commonsense reasoning and argumentation 14/15 hc 8 structured argumentation (1) henry prakken march...
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Commonsense Reasoning and Argumentation 14/15
HC 8Structured argumentation (1)
Henry PrakkenMarch 2, 2015
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We should lower taxes
Lower taxes increase productivity
Increased productivity is good
We should not lower taxes
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Prof. P says that …
Prof. P has political ambitions
People with political ambitions are not objective
Prof. P is not objective
Increased inequality is good
Increased inequality stimulates competition
Competition is good
USA lowered taxes but productivity decreased
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Two accounts of the fallibility of
arguments Plausible Reasoning: all fallibility located in the
premises Assumption-based argumentation (Kowalski, Dung,
Toni,… Classical argumentation (Cayrol, Besnard, Hunter,
…) Defeasible reasoning: all fallibility located in the
defeasible inferences Pollock, Loui, Vreeswijk, Prakken & Sartor, …
ASPIC+ combines these accounts John Pollock
Nicholas Rescher
Robert Kowalski
Tony Hunter
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Aspic+ framework: overview
Argument structure: Directed acyclic graphs where
Nodes are wff of a logical language L Links are applications of inference rules
Rs = Strict rules (1, ..., n ); or Rd= Defeasible rules (1, ..., n )
Reasoning starts from a knowledge base K L Defeat: attack on conclusion, premise or
inference, + preferences Argument acceptability based on Dung
(1995)
We should lower taxes
Lower taxes increase productivity
Increased productivity is good
We should not lower taxes
Lower taxes increase inequality
Increased inequality is bad
Attack on conclusion
We should lower taxes
Lower taxes increase productivity
Increased productivity is good
We should not lower taxes
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
USA lowered taxes but productivity decreased
Attack on premise …
We should lower taxes
Lower taxes increase productivity
Increased productivity is good
We should not lower taxes
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Prof. P says that …
USA lowered taxes but productivity decreased
… often becomes attack on intermediate conclusion
We should lower taxes
Lower taxes increase productivity
Increased productivity is good
We should not lower taxes
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Prof. P says that …
Prof. P has political ambitions
People with political ambitions are not objective
Prof. P is not objective
USA lowered taxes but productivity decreased
Attack on inference
We should lower taxes
Lower taxes increase productivity
Increased productivity is good
We should not lower taxes
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Prof. P says that …
Prof. P has political ambitions
People with political ambitions are not objective
Prof. P is not objective
USA lowered taxes but productivity decreased
We should lower taxes
Lower taxes increase productivity
Increased productivity is good
We should not lower taxes
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Prof. P says that …
Prof. P has political ambitions
People with political ambitions are not objective
Prof. P is not objective
Increased inequality is good
Increased inequality stimulates competition
Competition is good
USA lowered taxes but productivity decreased Indirect
defence
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Argumentation systems (with symmetric negation) An argumentation system is a triple AS =
(L,R,n) where: L is a logical language with negation (¬) R = Rs Rd is a set of strict and defeasible inference
rules n: Rd L is a naming convention for defeasible rules
Notation: - = ¬ if does not start with a negation - = if is of the form ¬
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Knowledge bases A knowledge base in AS = (L,R,n) is a
set K L where K is a partition Kn Kp with: Kn = necessary premises Kp = ordinary premises
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Argumentation theories An argumentation theory is a pair AT
= (AS, K) where AS is an argumentation system and K a knowledge base in AS.
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Structure of arguments
An argument A on the basis of an argumentation theory is:
if K with Prem(A) = {}, Conc(A) = , Sub(A) = {}, DefRules(A) =
A1, ..., An if there is a strict inference rule Conc(A1), ..., Conc(An)
Prem(A) = Prem(A1) ... Prem(An) Conc(A) = Sub(A) = Sub(A1) ... Sub(An) {A} DefRules(A) = DefRules(A1) ... DefRules(An)
A1, ..., An if there is a defeasible inference rule Conc(A1), ..., Conc(An)
Prem(A) = Prem(A1) ... Prem(An) Conc(A) = Sub(A) = Sub(A1) ... Sub(An) {A} DefRules(A) = DefRules(A1) ... DefRules(An) {A1, ..., An }
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Rs: Rd:
p,q s p tu,v w s,r,t v
Kn = {q} Kp = {p,r,u}
w
v u
s r t
p q p
p
pnp
p
u, v w Rs
p, q s Rs
s,r,t v Rd
p t Rd
A1 = p A5 = A1 t
A2 = q A6 = A1,A2 s
A3 = r A7 = A5,A3,A6 v
A4 = u A8 = A7,A4 w
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Types of arguments An argument A is:
Strict if DefRules(A) = Defeasible if not strict Firm if Prem(A) Kn Plausible if not firm
S |- means there is a strict argument A s.t.
