Vytautas ČYRAS Vilnius University

Faculty of Mathematics and Informatics Vilnius, Lithuania

Vytautas.Cyras@mif.vu.lt 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

Thank you

Vytautas.Cyras@mif.vu.lt Vytautas.Cyras@mif.vu.lt 19