1. Introduction to Argumentation
Theory
Federico Cerutti
DL Lunch
Tuesday 18
th
December, 2012
c 2012 Federico Cerutti <f.cerutti@abdn.ac.uk>

2. Non Monotonic Logics
Classical logic is monotonic: whenever a sentence A is a logical
consequence of a set of sentences T (T A), then A is also a
consequence of an arbitrary superset of T;
Commonsense reasoning is dierent: we often draw plausible
conclusions based on the assumption that the world is normal
and as expected;
This is farm from being irrational: it is the best we can do in
situations in which we have only incomplete information;
It can happen that our normality assumptions turn out to be
wrong: in this case we may have to revise our conclusions.
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 2

3. Answer Set Programming: the Tweety Example
f l i e s (X) :− bird (X) , not abnormal (X) .
abnormal (X) :− penguin (X) .
bird (X) :− penguin (X) .
bird ( tweety ) .
penguin ( tux ) .
Resulting Answer Sets:
{penguin ( tux ) , f l i e s ( tweety ) , bird ( tweety ) ,
bird ( tux ) , abnormal ( tux )}
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 3

4. Answer Set Programming: the Nixon Diamond
Usually, Quakers are pacist
Usually, Republicans are not pacist
Richard Nixon is both a Quaker and a Republican
quaker ( nixon ) .
republican ( nixon ) .
p a c i f i s t (X) :− quaker (X) , not −p a c i f i s t (X) .
−p a c i f i s t (X) :− republican (X) , not p a c i f i s t (X) .
Resulting Answer Sets:
{quaker ( nixon ) , republican ( nixon ) , p a c i f i s t ( nixon )}
{quaker ( nixon ) , republican ( nixon ) , −p a c i f i s t ( nixon )}
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5. Argumentation: an Informal Example (Courtesy
of M. Giacomin)
The reason
The conclusion
We are justified in believing that we should run LHC 
We should run Large Hadron Collider
LHC allows us to
understand the Laws
of the Universe
Understanding
the Laws of the
Universe is good
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 5

6. Argumentation: an Informal Example (Courtesy
of M. Giacomin)
The reason
The conclusion
We are justified in believing that we should run LHC 
We should run Large Hadron Collider
LHC allows us to
understand the Laws
of the Universe
Understanding
the Laws of the
Universe is good
In Argumentation (and in real life as well):
- reasons are not necessary “conclusive”
(they don’t logically entail conclusions)
- arguments and conclusions can be “retracted”
in front of new information, i.e. counterarguments
BUT
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 5

7. Argumentation: an Informal Example (Courtesy
of M. Giacomin)
We should run Large Hadron Collider
LHC allows us to
understand the Laws
of the Universe
Understanding
the Laws of the
Universe is good
We should not run LHC
LHC will generate
black holes
destroying Earth
Destroying
Earth
is bad
Now we are justified in believing that we should not run LHC 
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 5

8. Argumentation: an Informal Example (Courtesy
of M. Giacomin)
We should run Large Hadron Collider
LHC allows us to
understand the Laws
of the Universe
Understanding
the Laws of the
Universe is good
We should not run LHC
LHC will generate
black holes
destroying Earth
Destroying
Earth
is bad
Black holes will
not destroy Earth
Black holes will
evaporate because
of Hawking radiation
Now we are again justified in believing that we should run LHC 
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 5

9. Argumentation: an Informal Example (Courtesy
of M. Giacomin)
We should run Large Hadron Collider
LHC allows us to
understand the Laws
of the Universe
Understanding
the Laws of the
Universe is good
We should not run LHC
LHC will generate
black holes
destroying Earth
Destroying
Earth
is bad
Black holes will
not destroy Earth
Black holes will
evaporate because
of Hawking radiation
Hawking radiation
does not exist
Dr Azzeccagarbugli
says so
Now we are again justified in believing that we should not run LHC 
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 5

10. Argumentation: an Informal Example (Courtesy
of M. Giacomin)
We should run Large Hadron Collider
LHC allows us to
understand the Laws
of the Universe
Understanding
the Laws of the
Universe is good
We should not run LHC
LHC will generate
black holes
destroying Earth
Destroying
Earth
is bad
Black holes will
not destroy Earth
Black holes will
evaporate because
of Hawking radiation
Hawking radiation
does not exist
Dr Azzeccagarbugli
says so
Dr Azzeccagarbugli
is not expert in physics
He is a lawyer
Now we are again justified
in believing that we should
run LHC 
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 5

