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

3. 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

4. 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?

5. 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?

6. 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

7. 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

8. 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)

9. Curve-Fitting: assign error as required

10. Spline-Fitting: complexify as required

11. 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)

12. 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)

13. 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

14. More Nuanced Model of SFT Kyburg: –Corpus of accepted beliefs K –Probability of s given K: P K (s) –s is acceptable? P K (s) > 1-e –Theory is U: U  K = D-Thm(K0  U) –SFT: choose U* to “fit” K 0 Best fit of U* gives largest PI-Thm(K) PI-Thm(K) = K  {s | P K (s) > 1-e } –Trades power (simplicity) and error (fit) If U is too simple, it doesn’t fit, hence all P K small If U is too complicated, D-Thm(K0  U) small

15. More Nuanced Model of LFT Loui-Norman (Prakken-Sartor-Hage- Verheij-Lodder-Roth) –A case has arguments, A 1, … A k, B 1, … B k-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

16. Argument for h h p q ab c d

17. h p q a b c d

18. h pq abcd

19. h pq abcd

20. h pq abcd Defeasibly, a > __ p b c d > __ q p q > __ h

21. Dialectical Tree A1A1 A3A3 B2B2 A2A2 B1B1 petitionerrespondent

22. Dialectical Tree A1A1 A3A3 B2B2 A2A2 B1B1 Interferes defeats

23. Dialectical Tree A 1 (for h) A 3 for !q B 2 for !r A 2 for !q B 1 (for !p) Interferes defeats Defeats defeats

24. More Nuanced Model of LFT Loui-Norman (Prakken-Sartor-Hage- Verheij-Lodder-Roth) –A case has arguments, A 1, … A k, B 1, … B k-1 –Arguments have structure –Induced rules must be grounded in cases Δ (e.g. c 1 = ({a,b,c,d,e}, h, {(h,{(p,{a}),(q,{b,c,d})}, …) or background sources Ω (e.g., p q > __ h, r 17 = ({p,q},h) )

25. 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

26. 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

27. 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

28. 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.

29. Proper Elisions I: Argument Trees h pq abcd p b c d > __ h

30. Proper Elisions I: Argument Trees h pq abcd !q abf p b c d > __ h p b c d f > __ h ?

31. Proper Elisions II: Dialectical Trees A 1 (for h) A 3 for !q B 2 for !r A 2 for !q B 1 (for !p) Interferes defeats Defeats defeats

32. Proper Elisions II: Dialectical Trees A 1 (for h) A 3 for !q B 2 for !r A 2 for !q B 1 (for !p) Interferes defeats Defeats defeats

33. Proper Elisions II: Dialectical Trees A 1 (for h) A 3 for !q B 2 for !r A 2 for !q B 1 (for !p) Interferes defeats Defeats defeats

34. SFT vs. LFT 1.Defeasible 2.Differences distinguished 3.Cases summarized/organized 4.Argument is crucial 5.Justification obsessed 6.Loui: Arguments Grounding Proper Elision Principles 1.Deductive 2.Error quantified 3.Rules discovered 4.Probability is key 5.Simplicity biased 6.Kyburg: Acceptance Error Inference Coherence

35. More Nuanced Model of SFT Kyburg: –Corpus of accepted beliefs K –Probability of s given K: P K (s) –s is acceptable? P K (s) > 1-e –Theory is U: U  K = D-Thm(K0  U) –SFT: choose U* to “fit” K 0 Best fit of U* gives largest PI-Thm(K) PI-Thm(K) = K  {s | P K (s) > 1-e } –Trades power (simplicity) and error (fit) If U is too simple, it doesn’t fit, hence all P K small If U is too complicated, D-Thm(K0  U) small

36. More Nuanced Model of LFT Loui-Norman (Prakken-Sartor-Hage- Verheij-Lodder-Roth) –A case has arguments, A 1, … A k, B 1, … B k-1 –Arguments have structure –Induced rules must be grounded in cases Δ (e.g. c 1 = ({a,b,c,d,e}, h, {(h,{(p,{a}),(q,{b,c,d})}, …) or background sources Ω (e.g., p q > __ h, r 17 = ({p,q},h) ) –And proper elisions

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

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