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Ronald P. Loui St. Louis USA

A Mathematical Comment on the Fundamental Difference Between Scientific Theory Formation and Legal Theory Formation. Ronald P. Loui St. Louis USA. Why? Who?.

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Ronald P. Loui St. Louis USA

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  1. A Mathematical Comment on the Fundamental Difference Between Scientific Theory Formation and Legal Theory Formation Ronald P. Loui St. Louis USA

  2. Why? Who? • Philosophers of science (students of generalized inductive reasoning) should find the legal theory formation problem (generalized moral reasoning) interesting now that there are new tools: • Defeasible conditional • A record of arguments • Models of procedures • Diachronic models: confirmational conditionalization, belief revision, derogation, contraction, choice • Models of legislative compromise, linguistic interpretation and determination

  3. Why? Who? • What are the similarities, dissimilarities? • Obviously: attitude toward error • What else? • What formal ramifications? • Could the LTF problem be expressed as simply as the STF problem?

  4. Further Motivation • Is Machine Learning too quick to simplify the problem? • Can the important nuances of LTF and STF be written in a mathematically brief way?

  5. Legal Theory Formation: LTF • Case 1: • Facts: a b c d e • Decision: h • Case 2: • Facts: a b c d e f • Decision: !h • Induced rule(s): • Defeasibly, a b c d e >__ h • Defeasibly, a b c d e f >__!h Why not: a >__ h a f >__!h

  6. Scientific Theory Formation: STF • Case 1: • Facts: a b c d e • Decision: h • Case 2: • Facts: a b c d e f • Decision: !h • Induced rule(s): • Deductively, a b c d e !f  h • Deductively, a b c d e f !h Why not: !f  h f  h

  7. SFT vs. LFT • Conditionals: • Deductive vs. • Defeasible • Bias: • What is simpler? vs. • What is motivated by argument? • Input: • State (complete closed world) vs. • Partial (incomplete) Description • STF, LFT vs: Belief revision (AGM) • too much (=epistemic state + constraints on chance) vs. • too little (=not enough guidance among choices)

  8. Curve-Fitting: assign error as required

  9. Spline-Fitting: complexify as required

  10. 2-DNF Fitting • Data: • Case 1: a b c d • Case 2: !a b c !d • Case 3: a !b !c d • Formula: • (a v b) ^ (c v d)

  11. Transitive  fitting • Reports of indifference, preference • A ~ B • B > C • A ~ C • C ~ D • A ~ D • Error: remove B > C, actually B ~ C (1 of 5)

  12. SFT vs. LFT • Fit: • Quantify error (like overturning precedent in LFT) vs. • Distinguish as needed (like auxiliary hypotheses in SFT) • SO FAR, ALL THIS IS OBVIOUS

  13. More Nuanced Model of SFT • Kyburg: • Corpus of accepted beliefs K • Probability of s given K: PK(s) • s is acceptable? PK(s) > 1-e • Theory is U: U K = D-Thm(K0  U) • SFT: choose U* to “fit” K0 • Best fit of U* gives largest PI-Thm(K) • PI-Thm(K) = K  {s | PK(s) > 1-e } • Trades power (simplicity) and error (fit) • If U is too simple, it doesn’t fit, hence all PK small • If U is too complicated, D-Thm(K0  U) small

  14. More Nuanced Model of LFT • Loui-Norman (Prakken-Sartor-Hage-Verheij-Lodder-Roth) • A case has arguments, A1, … Ak , B1, … Bk-1 • Arguments have structure • Trees, labeled with propositions • Argument for h, h is root • Leaves are uncontested “facts” • Internal nodes are “intermediary conclusions” • Defeasible rules: Children(p) >__ p

