Sequent-Based Argumentation for NormativeReasoning
Christian Straßer Ofer Arieli
Centre for Logic and Philosophy of ScienceGhent University, Belgium
School of Computer Science,The Academic College of Tel-Aviv, Israel
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Some Proposals to Tackle Conflicts
The Weakening Approach
aggregation: OA,OB ⊢ O(A ∧ B)
inheritance: If A ⊢ B then OA ⊢ OB
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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.
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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.
Solution: Strengthen non-monotonically.
e.g., via adaptive logics [Goble (2013), etc.]
assume OA ∧ O¬A to be false “as much as possible”
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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.
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Some Annoying Problems in Deontic Logic
2. contrary-to-duties
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Some Annoying Problems in Deontic Logic
3. specificity
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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))
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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?
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
A Change of Perspective: Enters Argumentation Theory
Premises Conclusion
backing warrant rebuttal
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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 . . .
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Formal Argumentation: the Abstract Perspective
a
c d
b
arguments: abstract, points in a directed graph
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Formal Argumentation: the Abstract Perspective
a
c d
b
arguments: abstract, points in a directed grapharrows: argumentative attacks
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Formal Argumentation: the Abstract Perspective
aa
c dd
b
arguments: abstract, points in a directed grapharrows: argumentative attacks
Argumentation Semantics
Idea: select sets of arguments that represent rational stances
conflict-free
defended
maximality (?)
This gives rise to different semantics.
a
d
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Formal Argumentation: the Abstract Perspective
aa
c dd
b
a
c dd
bb
arguments: abstract, points in a directed grapharrows: argumentative attacks
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Structured / Logical Argumentation
a
c d
b
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Structured / Logical Argumentation
a
〈Γ1, ϕ〉
c
〈Γ3,¬ϕ, ψ〉
d
〈Γ4,¬ψ, ψ′〉b
〈Γ2,¬ϕ〉
structure arguments
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Structured / Logical Argumentation
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’)
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation
Arieli (CLIMA’13), A&S (A&C, subm.), A&S (COMMA’14)
a
c d
b
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation
Arieli (CLIMA’13), A&S (A&C, subm.), A&S (COMMA’14)
a
. . .
Γ1 ⇒ ϕ
c
. . .
Γ3,¬ϕ⇒ ψ
d
. . .
Γ4,¬ψ ⇒ ψ′b
. . .
Γ2 ⇒ ¬ϕarguments are C-proofs of sequents, where
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation
Arieli (CLIMA’13), A&S (A&C, subm.), A&S (COMMA’14)
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation
Arieli (CLIMA’13), A&S (A&C, subm.), A&S (COMMA’14)
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation
Arieli (CLIMA’13), A&S (A&C, subm.), A&S (COMMA’14)
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation: Sequent-Elimination Rules
Scheme
Attacker Sequent Conditions Attacked Sequent
Eliminated Sequent
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation: Sequent-Elimination Rules
Scheme
Attacker Sequent Conditions Attacked Sequent
Eliminated Sequent
Examples
Undercut:
Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬∧
Γ′2 Γ2, Γ′2 ⇒ ψ2
Γ2, Γ′2 6⇒ ψ2
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation: Sequent-Elimination Rules
Scheme
Attacker Sequent Conditions Attacked Sequent
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation Frameworks
〈Arg(Σ),Attack〉
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation Frameworks
〈Arg(Σ),Attack〉
input: Σ (a set of wff in the language of the base logic L)
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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
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 .
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation for Normative Reasoning
What we need:
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation for Normative Reasoning
What we need:
1. a base logic
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation for Normative Reasoning
What we need:
1. a base logic
2. attack rules
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation for Normative Reasoning
1. a base logic: SDL
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation for Normative Reasoning
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation for Normative Reasoning
2. sequent-elimination rules
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Sequent-Based Argumentation for Normative Reasoning
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Asparagus, anybody?
