Sequent-Based Argumentation for Normative Reasoning

Sequent-Based Argumentation for NormativeReasoning

Christian Straßer Ofer Arieli

Centre for Logic and Philosophy of ScienceGhent University, Belgium

[email protected]

School of Computer Science,The Academic College of Tel-Aviv, Israel

[email protected]

July 13, 2014

Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning

Some Annoying Problems in Deontic Logic

1. deontic conflicts: OA ∧ O¬A


Some Proposals to Tackle Conflicts

The Weakening Approach

aggregation: OA,OB ⊢ O(A ∧ B)

inheritance: If A ⊢ B then OA ⊢ OB






Problem

Logics get very weak.






Problem


Solution: Strengthen non-monotonically.

e.g., via adaptive logics [Goble (2013), etc.]

assume OA ∧ O¬A to be false “as much as possible”






Problem


Solution: Strengthen non-monotonically.

e.g., via adaptive logics [Goble (2013), etc.]

assume OA ∧ O¬A to be false “as much as possible”

Problem [Goble (2013)]

O¬f ,O(f ∨ s),O¬s 0 Os



2. contrary-to-duties



3. specificity


Some Proposals to Tackle Specificity

Go Dyadic and Weaken (Rational) Monotonicity

Goble: WRM(

O(B |A) ∧ P(B ∧ C |A))

⊃ O(B |A ∧ C )

still problematic for some examples (see Goble (2004), Straßer(2010))


Some Proposals to Tackle Specificity

Go Dyadic and Weaken (Rational) Monotonicity

Goble: WRM(

O(B |A) ∧ P(B ∧ C |A))

⊃ O(B |A ∧ C )

still problematic for some examples (see Goble (2004), Straßer(2010))

General Problem

How to weaken best?

How to save the intuitive inferences that are invalidated bythe weakening?


A Change of Perspective: Enters Argumentation Theory

Premises Conclusion

backing warrant rebuttal


A Change of Perspective: Enters Argumentation Theory

A mealis served.

You are notto eat

with yourfingers.

Referenceto a bookof tablemanners

(In general),if a meal is servedyou ought not to

eat with your fingers.

. . . except . . .


Formal Argumentation: the Abstract Perspective

a

c d

b

arguments: abstract, points in a directed graph



a

c d

b

arguments: abstract, points in a directed grapharrows: argumentative attacks



aa

c dd

b


Argumentation Semantics

Idea: select sets of arguments that represent rational stances

conflict-free

defended

maximality (?)

This gives rise to different semantics.

a

d



aa

c dd

b

a

c dd

bb


Argumentation Semantics

Idea: select sets of arguments that represent rational stances

conflict-free

defended

maximality (?)

This gives rise to different semantics.

a

d d

b


Structured / Logical Argumentation

a

c d

b



a

〈Γ1, ϕ〉

c

〈Γ3,¬ϕ, ψ〉

d

〈Γ4,¬ψ, ψ′〉b

〈Γ2,¬ϕ〉

structure arguments



a

〈Γ1, ϕ〉

c

〈Γ3,¬ϕ, ψ〉

d

〈Γ4,¬ψ, ψ′〉b

〈Γ2,¬ϕ〉

structure arguments

define attacks relative to this structure

rebuttalpremise-attack (sometimes ‘undercut’)


Sequent-Based Argumentation

Arieli (CLIMA’13), A&S (A&C, subm.), A&S (COMMA’14)

a

c d

b




a

. . .

Γ1 ⇒ ϕ

c

. . .

Γ3,¬ϕ⇒ ψ

d

. . .

Γ4,¬ψ ⇒ ψ′b

. . .

Γ2 ⇒ ¬ϕarguments are C-proofs of sequents, where




a

. . .

Γ1 ⇒ ϕ

c

. . .

Γ3,¬ϕ⇒ ψ

d

. . .

Γ4,¬ψ ⇒ ψ′b

. . .

Γ2 ⇒ ¬ϕarguments are C-proofs of sequents, whereC is a sound and complete sequent-calculus




a

. . .

Γ1 ⇒ ϕ

c

. . .

Γ3,¬ϕ⇒ ψ

d

. . .

