Vytautas ČYRAS Vilnius University
Faculty of Mathematics and Informatics Vilnius, Lithuania
[email protected] http://www.mif.vu.lt/~cyras/
On legal reasoning, legal informatics and visualization
ERASMUS Teaching Assignment, University of Salzburg, February 2013
Transforming the problem of infeasibility of
achieving several goals into a weighing problem
1. Legal reasoning.
An example
2
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
An example problem: legal punishment
• 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 }
3
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
Final state ( pu, de, pt, re )
pr fi cs
( )
• Actions:
1. imprisonment (pr),
2. fine (fi)
3. community service (cs)
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.
4
R1: pr pu
R2: fi pu
R3: cs pu
R4: pr de
R5: pr pt
R6: pr re
R7: fi de
R8: cs de
R9: cs re
Causal rules (between actions and goals):
fine (fi) imprisonment (pr) community
service (cs) 3 actions:
protection of
society (pt) deterrence (de) punishment (pu) rehabilitation (re)
R5 R4
R8 R6 R1
R2
R9 R3
4 goals:
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
5
R1: pr pu
R2: fi pu
R3: cs pu
R4: pr de
R5: pr pt
R6: pr re
R7: fi de
R8: cs de
R9: cs re
Value (promoted, demoted) Score { pu, de, pt, re }
v(pr +) = ( {pu, de, pt} , {re} ) 3:1 (1, 1, 1, -1)
v(fi +) = ( {pu, de} , ) 2:0 (1, 1, 0, 0)
v(cs+) = ( {pu, re} , {de} ) 2:1 (1, -1, 0, 1)
pr +
fi +
cs+
A sketch of reasoning for cs
• Step 1
– pr + >1 pr – Reason: pu sways
– cs+ >2 cs – – ’’ –
6
pr +
fi +
cs+
pr – >1
cs – >2
>3
>4
winner
• Step 3: >4
– Extralogical choice for rehabilitation:
re – de >4 de
• Step 2: >3
– Extralogical choice: re is next to pu
– Hence re >3 de + pt
Arguments on imprisonment Practical syllogism, originally see Aristotle
7
1. Agent P wishes to realise goal G.
2. If P performs action A, G will be realised.
3. Therefore, P should perform A.
D pr
pr pu D pu
R1
l1 D pr
pr de D de
R4
l2 D pr
pr pt D pt
R5
l3 D pr
pr re D re
R6
l4
Individual defeat
D G
A G
–––––
D A
(Positive practical
syllogism, PPS)
Abduction
positive
(Negative practical
syllogism, NPS)
D G
A G
–––––
D A
Abduction
negative
Both PPS and NPS are defeasible.
Abduction and deduction
Abduction
• Reasoning from goals to facts
• Abductive reasoning
• Backward-chaining in Artificial Intelligence
Deduction
• Reasoning from facts to goals
• Deductive reasoning
• Forward-chaining in Artificial Intelligence
8
Facts
R1 R2
Goal
R3
R4
G
A G
––––– Abduction
A
A
A G
–––––
G
modus
ponens
D G
A G
–––––
D A
Abduction
positive
Accruals on imprisonment.
Then defeat
9
D pr
pr pu D pu
R1
l1 D pr
pr de D de
R4
l2 D pr
pr pt D pt
R5
l3 D pr
pr re D re
R6
l4
D pr Accrual : pr + D pr pr −
Defeat >1 :
pu sways
Accrual on fining
10
D fi
fi pu D pu
R2
l5 D fi
fi de D de
R7
l6
D fi Accrual : fi +
Accruals on community service
11
D cs
cs de D de
R8
l9
Defeat >2 :
pu sways
D cs cs −
D cs
cs pu D pu
R3
l7 D cs
cs re D re
R9
l8
D cs Accrual : cs +
The attack graph
• Step 1. >1 and >2 proved above.
• Step 2. >3
12
D pr
pr pu D pu
R1
l1 D pr
pr de D de
R4
l2 D pr
pr pt D pt
R5
l3
D pr pr + Defeat: >3
D cs
cs pu D pu
R3
l7 D cs
cs re D re
R9
l8
D cs cs +
Value (promoted, demoted)
v(pr +) = ( {pu, de, pt} , {re} ) 3:1
v(cs+) = ( {pu, re} , {de} ) 2:1
re >3 de + pt
More precisely, re – de >3 de + pt – re
• Extralogical choice: re is next to pu
• Thus we (judge) make pu the second most important goal
• Other choices, e.g. pro fine fi +
are possible
pr +
fi +
cs+
pr – >1
cs – >2
>3
>4
winner
Step 3. Defeat >4
• We chose promoting rehabilitation re while demoting deterrence de over promoting deterrence de.
• Formally, re – de >4 de .
13
pr +
fi +
cs+
pr – >1
cs – >2
>3
>4
winner
Value (promoted, demoted)
v(fi +) = ( {pu, de} , ) 2:0
v(cs+) = ( {pu, re} , {de} ) 2:1
re – de >4 de
Justification: given that we must punish, we choose to do so in a way which will aid rehabilitation.
Defeat: >4
D cs
cs pu D pu
R3
l7 D cs
cs re D re
R9
l8
D cs cs +
D fi
fi pu D pu
R2
l5 D fi
fi de D de
R7
l6
D fi fi +
Conclusions
• Limitations
– of the this formalisation
• See Bench-Capon & Prakken
– of artificial intelligence in law
• formalising choice
• algorithmically undecidable problems
• NP-problems
– of mathematics
14
2. On the problem of
infeasibility of achieving
several goals
15
Example: drink vs. roll
• You have one coin and want to buy two
items:
– drink
– roll of bread
• Buy one item for a coin.
You cannot buy both items.
– “One cannot eat it and keep it”
16
Initial state
Final state ( drink, roll )
chooseDrink chooseRoll
( )
Impossible
Choice is extralogical • Which action to choose:
to buy a drink or alternatively a roll?
– (drink, 0) > (0, roll) or alternatively
(drink, 0) < (0, roll) ?
– No ordering of vectors, i.e.
neither (1,0) > (0,1)
nor (1,0) < (0,1)
• Formal logic cannot help here to make a choice – Extralogical reasons have to be involved
– 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
• 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 wins.
17
0 1
roll
drink
Impossible
(1,1)
0
1
(1,0)
0 2
roll
drink
(2,1)
0
1
winner
The landscape metaphor in means-
ends analysis
18
“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
bringsAboutTheEnd
positive 1
negative 0
1
true
0
false
mright = 1,1
mweak = 0,1
mwrong = 1,0
3 means mwrong, mweak and mright