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Asparagus, anybody?
being served a meal you ought not to eat with your fingers
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Asparagus, anybody?
being served a meal you ought not to eat with your fingers
however, being served asparagus you are obliged to eat withyour fingers
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Asparagus, anybody?
being served a meal you ought not to eat with your fingers
however, being served asparagus you are obliged to eat withyour fingers
Σ = {m, a,m ⊃ O¬f , (m ∧ a) ⊃ Of }
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Asparagus, anybody?
being served a meal you ought not to eat with your fingers
however, being served asparagus you are obliged to eat withyour fingers
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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⊥
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
. . . don’t be a rude host
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
. . . don’t be a rude host
being served a meal you ought not to eat with your fingers
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
. . . don’t be a rude host
being served a meal you ought not to eat with your fingers
however, being served asparagus you are permitted to eat withyour fingers
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
. . . don’t be a rude host
being served a meal you ought not to eat with your fingers
however, being served asparagus you are permitted to eat withyour fingers
except, you have a guest who considers this rude
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
. . . don’t be a rude host
being served a meal you ought not to eat with your fingers
however, being served asparagus you are permitted to eat withyour fingers
except, you have a guest who considers this rude
Σ3 = {a,m, c ,m ⊃ O¬f , (m ∧ a) ⊃ Pf , (m ∧ a ∧ c) ⊃ O¬f }
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
. . . don’t be a rude host
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
. . . don’t be a rude host
Σ3 = {a,m, c ,m ⊃ O¬f , (m ∧ a) ⊃ Pf , (m ∧ a ∧ c) ⊃ 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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
The snoring professor (Caminada/Prakken)
Σpro = {s, p, s ⊃ m, p ⊃ ¬Pr ,m ⊃ Pr ,m ⊃ Of }
snoring is a misbehavior (s ⊃ m),
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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),
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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),
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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),
people who misbehave are subject to a fine (m ⊃ Of )
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Snoring Prof: a problem with contra-position
Σpro = {s, p, s ⊃ m, p ⊃ ¬Pr ,m ⊃ Pr ,m ⊃ Of }
p
p ⊃ O¬r
O¬r
interdef.
¬Pr
Contrapos: m ⊃ Pr
¬m
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Snoring Prof: a problem with contra-position
Σpro = {s, p, s ⊃ m, p ⊃ ¬Pr ,m ⊃ Pr ,m ⊃ Of }
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.
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
The snoring professor (Caminada/Prakken)
Also the sequent p, p ⊃ O¬r ,m ⊃ Pr ⇒ ¬m is provable (from
Σpro) and conflicts with s, s ⊃ m ⇒ m.
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
The snoring professor (Caminada/Prakken)
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
The snoring professor (Caminada/Prakken)
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
The snoring professor (Caminada/Prakken)
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)
A5A5 A6 A7A7
A1A1 A2 A4
A8A8 A3A3
A5 A6 A7
A1 A2 A4
A8 A3
A5 A7
A1
A8 A3
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
The snoring professor (Caminada/Prakken)
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)
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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⊥
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Relation to Input/Output Logic
Let out(ΣF ,ΣO) = {ψ | (φ, ψ) ∈ ΣO , ΣF ⊢CPL φ}.
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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}).
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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
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.
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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
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.
ψ ∈ out∪2 (ΣF ,ΣO) iff ψ ∈
⋃
ΓO∈maxfamily(ΣF ,ΣO)out2(ΣF , ΓO).
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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
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.
ψ ∈ out∪2 (ΣF ,ΣO) iff ψ ∈
⋃
ΓO∈maxfamily(ΣF ,ΣO)out2(ΣF , ΓO).
ψ ∈ out∩2 (ΣF ,ΣO) iff ψ ∈
⋂
ΓO∈maxfamily(ΣF ,ΣO)out2(ΣF , ΓO).
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Relation to Input/Output Logic
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.
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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)
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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.
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Wrapping up
other approaches (Gabbay (CLIMA’12), Oren et. al(AAMAS’08)): use bipolar nets
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Wrapping up
other approaches (Gabbay (CLIMA’12), Oren et. al(AAMAS’08)): use bipolar nets
corrective approach with core logic SDL
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Wrapping up
other approaches (Gabbay (CLIMA’12), Oren et. al(AAMAS’08)): use bipolar nets
corrective approach with core logic SDL
some open tasks
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Wrapping up
other approaches (Gabbay (CLIMA’12), Oren et. al(AAMAS’08)): use bipolar nets
corrective approach with core logic SDL
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
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Wrapping up
other approaches (Gabbay (CLIMA’12), Oren et. al(AAMAS’08)): use bipolar nets
corrective approach with core logic SDL
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
express other deontic principles such as lex posterior derogat
legi priori
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Wrapping up
other approaches (Gabbay (CLIMA’12), Oren et. al(AAMAS’08)): use bipolar nets
corrective approach with core logic SDL
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
express other deontic principles such as lex posterior derogat
legi priori
what about other I/O-logics?
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
Thanks! . . . and sorry for keeping you away from lunch.
Christian Straßer, Ofer Arieli Sequent-Based Argumentation for Normative Reasoning
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