11. What is Argumentation?
[Prakken, 2011] Argumentation is the process of supporting
claims with grounds and defending them against attack.
[van Eemeren et al., 1996] Argumentation is a verbal and social
activity of reason aimed at increasing (or decreasing) the
acceptability of a controversial standpoint for the listener or
reader, by putting forward a constellation of propositions
intended to justify (or refute) the standpoint before a rational
judge.
A framework for practical and uncertain reasoning able to cope
with partial and inconsistent knowledge.
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12. The Elements of an Argumentation System
[Prakken and Vreeswijk, 2001]
1 The denition of an argument (possibly including an underlying
logical language + a notion of logical consequence)
2 The notion of attack and defeat (successful attack) between
arguments;
3 An argumentation semantics selecting acceptable (justied)
arguments
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13. Classical logic and argumentation
[Besnard and Hunter, 2008]
Let ∆ be set of formulae in classical logic.
An argument is a pair Φ, α such that:
1 Φ ⊥
2 Φ α
3 Φ is a minimal subset of ∆ satisfying 2.
A defeater for Φ, α is an argument Ψ, β such that
β ¬(φ1 ∧ . . . ∧ φn) for some {φ1, . . . , φn} ⊆ Φ
A rebuttal for Φ, α is an argument Ψ, β where β ¬α
An undercut for Φ, α is an argument Ψ, ¬(φ1 ∧ . . . ∧ φn) where
{φ1, . . . , φn} ⊆ Ψ
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14. Arguments and Attacks: Argument Schemes
[Walton, 1996]
An argument scheme is a reasoning pattern giving us the
presumption in favour of its conclusion.
A critical question is a question that can be posed by an opponent
in order to undermine the validity of the stated argument.
There are several argument schemes in literature.
Expert testimony
Premise 1: E is expert on D
Premise 2: E says P
Premise 3: P is in D
Conclusion: P is the case
Critical questions:
1 Is E biased?
2 Is P consistent with what other experts say?
3 Is P consistent with known evidence?
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15. Abstract argumentation: Nixon Diamond
An abstract argumentation framework AF is a tuple A, R , where A
is a set of argument (whose origin and structure is not specied), and
R ⊆ A × A is a set of attack (or defeat) relations.
AFN = AN , RN , where AN = {A1, A2}, RN = { A1, A2 , A2, A1 },
and
A1: since Nixon is a quaker, then he is also a pacist;
A2: since Nixon is a republican, he is not a pacist.
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16. Nixon: from Prolog to Arguments (the Dung's
way)
AR = {(K, k)|∃C ∈ Gp : head(C) = k, and body(C) =
K} ∪ {({¬k}, ¬k)|k is a ground atom}
(K, h) attacks (K , h ) i h∗ ∈ K
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17. Nixon: from Prolog to Arguments (the Dung's
way)
A1 ({¬quaker(nixon)}, ¬quaker(nixon))
A2 ({¬republican(nixon)}, republican(nixon))
A3 ({}, quaker(nixon))
A4 ({}, republican(nixon))
A5 ({pacifist(nixon), quaker(nixon)}, pacifist(nixon))
A6 ({¬pacifist(nixon), republican(nixon)}, ¬pacifist(nixon))
A7 ({¬pacifist(nixon)}, ¬pacifist(nixon))
A5 A6
A3 A1 A4 A2
A7
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 11

18. Tweety: from Prolog to Arguments (the Dung's
way)
A1 ({¬penguin(tux)}, ¬penguin(tux))
A2 ({}, penguin(tux))
A3 ({¬bird(tux)}, ¬bird(tux))
A4 ({¬bird(tweety)}, ¬bird(tweety))
A5 ({}, bird(tweety))
A6 ({¬penguin(tweety)}, ¬penguin(tweety))
A7 ({¬abnormal(tux)}, ¬abnormal(tux))
A8 ({¬abnormal(tweety)}, ¬abnormal(tweety))
A9 ({¬flies(tux)}, ¬flies(tux))
A10 ({¬flies(tweety)}, ¬flies(tweety))
A11 ({penguin(tweety)}, bird(tweety))
A12 ({penguin(tweety)}, abnormal(tweety))
A13 ({bird(tweety), ¬abnormal(tweety)}, flies(tweety))
A14 ({penguin(tux)}, bird(tux))
A15 ({penguin(tux)}, abnormal(tux))
A16 ({bird(tux), ¬abnormal(tux)}, flies(tux))
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 12

