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A Mathematical Comment on the Fundamental Difference Between Scientific
Theory Formation and Legal Theory Formation
Ronald P. Loui
St. Louis
USA
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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
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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?
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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?
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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
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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
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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)
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Curve-Fitting: assign error as required
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Spline-Fitting: complexify as required
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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)
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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)
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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
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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
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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
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Argument for h
hp qa bc
d
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Argument for h
h
p q
a
bc d
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Argument for h
h
p q
a b c d
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Argument for h
h
p q
a b c d
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Argument for h
h
p q
a b c d
Defeasibly,
a >__ p
b c d >__ q
p q >__ h
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Dialectical Tree
A1
A3
B2A2
B1
petitioner respondent
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Dialectical Tree
A1
A3
B2
A2
B1
Interferes
defeats
defeats
defeats
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Dialectical Tree
A1 (for h)
A3 for !q
B2 for !r
A2 for !q
B1 (for !p)
Interferes
defeats
Defeats
defeats
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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) )
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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
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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
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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
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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.
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Proper Elisions I: Argument Trees
h
p q
a b c d
p b c d >__ h
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Proper Elisions I: Argument Trees
h
p q
a b c d
!q
a b f
p b c d >__ h p b c d f >__ h ?
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Proper Elisions II: Dialectical Trees
A1 (for h)
A3 for !q
B2 for !r
A2 for !q
B1 (for !p)
Interferes
defeats
Defeats
defeats
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Proper Elisions II: Dialectical Trees
A1 (for h)
A3 for !q
B2 for !r
A2 for !q
B1 (for !p)
Interferes
defeats
Defeats
defeats
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Proper Elisions II: Dialectical Trees
A1 (for h)
A3 for !q
B2 for !r
A2 for !q
B1 (for !p)
Interferes
defeats
Defeats
defeats
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SFT vs. LFT
1. Defeasible2. Differences distinguished3. Cases
summarized/organized4. Argument is crucial5. Justification obsessed6. Loui:
ArgumentsGroundingProper ElisionPrinciples
1. Deductive
2. Error quantified
3. Rules discovered
4. Probability is key
5. Simplicity biased
6. Kyburg:
Acceptance
Error
Inference
Coherence
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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
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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
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Machine Learning?
• Models are too simple
• The problem is in the modeling, not the algorithm
• SVM is especially insulting
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Acknowledgements
• Henry Kyburg
• Ernest Nagel, Morris Cohen
• Jeff Norman
• Guillermo Simari
• AnaMaguitman, Carlos Chesñevar, Alejandro Garcia
• John Pollock, Thorne McCarty, Henry Prakken