### On the Acceptability Semantics of Argumentation

On the Semantics of Argumentation
1
Antonis Kakas
Francesca Toni
Paolo Mancarella
Department of Computer Science
University of Cyprus
Department of Computing
Imperial College
Dipartimento di Informatica
Universita di Pisa
20 April, 2012
London Argumentation Forum
Contents
2
 Part
1: Acceptability Semantics for Abstract
Argumentation
 Generalizing
old work on the Argumentation
Based Acceptability Semantics for Logic
Programming
 Part
2: Argumentation Logic
 Recent
work on the application of Acceptability
Semantics to reformulate (Propositional) Logic in
terms of Argumentation
Acceptability Semantics for
Abstract Argumentation
3

Abstract Argumentation: <Args, Attack>
 Args:
a set of arguments
 Attack: a binary relation on Args
 Attack: the argument “a” attacks the argument “b”
 A attacks B iff  a  A and b  B s.t. (a,b)  Attack, for
 (a,b)
any A,B  P(Args)

Acceptability semantics is defined via a relative
Acceptability Relation between (sets of) arguments:
 Acc(,0):
Given 0 the set  can be accepted.
Acceptability Semantics
Informal Motivation
4

Acceptability Relation: Follow the “universal” intuition:
An argument (or a set of arguments) can be accepted iff
all its counter-arguments can be rejected.

Can we formalize directly this intuition?
How are we to understand the “Rejection of Argument”?
 As “Can not be Accepted”?
 The argument can play a role in rejecting its counter-arguments


Hence Acceptance is a RELATIVE notion.
An argument (or a set of arguments) is acceptable iff it
renders all its counter-arguments non-acceptable.
Acceptability Semantics
Informal Motivation
5
An argument (or a set of arguments), , is acceptable iff all its
counter-arguments, A, are rendered non-acceptable.

How do we understand “non-acceptable” or more generally
“non-acceptable relative to ”?


This is an approximation of the negation of acceptable!
Negation of Acceptance: An argument (or a set of arguments) A
is non-acceptable iff there exists a set of arguments D that
attacks A such that D is acceptable (relative to A).
Acceptability Semantics
Definition
DEFINITION A set  is acceptable relative to 0, i.e. Acc(, 0) holds, iff
(i)  is a subset of 0 or (ii) for any set A that attacks  there exists a set
D that attacks A – D defends against A - such that D is acceptable
relative to 0   A, i.e. Acc(D, 0   A) holds.
FACC: 2P(Args)xP(Args)  2P(Args)xP(Args)
(P(Args) is the power set 2Argsof Arg)
FACC(acc)(Δ,Δ’) iff Δ  Δ’, or
for any A s.t. A attacks Δ:
there exists D s.t. D attacks A
and acc(D, Δ’  Δ  A).


The operator FACC is monotonic.
Acceptability, Acc(-,-), is defined as the least fixed point of FACC
Definition of SEMANTICS: Δ is acceptable iff Acc(Δ,{}) holds.
Acceptability Semantics
Some Results
7
 Acyclic AF: Acceptability Semantics = Grounded Semantics
 In general, it captures the well known semantic notions.

Does it give anything else?

Captures semantic notions of self-defeating (set of) argument(s):
S is self-defeating iff there exists an attacking set, A, against S such that
¬Acc(A, {}) and Acc(A, S) hold.
 Hence S renders one of its attacks acceptable!


Acceptable sets do not need to defend against such selfdefeating attacking sets by counter-attacking them back.

Acceptability Semantics
8
 Example
of Self-Defeating: Odd Loops
 Elements
of (any length) odd loops are not acceptable.
 But also arguments that are attacked only by elements of
(isolated) odd loops are acceptable.
a
a1
{a} is Acceptable
a1
a
a1
a2
a3
a1
Acceptability Semantics
9
 Self-defeat
emerges implicitly as a semantic notion from
the minimal formulation of the acceptability semantics.
 C.f.
other semantics where this is explicit and syntactic.
 Self-defeating
 This
S: renders one of its attacks acceptable
is a kind of Reductio ad Absurdum Principle!
Part2: Argumentation Logic
10
 Can
we understand Reductio ad Absurdum in Logic as
a case of self-defeating under acceptability?
 Can
this help to formulate (Propositional) Logic in
terms of Argumentation?
 Originally,
logic was developed to formulated human
argumentation.
 PL
can be reformulated as a realization of abstract
argumentation under an acceptability type semantics.
 Argumentation
Logic
 Naturally extends PL for (classically) inconsistent theories.
Argumentation Logic
11
 Consider
Propositional Logic (PL) and its Natural
Deduction (ND) proof system.
 Take
out the Reductio ad Absurdum rule (¬I rule) from ND
Direct Proofs, ├MRA .
 RA will be recovered at the semantic level by Acceptability
 Only
 Argumentation
 Args:
 Att:
Framework for AL: <Args, Att>
Sets of Propositional Formulae: ∆
(Direct proofs from ∆ and the given theory, T)
A attacks ∆ : T ∪  ∪  ├MRA 
D defends A: D={¬} for some  in A
D={} when T ∪  ├MRA 
Argumentation Logic = Propositional Logic
Sketch Proof
12
 ¬Acc({},{})

