) (unpublished conference paper), available at http://ssrn.com/abstract=1830344.
180. See Craig R. Callen, Commentary, Kicking Rocks with Dr. Johnson: A Comment on Professor Allen's Themy, 13 CARDOZO L. REV. 423
181. See, e.g., Michael S. Pardo, Second-Order Proof Rules, 61 FLA. L. REV. 1083 (2009)
182. Edmund M. Morgan, Instructing the Jury upon Presumptions and Burden of Proof 47 HARV. L. REV. 59, 67 (1933)
183. See Michael J. Saks & Robert F. Kidd, Human Information Processing and Adjudication: Trial by Heuristics, 15 LAW & Soc'y REV. 123, 126 (1981)
184. Milton Dawes, Multiordinality: A Point of View, ETC, Summer 1986, at 128, 131
185. cf Neal Gabler, The Elusive Big Idea, N.Y. TIMES, Aug. 14, 2011 (Sunday Review), at 1, available at http://www.nytimes.com/2011/08/14/opinion/sunday/ the-elusive-big-idea.html
186. See, e.g., Daniel Greco, Probability and Prodigality, 4 OXFORD STUD. EPISTEMOLOGY (forthcoming 2013), available at http://web.mit.edu/dlgreco/www/ ProbAndProd.pdf
187. Gary Lawson, Proving the Law, 86 Nw. U. L. REV. 859,871-74 (1992)
188. See Ronald J. Ailen & Larry Laudan, Deadly Dilemmas, 41 TEX. TECH. L. REV. 65 (2008)
189. Larry Laudan & Harry D. Saunders, Re-thinking the Criminal Standard of Proof Seeking Consensus About the Utilities of Trial Outcomes, 7 INT'L COMMENT. ON EVIDENCE iss. 2, art. 1 (2009).
190. Nissho-Iwai Co. v. M/T Stolt Lion, 719 F.2d 34, 38 (2d Cir. 1983)
191. See Dominique Demougin & Claude Fluet, Deterrence Versus Judicial Error: A Comparative View of Standards of Proof, 161 J. INTITUTIONAL & THEORETICAL ECON. 193 (2005)
192. See Richard A. Posner, An Economic Approach to Legal Procedure and Judicial Administration, 2 J. LEGAL STUD. 399, 414-16 (1973)
193. cf Richard D. Friedman, Standards of Persuasion and the Distinction Between Fact and Law, 86 Nw. U. L. REV. 916, 926 (1992)
194. See D.H. Kaye, The Error of Equal Error Rates, 1 LAW PROBABILITY & RISK 3, 7 (2002)
195. David Hamer, Probabilistic Standards of Proof Their Complements and the Errors That Are Expected to Row from Them, 1 U. NEW ENG. L.J. 71 (2004)
196. cf Neil Orloff & Jery Stedinger, A Framework for Evaluating the Preponderance-of-the-Evidence Standard, 131 U. PA. L. REV. 1159 (1983)
197. See Louis Kaplow, Burden of Proof, 121 YALE L.J. 738 (2012)
198. see also Fredrick E. Vars, Toward a General Theory of Standards of Proof, 60 CATH. U. L. REV. 1 (2010)