  15. Argument for h h p q a b c d

  16. Argument for h h q p b c d a

  17. Argument for h h p q a b c d

  18. Argument for h h p q a b c d

  19. Argument for h h p q a b c d Defeasibly, a >__ p b c d >__ q p q >__ h

  20. Dialectical Tree petitioner respondent B1 A1 A2 B2 A3

  21. Dialectical Tree A1 Interferes B1 defeats defeats A2 A3 defeats B2

  22. Dialectical Tree A1 (for h) Interferes B1 (for !p) Defeats defeats A2 for !q A3 for !q defeats B2 for !r

  23. More Nuanced Model of LFT • Loui-Norman (Prakken-Sartor-Hage-Verheij-Lodder-Roth) • A case has arguments, A1, … Ak , B1, … Bk-1 • Arguments have structure • Induced rules must be grounded in • cases Δ (e.g. c1 = ({a,b,c,d,e}, h, {(h,{(p,{a}),(q,{b,c,d})}, …) or • background sources Ω (e.g., p q >__ h, r17 = ({p,q},h) )

  24. SFT vs. LFT • Invention: • Out of (mathematical) thin air vs. • Possible interpretations of cases • Purpose: • To discover rules from cases • To summarize cases as rules

  25. SFT vs. LFT • Invention: • Out of (mathematical) thin air vs. • Possible interpretations of cases • Purpose: • To discover (nomological) rules from cases • To summarize cases as (linguistic) rules

  26. SFT vs. LFT • Invention: • Out of (mathematical) thin air vs. • Possible interpretations of cases • Purpose: • To discover (nomological) rules from (accident of) cases • To summarize (wisdom of) cases as (linguistic) rules

  27. What is grounded? • Case: a b c d e ]__ h • φ = {a, b, c, d, e} • Any C  φ as lhs for rule for h? • What if d was used only to argue against h? • d >__ h • Really? (Even Ashley disallows this) • What if e was used only to rebut d-based argument? • a b c e >__ h • Really? e isn't relevant except to undercut d.

  28. Proper Elisions I: Argument Trees h p q a b c d p b c d >__ h

  29. !q a b f Proper Elisions I: Argument Trees h p q a b c d p b c d >__ h p b c d f >__ h ?

  30. Proper Elisions II: Dialectical Trees A1 (for h) Interferes B1 (for !p) Defeats defeats A2 for !q A3 for !q defeats B2 for !r

  31. Proper Elisions II: Dialectical Trees A1 (for h) Interferes B1 (for !p) Defeats defeats A2 for !q YES! A3 for !q defeats B2 for !r

  32. Proper Elisions II: Dialectical Trees A1 (for h) Interferes B1 (for !p) Defeats defeats A2 for !q NO! A3 for !q defeats B2 for !r

  33. SFT vs. LFT • Defeasible • Differences distinguished • Cases summarized/organized • Argument is crucial • Justification obsessed • Loui: Arguments Grounding Proper Elision Principles • Deductive • Error quantified • Rules discovered • Probability is key • Simplicity biased • Kyburg: • Acceptance • Error • Inference • Coherence

  34. More Nuanced Model of SFT • Kyburg: • Corpus of accepted beliefs K • Probability of s given K: PK(s) • s is acceptable? PK(s) > 1-e • Theory is U: U K = D-Thm(K0  U) • SFT: choose U* to “fit” K0 • Best fit of U* gives largest PI-Thm(K) • PI-Thm(K) = K  {s | PK(s) > 1-e } • Trades power (simplicity) and error (fit) • If U is too simple, it doesn’t fit, hence all PK small • If U is too complicated, D-Thm(K0  U) small

  35. More Nuanced Model of LFT • Loui-Norman (Prakken-Sartor-Hage-Verheij-Lodder-Roth) • A case has arguments, A1, … Ak , B1, … Bk-1 • Arguments have structure • Induced rules must be grounded in • cases Δ (e.g. c1 = ({a,b,c,d,e}, h, {(h,{(p,{a}),(q,{b,c,d})}, …) or • background sources Ω (e.g., p q >__ h, r17 = ({p,q},h) ) • And proper elisions

  36. Machine Learning? • Models are too simple • The problem is in the modeling, not the algorithm • SVM is especially insulting

  37. Acknowledgements • Henry Kyburg • Ernest Nagel, Morris Cohen • Jeff Norman • Guillermo Simari • AnaMaguitman, Carlos Chesñevar, Alejandro Garcia • John Pollock, Thorne McCarty, Henry Prakken

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