 Genuine RA derivation for 
Genuine RA derivations:
[
.
[’
.
c()
.
[
.
.
¬
]
T    ¬ ├MRA 
’ is necessary for the
direct derivation of 
]
 Technical Lemma: For classically consistent theories if there
exists a RA derivation for  the there exists a Genuine RA
derivation for .
Natural Deduction (RA) as Argumentation
Example: ¬Acc({},{}) Genuine RA derivation for 
13

Θ = {¬ (θεός  θνητός), ¬ θνητός  ¬ πεθάνει, πεθάνει}
Θεός
[θεός
[¬
θνητός
¬ θνητός
¬ θνητός
θνητός
¬πεθάνει
πεθάνει
¬ (¬θνητός)
]
θνητός
θεός  θνητός
¬(θεός  θνητός)
]
¬ θεός
The argument, ¬ θνητός, that can defend against the
attack θνητός cannot do so as it is self-defeating.
Hence the argument, θεός, is not acceptable.
Natural Deduction (RA) as Argumentation
Example: ¬Acc({},{}) Genuine RA derivation for 
15

Θ = {¬ (θεός  θνητός), ¬ θνητός -> ¬ πεθάνει, πεθάνει}
[θεός
[θνητός
¬ θνητός
¬πεθάνει
πεθάνει
Θεός
θεός (copy)
θεός  θνητός
¬(Θεός  θνητός)
]
Attack ???
¬ θνητός
θνητός
]
¬ θεός
{}
Violates the Genuine property!
θεός
Argumentation Logic
Results (1)
16

T classically consistent
 AL
= PL (for the restricted language of ¬ and )
 AL
entails  iff ¬Acc({¬},{}) holds.
 Interpretation
 Both

of implication in AL differs from PL, e.g.
ab and ¬(ab) are acceptable w.r.t. to T={¬a}
AL distinguishes two forms of Inconsistency of T

Classically inconsistent but directly consistent (under├MRA )
Violation of rule of «Excluded Middle».
 For some, φ, neither φ nor ¬φ is acceptable, e.g. T = { φ  , ¬φ  }


Directly inconsistent

For some φ, T has a direct argument for φ and ¬φ, e.g. T = { φ , ¬φ }
Argumentation Logic
Results (2)
17

AL extends PL when T is (classically) inconsistent
 Directly
consistent
 AL
does not trivialize
 AL entails  iff ¬Acc({¬},{}) and Acc({},{}) hold.
 AL isolates out the non-relevant use of Reductio ad Absurdum
 Example: Logical Paradoxes (T = { φ  , ¬φ  } )
 Directly
 Use
inconsistent
“Belief Revision Type” approach
1. Close T under direct consequence: C(T),
 2. Maximally directly consistent subsets of C(T).

Example of Argumentation Logic

“A barber shaves anyone that does not shave himself”
¬
ShavesHimself(Person) ShavedByBarber(Person)
 ShavesHimself(Person)  ¬ ShavedByBarber(Person)

Self-reference: When Person = barber
 ShavesHimself(barber)
 ¬ ShavedByBarber(barber)  ¬ ShavesHimself(barber)
 ShavedByBarber(barber)
Example – Classical Logic


¬ SH(P)  SB(P)
SB(b)  SH(b)
SH(P)  ¬ SB(P)
¬ SB(b)  ¬ SH(b)
SB(b) |- SH(b) |- ¬ SB(b)
 ¬ SB(b) |- ¬ SH(b) |- SB(b)
i.e. SB(b) |- 
i.e. ¬ SB(b)|- 


Problem arises due to the excluded middle law
SB(P) or ¬ SB(P) , for any person P, even for P=barber.
 This makes the theory inconsistent and therefore non meaningful
(even for any other person than the barber).


Problem arises as SB(b) must take a truth value (in model
theory).
Example – Argumentation Logic


¬ SH(P)  SB(P)
SB(b)  SH(b)



¬ ACC(SB(b))
¬ ACC(¬ SB(b))
SB(b) is a non-acceptable argument
¬ SB(b) is a non-acceptable argument
Law of excluded middle for SB(b)?

The law (SB(b) or ¬ SB(b)) is non-acceptable.



SH(P)  ¬ SB(P)
¬ SB(b)  ¬ SH(b)
Each one of SB(b) and ¬ SB(b) is directly inconsistent and so non-acceptable
The negation of the law, ¬ (SB(b) or ¬ SB(b)), is acceptable (?)
Gives up the law of excluded middle!

Give up (two valued) model theory?
Conclusions
21
 Acceptability
is a direct, minimal and natural formulation
of the semantics of abstract argumentation.



Let the “Formalism Tell” vs “Telling the Formalism”.


Approach is a Synthesis of Labelling and Extension based approaches
Is it “complete”?
Acceptability “tells us” (encapsulates) a Reductio ad Absurdum
Principle in argumentation.
This enables a reformulation of PL in terms of argumentation
=> Argumentation Logic (AL)
AL is a conservative extension of PL into a type of Relevance Paraconsistent Logic -- Only genuine use of Reductio ad Absurdum
 Looking for the analogue of a “model theory” for AL.