Γ4,¬ψ ⇒ ψ′b

. . .

Γ2 ⇒ ¬ϕarguments are C-proofs of sequents, whereC is a sound and complete sequent-calculusof a (Tarskian) core logic L




a

. . .

Γ1 ⇒ ϕ

c

. . .

Γ3,¬ϕ⇒ ψ

d

. . .

Γ4,¬ψ ⇒ ψ′b

. . .

Γ2 ⇒ ¬ϕarguments are C-proofs of sequents, whereC is a sound and complete sequent-calculusof a (Tarskian) core logic L

a sequent Γ ⇒ ψ is a subsequent of an argument A for Σ ⇒ φ

if it is contained in A, and Σ ⇒∧

Γ is provable


Sequent-Based Argumentation: Sequent-Elimination Rules

Scheme

Attacker Sequent Conditions Attacked Sequent

Eliminated Sequent



Scheme


Eliminated Sequent

Examples

Undercut:

Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬∧

Γ′2 Γ2, Γ′2 ⇒ ψ2

Γ2, Γ′2 6⇒ ψ2



Scheme


Eliminated Sequent

Examples

Undercut:

Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬∧

Γ′2 Γ2, Γ′2 ⇒ ψ2

Γ2, Γ′2 6⇒ ψ2

Rebuttal:

Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬ψ2 Γ2 ⇒ ψ2

Γ2 6⇒ ψ2


Sequent-Based Argumentation Frameworks

〈Arg(Σ),Attack〉




input: Σ (a set of wff in the language of the base logic L)





Arg(Σ) is the set of C-proofs of sequents of the form Γ ⇒ ψ

for some Γ ⊆ Σ






for some Γ ⊆ Σ

Attack: determined by sequent-elimination rules (such asrebuttal, undercut, etc.):An argument A ∈ Arg(Σ) for Γ ⇒ φ R-attacks an argumentB ∈ Arg(Σ) if Γ ⇒ φ R-attacks some subsequent of B .






for some Γ ⊆ Σ

Attack: determined by sequent-elimination rules (such asrebuttal, undercut, etc.):An argument A ∈ Arg(Σ) for Γ ⇒ φ R-attacks an argumentB ∈ Arg(Σ) if Γ ⇒ φ R-attacks some subsequent of B .

Defining a consequence relation:

Σ |∼∩

pr ψ [Σ |∼∪

pr ψ] if in every [some] preferred extension ofAFR(Σ) there is A ∈ Arg(Σ) where A proves Γ ⇒ ψ


Sequent-Based Argumentation for Normative Reasoning

What we need:



What we need:

1. a base logic



What we need:

1. a base logic

2. attack rules



1. a base logic: SDL



1. a base logic: SDL

Axioms: ψ ⇒ ψ

Structural Rules:

Weakening:Γ ⇒ ∆

Γ, Γ′ ⇒ ∆,∆′Cut:

Γ1 ⇒ ∆1, ψ Γ2, ψ ⇒ ∆2

Γ1, Γ2 ⇒ ∆1,∆2

Logical Rules:

[∧⇒]Γ, ψ, ϕ⇒ ∆

Γ, ψ ∧ ϕ⇒ ∆[⇒∧]

Γ ⇒ ∆, ψ Γ ⇒ ∆, ϕ

Γ ⇒ ∆, ψ ∧ ϕ

[∨⇒]Γ, ψ ⇒ ∆ Γ, ϕ⇒ ∆

Γ, ψ ∨ ϕ⇒ ∆[⇒∨]

Γ ⇒ ∆, ψ, ϕ

Γ ⇒ ∆, ψ ∨ ϕ

MP:Γ, φ, φ ⊃ ψ ⇒ ψ

[⇒⊃]Γ, ψ ⇒ ϕ,∆

Γ ⇒ ψ ⊃ ϕ,∆

[¬⇒]Γ ⇒ ∆, ψ

Γ,¬ψ ⇒ ∆[⇒¬]

Γ, ψ ⇒ ∆

Γ ⇒ ∆,¬ψ

KR:Γ ⇒ φ

OΓ ⇒ OφDR:

Γ ⇒ φ

OΓ ⇒ ¬O¬φ



2. sequent-elimination rules



2. sequent-elimination rules

CON⇒ ¬

∧Γ Γ, Γ′ ⇒ ψ

Γ, Γ′ 6⇒ ψNIC

Γ ⇒ ¬φ Γ′ ⇒ Nφ

Γ′ 6⇒ Nφ

NN′CONFUΓ, φ ⊃ Nψ ⇒ Nψ Γ ⇒ φ ψ ⇒ ¬ψ′ Γ′, φ′ ⊃ N′ψ′ ⇒ ψ′′

Γ′, φ′ ⊃ N′ψ′ 6⇒ ψ′′

NCONFU′Γ ⇒ ¬(φ ⊃ Nψ) Γ′, φ ⊃ Nψ ⇒ ψ′

Γ′, φ ⊃ Nψ 6⇒ ψ′

NCTD

Γ, φ ⊃ Nψ⇒ Nψ

Γ ⇒ φ Γ′ ⇒ φ′ φ⇒ φ′ ψ ⇒ ¬ψ′Γ′, φ′ ⊃ Oψ′

⇒ Oψ′

Γ′, φ′ ⊃ Oψ′ 6⇒ Oψ′

NN′SPECU

Γ, φ ⊃ Nψ⇒ ¬(φ′ ⊃ N′ψ′)

Γ ⇒ φ φ⇒ φ′ ψ ⇒ ¬ψ′Γ′, φ′ ⊃ N′ψ′

⇒ ψ′′

Γ′, φ′ ⊃ N′ψ′ 6⇒ ψ′′

FCONFΓ ⇒ ¬

∧n

i=1 φi ⊃ Niψi Γ′, φ1 ⊃ N1ψ1, . . . , φn ⊃ Nnψn ⇒ ψ

Γ′, φ1 ⊃ N1ψ1, . . . , φn ⊃ Nnψn 6⇒ ψ


Asparagus, anybody?


Asparagus, anybody?

being served a meal you ought not to eat with your fingers


Asparagus, anybody?


however, being served asparagus you are obliged to eat withyour fingers


Asparagus, anybody?



Σ = {m, a,m ⊃ O¬f , (m ∧ a) ⊃ Of }


Asparagus, anybody?



Σ = {m, a,m ⊃ O¬f , (m ∧ a) ⊃ Of }

Attack rule: NN′SPECU (where NN′ ∈ {OO,OP,PO}):

Γ, φ ⊃ Nψ⇒ ¬(φ′ ⊃ N′ψ′)

Γ ⇒ φ φ⇒ φ′ ψ ⇒ ¬ψ′ Γ′, φ′ ⊃ N′ψ′

⇒ ψ′′

Γ′, φ′ ⊃ N′ψ′ 6⇒ ψ′′

m, a,m ∧ a ⊃ Of⇒ ¬(m ⊃ O¬f )

m, a ⇒m ∧ a

m ∧ a

⇒ m

f ⇒¬¬f

m,m ⊃ O¬f⇒ O¬f

m,m ⊃ O¬f 6⇒ O¬f


Asparagus, anybody?

I H

G

E F

A B C D

A = m ⊃ O¬f ⇒ m ⊃ O¬fB = m ⇒ m

C = a ⇒ a

D = (m ∧ a) ⊃ Of ⇒ (m ∧ a) ⊃ OfE = m,m ⊃ O¬f ⇒ O¬fF = m, a ⇒ m ∧ a

G = m, a, (m ∧ a) ⊃ Of ⇒ OfH = m, a, (m ∧ a) ⊃ Of ⇒ ¬(m ⊃ O¬f )I = m, a,m ⊃ O¬f , (m ∧ a) ⊃ Of ⇒ O⊥


Asparagus, anybody?

