A Mathematical Comment on the Fundamental Difference Between Scientific
Theory Formation and Legal Theory Formation
Ronald P. Loui
St. Louis
USA
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
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?
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?
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
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
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)
Curve-Fitting: assign error as required
Spline-Fitting: complexify as required
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)
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)
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
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
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
Argument for h
hp qa bc
d
Argument for h
h
p q
a
bc d
Argument for h
h
p q
a b c d
Argument for h
h
p q
a b c d
Argument for h
h
p q
a b c d
Defeasibly,
a >__ p
b c d >__ q
p q >__ h
Dialectical Tree
A1
A3
B2A2
B1
petitioner respondent
Dialectical Tree
A1
A3
B2
A2
B1
Interferes
defeats
defeats
defeats
Dialectical Tree
A1 (for h)
A3 for !q
B2 for !r
A2 for !q
B1 (for !p)
Interferes
defeats
Defeats
defeats
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) )
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
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
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
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.
Proper Elisions I: Argument Trees
h
p q
a b c d
p b c d >__ h
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 ?
Proper Elisions II: Dialectical Trees
A1 (for h)
A3 for !q
B2 for !r
A2 for !q
B1 (for !p)
Interferes
defeats
Defeats
defeats
Proper Elisions II: Dialectical Trees
A1 (for h)
A3 for !q
B2 for !r
A2 for !q
B1 (for !p)
Interferes
defeats
Defeats
defeats
Proper Elisions II: Dialectical Trees
A1 (for h)
A3 for !q
B2 for !r
A2 for !q
B1 (for !p)
Interferes
defeats
Defeats
defeats
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
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
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
Machine Learning?
• Models are too simple
• The problem is in the modeling, not the algorithm
• SVM is especially insulting
Acknowledgements
• Henry Kyburg
• Ernest Nagel, Morris Cohen
• Jeff Norman
• Guillermo Simari
• AnaMaguitman, Carlos Chesñevar, Alejandro Garcia
• John Pollock, Thorne McCarty, Henry Prakken