Conc(A) = Prem(A) S
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Rs: Rd:
p,q s p tu,v w s,r,t v
w
v u
s r t
p q p
p
pnp
p
A1 = p A5 = A1 t
A2 = q A6 = A1,A2 s
A3 = r A7 = A5,A3,A6 v
A4 = u A8 = A7,A4 w
Kn = {q} Kp = {p,r,u}
An argument A is:- Strict if DefRules(A) = - Defeasible if not strict- Firm if Prem(A) Kn - Plausible if not firm
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Example
R: d1: p q d2: s t d3: t ¬d1 d4: u v d5: v,x ¬t d6: s ¬p s1: p,q r s2: v ¬sKn = {p}, Kp = {s,u,x}
n(p q ) = d1
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Attack A undermines B (on ) if
Conc(A) = - for some Prem(B )/ Kn; A rebuts B (on B’ ) if
Conc(A) = -Conc(B’ ) for some B’ Sub(B) with a defeasible top rule
A undercuts B (on B’ ) if Conc(A) = -n(r ) ’for some B’ Sub(B ) with
defeasible top rule r
A attacks B iff A undermines or rebuts or undercuts B.
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Rs: Rd:
p,q s p tu,v w s,r,t v
w
v u
s r t
p q p
p
pnp
p
A1 = p A5 = A1 t
A2 = q A6 = A1,A2 s
A3 = r A7 = A5,A3,A6 v
A4 = u A8 = A7,A4 w
Kn = {q} Kp = {p,r,u}
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Structured argumentation frameworks
Let AT = (AS,K) be an argumentation theory A structured argumentation framework (SAF)
defined by AT is a triple (Args,C, a) where Args = {A | A is a finite argument on the basis of K in
AS } C is the attack relation on Args a is an ordering on Args (A <a B iff A a B and not B
a A)
A c-SAF is a SAF in which all arguments have indirectly consistent premises (to be defined later)
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Defeat A undermines B (on ) if
Conc(A) = - for some Prem(B )/ Kn;
A rebuts B (on B’ ) if Conc(A) = -Conc(B’ ) for some B’ Sub(B ) with a defeasible top
rule A undercuts B (on B’ ) if
Conc(A) = -n(r) ’for some B’ Sub(B ) with defeasible top rule r
A defeats B iff for some B’ A undermines B on B’ = and not A < ; or A rebuts B on B’ and not A < B’ ; or A undercuts B on B’
Direct vs. subargument attack/defeatPreference-dependent vs. preference-independent attacks
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Example cont’d
R: d1: p q d2: s t d3: t ¬d1 d4: u v d5: v,x ¬t d6: s ¬p s1: p,q r s2: v ¬sKn = {p}, Kp = {s,u,x}
n(p q ) = d1
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Abstract argumentation frameworks corresponding to
SAFs
An abstract argumentation framework corresponding to a SAF = (Args,C, ) is a pair (Args,D) where D is the defeat relation on Args defined by C and .
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The ultimate status of conclusions
With grounded semantics: A is justified if A g.e. A is overruled if A g.e. and A is defeated by g.e. A is defensible otherwise
With preferred semantics: A is justified if A p.e for all p.e. A is defensible if A p.e. for some but not all p.e. A is overruled otherwise (?)
In all semantics: is justified if is the conclusion of some justified argument (Alternative: if all extensions contain an argument for ) is defensible if is not justified and is the conclusion of
some defensible argument is overruled if is not justified or defensible and there
exists an overruled argument for
We should lower taxes
Lower taxes increase productivity
Increased productivity is good
We should not lower taxes
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Prof. P says that …
Prof. P has political ambitions
People with political ambitions are not objective
Prof. P is not objective
Increased inequality is good
Increased inequality stimulates competition
Competition is good
USA lowered taxes but productivity decreased
C
A B
E
D