I HH

GG

E FF

A BB CC DD

A = m ⊃ O¬f ⇒ m ⊃ O¬fB = m ⇒ m

C = a ⇒ a

D = (m ∧ a) ⊃ Of ⇒ (m ∧ a) ⊃ OfE = m,m ⊃ O¬f ⇒ O¬fF = m, a ⇒ m ∧ a

G = m, a, (m ∧ a) ⊃ Of ⇒ OfH = m, a, (m ∧ a) ⊃ Of ⇒ ¬(m ⊃ O¬f )I = m, a,m ⊃ O¬f , (m ∧ a) ⊃ Of ⇒ O⊥

H

G

F

B C D


. . . don’t be a rude host




however, being served asparagus you are permitted to eat withyour fingers





except, you have a guest who considers this rude





except, you have a guest who considers this rude

Σ3 = {a,m, c ,m ⊃ O¬f , (m ∧ a) ⊃ Pf , (m ∧ a ∧ c) ⊃ O¬f }




M

I H N

G L

E F

A B C D K J

D = (m ∧ a) ⊃ Pf ⇒ (m ∧ a) ⊃ PfG = m, a, (m ∧ a) ⊃ Pf ⇒ PfH = m, a, (m ∧ a) ⊃ Pf ⇒ ¬(m ⊃ O¬f )I = m, a,m ⊃ O¬f , (m ∧ a) ⊃ Pf ⇒ O⊥J = c ⇒ c

K = (m ∧ a ∧ c) ⊃ O¬f ⇒ (m ∧ a ∧ c) ⊃O¬fL = m, a, c ⇒ m ∧ a ∧ c

M = m, a, c , (m∧ a∧ c) ⊃ O¬f ⇒ ¬((m∧a) ⊃ Pf ))N = m, a, c , (m ∧ a ∧ c) ⊃ O¬f ⇒ O¬f




MM

I H NN

G LL

EE FF

AA BB CC D KK JJ

D = (m ∧ a) ⊃ Pf ⇒ (m ∧ a) ⊃ PfG = m, a, (m ∧ a) ⊃ Pf ⇒ PfH = m, a, (m ∧ a) ⊃ Pf ⇒ ¬(m ⊃ O¬f )I = m, a,m ⊃ O¬f , (m ∧ a) ⊃ Pf ⇒ O⊥J = c ⇒ c

K = (m ∧ a ∧ c) ⊃ O¬f ⇒ (m ∧ a ∧ c) ⊃O¬fL = m, a, c ⇒ m ∧ a ∧ c

M = m, a, c , (m∧ a∧ c) ⊃ O¬f ⇒ ¬((m∧a) ⊃ Pf ))N = m, a, c , (m ∧ a ∧ c) ⊃ O¬f ⇒ O¬f

M

N

L

E F

A B C K J


The snoring professor (Caminada/Prakken)

Σpro = {s, p, s ⊃ m, p ⊃ ¬Pr ,m ⊃ Pr ,m ⊃ Of }

snoring is a misbehavior (s ⊃ m),





it is allowed to remove misbehaving people from the library(m ⊃ Pr),






it is obliged not to remove a professor from the library(p ⊃ O¬r),






it is obliged not to remove a professor from the library(p ⊃ O¬r),

people who misbehave are subject to a fine (m ⊃ Of )


Snoring Prof: a problem with contra-position


p

p ⊃ O¬r

O¬r

interdef.

¬Pr

Contrapos: m ⊃ Pr

¬m


Snoring Prof: a problem with contra-position


p

p ⊃ O¬r

O¬r

interdef.

¬Pr

Contrapos: m ⊃ Pr

¬m

Caminada argues:

this violates the principle to keep conflicts as local as possible

and so deontic conditionals are not to be contrapositive.



Also the sequent p, p ⊃ O¬r ,m ⊃ Pr ⇒ ¬m is provable (from

Σpro) and conflicts with s, s ⊃ m ⇒ m.



Also the sequent p, p ⊃ O¬r ,m ⊃ Pr ⇒ ¬m is provable (from

Σpro) and conflicts with s, s ⊃ m ⇒ m.