19. Tweety: from Prolog to Arguments (the Dung's
way)
A5 A4
A13A6 A11
A12 A8
A10
A2 A1
A14 A3
A15 A7
A16 A9
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 12

20. Argumentation Semantics (Courtesy of M.
Giacomin)
Argument evaluation: given an argumentation framework, determine the
justication state (defeat status) of arguments. In particular, what
argument emerge undefeated from the conict, i.e. are acceptable?
• Specification of a method for argument evaluation, or of
criteria to determine, given a set of arguments, their “defeat status”
Argumentation Framework
Semantics
Defeat status
Defeat status
Undefeated
Defeated
Provisionally Defeated
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 13

21. Extension-based Semantics (Courtesy of M.
Giacomin)
Set of extensions ℰS(AF)Argumentation framework AF
Semantics S
Defeat/Justification Status
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 14

22. Complete Semantics (Courtesy of M. Giacomin)
Acceptability
α acceptable w.r.t. (“defended by”) S
• all attackers of α are attacked by S
Admissible set S
• conflict-free
• every element acceptable w.r.t. S
(defends all of its elements)
α
S
IF
also includes all
acceptable elements
w.r.t. itself
Complete
extension
Complete semantics
All traditional semantics
select complete extensions
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23. Labelling Approach [Caminada and Gabbay, 2009]
(Courtesy of M. Caminada)
argument labels: in, out, undec
An argument is in
iff all its defeaters are out
An argument is out
iff it has a defeater that is in
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24. Labelling Approach: Examples (Courtesy of M.
Caminada)
A
B
C
A B
D
C
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25. Labelling Approach: Examples (Courtesy of M.
Caminada)
A
B
C
A B
D
C
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26. Labelling Approach: Examples (Courtesy of M.
Caminada)
A
B
C
A B
D
C
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27. Labelling Approach: Examples (Courtesy of M.
Caminada)
A
B
C
A B
D
C
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28. Labelling Approach: Examples (Courtesy of M.
Caminada)
A
B
C
A B
D
C
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29. Labelling Approach: Examples (Courtesy of M.
Caminada)
A
B
C
A B
D
C
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30. Labelling Approach: Examples (Courtesy of M.
Caminada)
A
B
C
A B
D
C
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31. Labelling Approach: Examples (Courtesy of M.
Caminada)
A
B
C
A B
D
C
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32. Labelling Approach: Examples (Courtesy of M.
Caminada)
D
BA
A B
C
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33. Labelling Approach: Examples (Courtesy of M.
Caminada)
D
BA
A B
C
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34. Labelling Approach: Examples (Courtesy of M.
Caminada)
D
BA
A B
C
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35. Labelling Approach: Examples (Courtesy of M.
Caminada)
D
BA
A B
C
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 18

36. Labelling Approach: Examples (Courtesy of M.
Caminada)
D
BA
A B
C
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 18

37. Labelling Approach: Examples (Courtesy of M.
Caminada)
D
BA
A B
C
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 18

38. Labelling Approach: Examples (Courtesy of M.
Caminada)
D
BA
A B
C
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 18

39. Semantics and Labelling (Courtesy of M.
Caminada)
restriction on Dung-style
compl. labeling semantics
no restrictions complete semantics
empty undec stable semantics
maximal in preferred semantics
maximal out preferred semantics
maximal undec grounded semantics
minimal in grounded semantics
minimal out grounded semantics
minimal undec semi-stable semantics
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 19

40. Nixon: Labellings
A5 A6
A3 A1 A4 A2
A7
quaker(nixon), republican(nixon), pacifist(nixon)
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 20

41. Nixon: Labellings
A5 A6
A3 A1 A4 A2
A7
quaker(nixon), republican(nixon), ¬pacifist(nixon)
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 20

42. Nixon: Labellings
A5 A6
A3 A1 A4 A2
A7
quaker(nixon), republican(nixon)
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 20

43. Tweety: Labellings
A5 A4
A13A6 A11
A12 A8
A10
A2 A1
A14 A3
A15 A7
A16 A9
bird(tweety), ¬penguin(tweety), ¬abnormal(tweety), flies(tweety)
penguin(tux), bird(tux), abnormal(tux), ¬flies(tux)
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 21

44. Conclusions
Argumentation as a way for encompassing common sense
reasoning
Argumentation as a way for encompassing non-monotonic
reasoning
Argumentation as a way for encompassing defeasible reasoning
Fundamental elements:
Structure of arguments;
Structure of attacks (notion of defeat);
Way for determining the outcome of the reasoning
(semantics/labellings).
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 22

