Abstract The course concerns three subjects: modelling legal argument, legal informatics, and visualization. Visualization makes legal reasoning more comprehensible. But exhaustive theories of these subjects can hardly be developed on the level of formal theories. However, I aim at a big picture of legal informatics, which is proposed by Friedrich Lachmayer. The following problem is examined as an introduction. Suppose several goals are pursued whereas the rules make it possible to achieve only some of them. Here I am inspired by an example of Trevor Bench-Capon and Henry Prakken (2006): a judge must determine the best way to punish a criminal. I address the infeasibility problem and simple formalisations. Suppose a judge who must determine the best way to punish (pu) a criminal found guilty. He has three options: imprisonment (pr), a fine (fi) and community service (cs). Besides punishment there are three more goals at stake, deterring the general public (de), rehabilitating the offender (re) and protecting society from crime (pt). The judge must ensure that the offender is punished, and so pu will be the most important goal, but the method of punishment chosen will depend on other goals than can be achieved by various methods of punishing the offender. The judge believes that imprisonment promotes both deterrence and protection of society, while it demotes rehabilitation of the offender. He believes that a fine promotes deterrence but has no effect on rehabilitation or the protection of society since the offender would remain free, and he believes that community service has a positive effect on rehabilitation of the offender but a negative effect on deterrence since this punishment is not feared. The judge’s goal base is G = {pu, pt, de, re}. An analysis of the alternative actions pr, fi and cs shows that two or three of the goals can be achieved but not four. Another example. Suppose I have one coin and want to buy two items: a drink and a roll of bread. The rule permits to buy one item for a coin. Therefore I cannot buy both items. Further I transform the infeasibility problem into the weighing problem. Which action to choose: to buy a drink or alternatively a roll? Formal logic cannot help here. Extra reasons have to be involved. E.g. I give the weight 2 to the drink and the weight 1 to the bun – I am thirsty. Therefore I choose a drink. In other circumstances, however, I might choose a bun. recall Hans Kelsen’s Is and Ought and Chapters 1 to 4 of his book “General Theory of Norms”. Kelsen formulates “No logical relation between willing the end and willing the means” when analysing Kant’s imperative of skill and the principle “The End Justifies the Means”. A decision maker can choose the means of the best weight. This can be formalised as a justified means which brings about a certain state of the world and minimises the distance to a desired state.

1. On legal reasoning, legal informatics and visualization
Transforming the problem of infeasibility of
achieving several goals into a weighing problem
Vytautas ČYRAS
Vilnius University
Faculty of Mathematics and Informatics
Vilnius, Lithuania
Vytautas.Cyras@mif.vu.lt
http://www.mif.vu.lt/~cyras/
ERASMUS Teaching Assignment, University of Salzburg, February 2013

2. 1. Legal reasoning.
An example
T. Bench-Capon, H. Prakken (2006) Justifying actions by
accruing arguments. In: Computational Models of Argument –
Proceedings of COMMA 2006, pp. 247–258. IOS Press.
http://www.booksonline.iospress.nl/Content/View.aspx?piid=89
Slides: http://www.cs.uu.nl/groups/IS/archive/henry/action.pdf
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3. An example problem: legal punishment
A judge must determine the best way to punish (pu) a criminal found guilty. He
has three options: imprisonment (pr), a fine (fi) and community service (cs).
Besides punishment there are three more goals at stake, deterring the general
public (de), rehabilitating the offender (re) and protecting society from crime
(pt).
So pu will be the most important goal, but the method of punishment chosen
(pr, fi or cs) will depend on other goals.
Initial state ()
• Actions:
1. imprisonment (pr), pr fi cs
2. fine (fi)
3. community service (cs) Final state ( pu, de, pt, re )
• Goals:
1. punishment (pu) – main goal
2. deterrence (de)
3. rehabilitation (re)
4. protecting society (pt)
Hence, the judge’s goal base G = { pu, de, pt, re }
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4. Causal knowledge
1. Imprisonment (pr) promotes both deterrence (de) [R4] and
protection of society (pt) [R5], but demotes rehabilitation (re) [R6]
of the offender.
2. Fine (fi) promotes deterrence (de) [R7] but has no effect on
rehabilitation (re) or the protection of society (pt) since the offender
would remain free.
3. Community service (cs) promotes rehabilitation (re) [R9] of the
offender, but demotes deterrence (de) [R8] since this punishment
is not feared.
Causal rules (between actions and goals):
R1: pr pu R4: pr de R7: fi de R8: cs de
R2: fi pu R5: pr pt
R3: cs pu R6: pr re R9: cs re
community
3 actions: imprisonment (pr) fine (fi)
service (cs)
R6 R2 R8
R1 R3
R5 R4 R9
4 goals: protection of deterrence (de) punishment (pu) rehabilitation (re)
society (pt) 4

5. Values of goals
• Judge’s goal base G = { pu, de, pt, re }
(more exactly, G = { D pu, D de, D pt, D re },
where D is a modality; standing for desire)
– A propositional modal logic is used
• All 4 goals cannot be achieved! See further
• Question: What is the best way to punish the offender?
• Answer: cs (see further)
– Reason: first, cs > pr, second, cs > fi
Value (promoted, demoted) Score { pu, de, pt, re }
pr +
v(pr +) = ( {pu, de, pt} , {re} ) 3:1 (1, 1, 1, -1)
fi +
v(fi +) = ( {pu, de} , ) 2:0 (1, 1, 0, 0)
v(cs+) = ( {pu, re} , {de} ) 2:1 (1, -1, 0, 1) cs+
R1: pr pu R4: pr de R7: fi de R8: cs de
R2: fi pu R5: pr pt
R3: cs pu R6: pr re R9: cs re 5