Solution: use an attack rule like FCONF:

Γ ⇒ ¬∧

n

i=1 φi ⊃ Niψi Γ′, φ1 ⊃ N1ψ1, . . . , φn ⊃ Nnψn ⇒ ψ

Γ′, φ1 ⊃ N1ψ1, . . . , φn ⊃ Nnψn 6⇒ ψ

since

s, s ⊃ m, p ⇒ ¬((m ⊃ Pr) ∧ (p ⊃ O¬r))

eliminates

p, p ⊃ O¬r ,m ⊃ Pr ⇒ ¬m



s1 = p, p ⊃ O¬r ⇒ O¬rs2 = s, s ⊃ m,m ⊃ Pr ⇒ Prs3 = s, s ⊃ m ⇒ m

s4 = p, p ⊃ O¬r ,m ⊃ Pr ⇒ ¬ms5 = s, s ⊃ m,m ⊃ Of ⇒ Ofs6 = p, p ⊃ O¬r ,m ⊃ Pr , s, s ⊃ m ⇒ O⊥s7 = s, s ⊃ m, p ⇒ ¬((m ⊃ Pr) ∧ (p ⊃ O¬r))s8 = p, p ⊃ O¬r ⇒ O(¬r ∨ x)

A5 A6 A7

A1 A2 A4

A8 A3





A5A5 A6 A7A7

A1A1 A2 A4

A8A8 A3A3

A5 A6 A7

A1 A2 A4

A8 A3

A5 A7

A1

A8 A3





A5A5 A6 A7A7

A1A1 A2 A4

A8A8 A3A3

A5A5 A6 A7A7

A1 A2A2 A4

A8 A3A3

A5 A7

A1

A8 A3

A5 A7

A2

A3


The Gentle Murderer: Σ = {k ,O¬k , k ⊃ O(k ∧ g)}

NN′CONFΓ ⇒ Nψ ψ ⇒ ¬ψ′ Γ′ ⇒ N′ψ′

Γ′ 6⇒ N′ψ′

CON⇒ ¬

∧

Γ Γ, Γ′ ⇒ ψ

Γ, Γ′ 6⇒ ψ

A F

D E

B C

A = ⇒ ¬(k ∧ (⊤ ⊃ O¬k) ∧ (k ⊃ O(k ∧ g)))B = k ⇒ k

C = k ⊃ O(k ∧ g) ⇒ k ⊃ O(k ∧ g)D = O¬k ⇒ O¬kE = k , k ⊃ O(k ∧ g) ⇒ O(k ∧ g)F = k ,O¬k , k ⊃ O(k ∧ g) ⇒ O⊥


Relation to Input/Output Logic

Let out(ΣF ,ΣO) = {ψ | (φ, ψ) ∈ ΣO , ΣF ⊢CPL φ}.




Definition

φ ∈ out2(ΣF ,ΣO) iff φ ∈ CnCPL (out(Ξ,ΣO)) for all CPL-maximalconsistent extensions Ξ of ΣF . In the degenerated case in whichΣF is CPL-inconsistent, we define out2(ΣF ,ΣO) to beCnCPL ({ψ | (ψ′, ψ) ∈ ΣO}).




Definition


Definition

We consider the following sets:

ΓO ∈ maxfamily(ΣF ,ΣO) iff out2(ΣF , ΓO) is CPL-consistentand for all (ψ, φ) ∈ ΣO \ ΓO , out2(ΣF , ΓO ∪ {(ψ, φ)}) is notCPL-consistent.




Definition


Definition



ψ ∈ out∪2 (ΣF ,ΣO) iff ψ ∈

⋃

ΓO∈maxfamily(ΣF ,ΣO)out2(ΣF , ΓO).




Definition


Definition



ψ ∈ out∪2 (ΣF ,ΣO) iff ψ ∈

⋃


ψ ∈ out∩2 (ΣF ,ΣO) iff ψ ∈

⋂




OCONFU′ Γ ⇒ ¬(φ ⊃ Oψ) Γ′, φ ⊃ Oψ ⇒ ψ′

Γ′, φ ⊃ Oψ 6⇒ ψ′

Let Σ∗O= {ψ ⊃ Oφ | (ψ, φ) ∈ ΣO}.

Theorem

If ΣF is CPL-consistent, then the set of all the preferred extensions

of AFOCONFU′(ΣF ∪ Σ∗O) is

{

Arg(ΣF ∪ Γ∗O) | ΓO ∈ maxfamily(ΣF ,ΣO)

}

.

Corollary

Where the only attack rule is OCONFU’, for every λ ∈ {∪,∩} it

holds that ψ ∈ outλ2(ΣF ,ΣO) iff ΣF ∪ Σ∗

O|∼λ

pr Oψ.