45. References I
[Alechina, 2011] Alechina, N. (2011).
Knowledge representation and reasoning 2011-2012: G53KRR course slides.
http://www.cs.nott.ac.uk/~nza/G53KRR/.
[Berners-Lee and Fischetti, 2000] Berners-Lee, T. and Fischetti, M. (2000).
Weaving the Web.
HarperBusiness.
[Besnard and Hunter, 2008] Besnard, P. and Hunter, A. (2008).
Elements of Argumentation.
The MIT Press.
[Black et al., 2009] Black, E., Hunter, A., and Pan., J. Z. (2009).
An Argument-based Approach to Using Multiple Ontologies.
In the Proc. of the 3rd International Conference on Scalable Uncertainty Management (SUM
2009).
[Bondarenko et al., 1993] Bondarenko, A., Toni, F., and Kowalski, R. (1993).
An assumption-based framework for non-monotonic reasoning.
In Nerode, A. and Pereira, L., editors, Proceedings Second International Workshop on Logic
Programming and Non-Monotonic Reasoning. MIT Press.
[Brachman and Levesque, 2004a] Brachman, R. and Levesque, H. (2004a).
Knowledge Representation and Reasoning.
Elsevier.
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 23

46. References II
[Brachman and Levesque, 2004b] Brachman, R. and Levesque, H. (2004b).
Knowledge representation and reasoning: Overhead slides.
http://www.cs.toronto.edu/~hector/PublicKRSlides.pdf.
[Caminada and Gabbay, 2009] Caminada, M. and Gabbay, D. M. (2009).
A logical account of formal argumentation.
Studia Logica, 93(2-3):109145.
[Dung, 1995] Dung, P. M. (1995).
On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic
programming, and n-person games.
Articial Intelligence, 77(2):321357.
[Flouris et al., 2006] Flouris, G., Huang, Z., Pan, J. Z., Plexousakis, D., and Wache, H. (2006).
Inconsistencies, negations and changes in ontologies.
In 21st AAAI Conf., pages 12951300.
[Gaertner and Toni, 2008] Gaertner, D. and Toni, F. (2008).
Hybrid argumentation and its properties.
In Proceedings of COMMA 2008.
[Herman, 2011] Herman, I. (2011).
Introduction to the semantic web.
http://www.w3.org/2011/Talks/0606-SemTech-Tut-IH/Talk.pdf.
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 24

47. References III
[Horrocks and Sattler, 2002] Horrocks, I. and Sattler, U. (2002).
Description logics - basics, applications, and more (tutorial at ecai-2002).
http://www.cs.man.ac.uk/~horrocks/Slides/ecai-handout.pdf.
[McCune, 2010] McCune, W. (20052010).
Prover9 and mace4.
http://www.cs.unm.edu/~mccune/prover9/.
[Prakken, 2011] Prakken, H. (2011).
An overview of formal models of argumentation and their application in philosophy.
Studies in Logic, 4:6586.
[Prakken and Vreeswijk, 2001] Prakken, H. and Vreeswijk, G. A. W. (2001).
Logics for defeasible argumentation.
In Gabbay, D. M. and Guenthner, F., editors, Handbook of Philosophical Logic, Second Edition.
Kluwer Academic Publishers, Dordrecht.
[Reiter, 1980] Reiter, R. (1980).
A logic for default reasoning.
Articial Intelligence, 13(1-2):81 132.
[van Eemeren et al., 1996] van Eemeren, F. H., Grootendorst, R., Johnson, R. H., Plantin, C., Walton,
D. N., Willard, C. A., Woods, J., and Zarefsky, D. (1996).
Fundamentals of Argumentation Theory. A Handbook of Historical Backgrounds and
Contemporary Developments.
Lawrence Erlbaum Associates.
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 25

48. References IV
[van Harmelen et al., 2007] van Harmelen, F., van Harmelen, F., Lifschitz, V., and Porter, B. (2007).
Handbook of Knowledge Representation.
Elsevier Science, San Diego, USA.
[W3C, 2012] W3C (2012).
Rdf tutorial.
http://www.w3schools.com/rdf/default.asp.
[Walton, 1996] Walton, D. N. (1996).
Argumentation Schemes for Presumptive Reasoning.
Lawrence Erlbaum Associates.
f.cerutti@abdn.ac.uk Tuesday 18th December, 2012 26