6. A sketch of reasoning for cs
>1
pr + pr –
• Step 1
fi +
– pr + >1 pr – Reason: pu sways >3
>4
– cs+ >2 cs – – ’’ – cs+
>2
cs –
• Step 2: >3
winner
– Extralogical choice: re is next to pu
– Hence re >3 de + pt
• Step 3: >4
– Extralogical choice for rehabilitation:
re – de >4 de
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7. Arguments on imprisonment
Practical syllogism, originally see Aristotle
1. Agent P wishes to realise goal G. DG DG
2. If P performs action A, G will be realised. A G A G
3. Therefore, P should perform A. ––––– Abduction ––––– Abduction
DA positive D A negative
(Positive practical (Negative practical
Both PPS and NPS are defeasible.
syllogism, PPS) syllogism, NPS)
Individual defeat
l1 D pr l2 D pr l3 D pr l4 D pr
pr pu D pu pr de D de pr pt D pt pr re D re
R1 R4 R5 R6
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8. Abduction and deduction
Abduction Goal Deduction
• Reasoning from goals • Reasoning from facts
to facts R3 to goals
• Abductive reasoning • Deductive reasoning
R4
• Backward-chaining in • Forward-chaining in
R1 R2 Artificial Intelligence
Artificial Intelligence
Facts
DG G A
A G A G A G
Abduction modus
––––– positive ––––– Abduction ––––– ponens
DA A G
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9. Accruals on imprisonment.
Then defeat
Defeat >1 :
pu sways
Accrual : pr + D pr pr − D pr
l1 D pr l2 D pr l3 D pr l4 D pr
pr pu D pu pr de D de pr pt D pt pr re D re
R1 R4 R5 R6
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10. Accrual on fining
Accrual : fi + D fi
l5 D fi l6 D fi
fi pu D pu fi de D de
R2 R7
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11. Accruals on community service
Defeat >2 :
pu sways
Accrual : cs + D cs cs − D cs
l7 D cs l8 D cs l9 D cs
cs pu D pu cs re D re cs de D de
R3 R9 R8
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12. The attack graph pr +
>1
pr –
• Step 1. >1 and >2 proved above. >3
fi +
winner
• Step 2. >3 >4
>2
Value (promoted, demoted) cs+ cs –
v(pr +) = ( {pu, de, pt} , {re} ) 3:1 • Extralogical choice: re is next to
v(cs+) = ( {pu, re} , {de} ) 2:1 pu
• Thus we (judge) make pu the
re >3 de + pt second most important goal
More precisely, re – de >3 de + pt – re • Other choices, e.g. pro fine fi +
are possible
cs + D cs Defeat: >3 pr + D pr
l7 D cs l8 D cs l1 D pr l2 D pr l3 D pr
cs pu D pu cs re D re pr pu D pu pr de D de pr pt D pt
R3 R9 R1 R4 R5
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13. Step 3. Defeat >4 >1
pr + pr –
• We chose promoting rehabilitation re
while demoting deterrence de over fi +
>3 winner
promoting deterrence de. >4
• Formally, re – de >4 de . >2
cs+ cs –
Value (promoted, demoted)
v(fi +) = ( {pu, de} , ) 2:0 Justification: given that we
v(cs+) = ( {pu, re} , {de} ) 2:1 must punish, we choose to do
so in a way which will aid
re – de >4 de rehabilitation.
cs + D cs Defeat: >4 fi + D fi
l7 D cs l8 D cs l5 D fi l6 D fi
cs pu D pu cs re D re fi pu D pu fi de D de
R3 R9 R2 R7
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14. Conclusions
• Limitations
– of the this formalisation
• See Bench-Capon & Prakken
– of artificial intelligence in law
• formalising choice
• algorithmically undecidable problems
• NP-problems
– of mathematics
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15. 2. On the problem of
infeasibility of achieving
several goals
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16. Example: drink vs. roll
• You have one coin and want to buy two
items:
– drink Initial state ()
– roll of bread chooseDrink chooseRoll
Final state ( drink, roll )
• Buy one item for a coin. Impossible
You cannot buy both items.
– “One cannot eat it and keep it”
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17. Choice is extralogical
• Which action to choose:
to buy a drink or alternatively a roll? roll
Impossible
– (drink, 0) > (0, roll) or alternatively (1,1)
1
(drink, 0) < (0, roll) ?
– No ordering of vectors, i.e.
neither (1,0) > (0,1)
nor (1,0) < (0,1) drink
0 (1,0)
• Formal logic cannot help here to make a choice
– Extralogical reasons have to be involved 0 1
– E.g., weight 2 to the drink to the roll – you are thirsty.
Therefore you choose the drink
– In other circumstances you might choose the roll roll (2,1)
1
• Reasoning with the distance to the goal
– Distance from drink: | (2,1) – (2,0) | = | (0,1) | = 1
– Distance from roll: | (2,1) – (1,0) | = | (0,1) | = 2
– Smaller distance to goal, 1, is better than 2. Therefore drink
drink wins. 0
0 2
winner
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18. The landscape metaphor in means-
ends analysis
“The end justifies the means” (Der Zweck heiligt das Mittel).
Kant’s imperative: “Who is willing the end, must be willing
the means” (Wer den Zweck will, muss das Mittel wollen).
evaluation
mweak = 0,1
mright = 1,1
positive 1
mwrong = 1,0
negative 0 bringsAboutTheEnd
0 1
false true
3 means mwrong, mweak and mright
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19. Thank you
Vytautas.Cyras@mif.vu.lt
Vytautas.Cyras@mif.vu.lt 19