Ex.: ΣO = {(p1, q1 ∧ q2), (p2,¬q1 ∧ q2)}, ΣF = {p1, p2}

I/O-logic

maxfamily(ΣF ,ΣO) = {{(p1, q1 ∧ q2)}, {(p2,¬q1 ∧ q2)}}

q2 ∈ out2(ΣF , {(p1, q1 ∧ q2)}) ∩ out2(ΣF , {(p2,¬q1 ∧ q2)})

thus: q2 ∈ out∩2 (ΣF ,ΣO)


Ex.: ΣO = {(p1, q1 ∧ q2), (p2,¬q1 ∧ q2)}, ΣF = {p1, p2}

I/O-logic

maxfamily(ΣF ,ΣO) = {{(p1, q1 ∧ q2)}, {(p2,¬q1 ∧ q2)}}

q2 ∈ out2(ΣF , {(p1, q1 ∧ q2)}) ∩ out2(ΣF , {(p2,¬q1 ∧ q2)})

thus: q2 ∈ out∩2 (ΣF ,ΣO)

AFOCONFU′(ΣF ,Σ∗O): two preferred extensions

one with e.g. arguments with top sequentsp1, p1 ⊃ O(q1 ∧ q2) ⇒ ¬(p2 ⊃ O(¬q1 ∧ q2)),p1, p1 ⊃ O(q1 ∧ q2) ⇒ O(q1 ∧ q2), andp1, p1 ⊃ O(q1 ∧ q2) ⇒ Oq2,

another one with e.g. arguments with top sequentsp2, p2 ⊃ O(¬q1 ∧ q2) ⇒ ¬(p1 ⊃ O(q1 ∧ q2)),p2, p2 ⊃ O(¬q1 ∧ q2) ⇒ O(¬q1 ∧ q2), andp2, p2 ⊃ O(¬q1 ∧ q2) ⇒ Oq2

thus, ΣF ∪ Σ∗O|∼∩

prOq2.


Wrapping up

other approaches (Gabbay (CLIMA’12), Oren et. al(AAMAS’08)): use bipolar nets


Wrapping up


corrective approach with core logic SDL


Wrapping up



some open tasks


Wrapping up



some open tasks

priorities among rules:

p1, p1 ⊃ O(q1 ∧ q2)⇒ O(q1 ∧ q2)

p2, p2 ⊃ O(¬q1 ∧ q2)⇒ O(¬q1 ∧ q2)

r1, r1 ⊃ O¬s ⇒ O¬s r1, r2, (r1 ∧ r2) ⊃ Os ⇒ Os


Wrapping up



some open tasks


p1, p1 ⊃ O(q1 ∧ q2)⇒ O(q1 ∧ q2)

p2, p2 ⊃ O(¬q1 ∧ q2)⇒ O(¬q1 ∧ q2)

r1, r1 ⊃ O¬s ⇒ O¬s r1, r2, (r1 ∧ r2) ⊃ Os ⇒ Os

express other deontic principles such as lex posterior derogat

legi priori


Wrapping up



some open tasks


p1, p1 ⊃ O(q1 ∧ q2)⇒ O(q1 ∧ q2)

p2, p2 ⊃ O(¬q1 ∧ q2)⇒ O(¬q1 ∧ q2)

r1, r1 ⊃ O¬s ⇒ O¬s r1, r2, (r1 ∧ r2) ⊃ Os ⇒ Os

express other deontic principles such as lex posterior derogat

legi priori

what about other I/O-logics?


Thanks! . . . and sorry for keeping you away from lunch.


Why MP?

Why

Γ, φ, φ ⊃ ψ ⇒ ψ

instead ofΓ ⇒ ψ,∆ Γ, ϕ⇒ ∆

Γ, ψ ⊃ ϕ⇒ ϕ?

Isolate and Attack.

m ⇒ mm ⇒ m,O(¬f ∨ k)

O¬f ⇒ O¬fm,O¬f ⇒ O¬f

...O¬f ⇒ O(¬f ∨ k)

m,O¬f ⇒ O(¬f ∨ k)

m,m ⊃ O¬f ⇒ O(¬f ∨ k)


Sequent-Based Argumentation for Normative Reasoning

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