Introduction to Argumentation
Theory
Federico Cerutti
DL Lunch
Tuesday 18th December, 2012
c© 2012 Federico Cerutti <[email protected]>
Non Monotonic Logics
Classical logic is monotonic: whenever a sentence A is a logicalconsequence of a set of sentences T (T � A), then A is also aconsequence of an arbitrary superset of T ;
Commonsense reasoning is di�erent: we often draw plausibleconclusions 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 insituations in which we have only incomplete information;
It can happen that our normality assumptions turn out to bewrong: in this case we may have to revise our conclusions.
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Answer Set Programming: the Tweety Example
f l i e s (X) :− bi rd (X) , not abnormal (X) .abnormal (X) :− penguin (X) .b i rd (X) :− penguin (X) .b i rd ( tweety ) .penguin ( tux ) .
Resulting Answer Sets:
{penguin ( tux ) , f l i e s ( tweety ) , b i rd ( tweety ) ,b i rd ( tux ) , abnormal ( tux )}
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Answer Set Programming: the Nixon Diamond
Usually, Quakers are paci�st
Usually, Republicans are not paci�st
Richard Nixon is both a Quaker and a Republican
quaker ( nixon ) .r epub l i can ( 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) :− r epub l i can (X) , not p a c i f i s t (X) .
Resulting Answer Sets:
{ quaker ( nixon ) , r epub l i can ( nixon ) , p a c i f i s t ( nixon )}{quaker ( nixon ) , r epub l i can ( nixon ) , −p a c i f i s t ( nixon )}
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Argumentation: an Informal Example (Courtesyof 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
Understandingthe Laws of the Universe is good
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Argumentation: an Informal Example (Courtesyof 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
Understandingthe 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
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Argumentation: an Informal Example (Courtesyof M. Giacomin)
We should run Large Hadron Collider
LHC allows us to understand the Laws
of the Universe
Understandingthe 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
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Argumentation: an Informal Example (Courtesyof M. Giacomin)
We should run Large Hadron Collider
LHC allows us to understand the Laws
of the Universe
Understandingthe 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
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Argumentation: an Informal Example (Courtesyof M. Giacomin)
We should run Large Hadron Collider
LHC allows us to understand the Laws
of the Universe
Understandingthe 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 radiationdoes not exist
Dr Azzeccagarbuglisays so
Now we are again justified in believing that we should not run LHC
<[email protected]> Tuesday 18th December, 2012 5
Argumentation: an Informal Example (Courtesyof M. Giacomin)
We should run Large Hadron Collider
LHC allows us to understand the Laws
of the Universe
Understandingthe 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 radiationdoes not exist
Dr Azzeccagarbuglisays so
Dr Azzeccagarbugliis not expert in physics
He is a lawyer
Now we are again justified in believing that we should
run LHC
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What is Argumentation?
[Prakken, 2011] Argumentation is the process of supportingclaims with grounds and defending them against attack.
[van Eemeren et al., 1996] Argumentation is a verbal and socialactivity of reason aimed at increasing (or decreasing) theacceptability of a controversial standpoint for the listener orreader, by putting forward a constellation of propositionsintended to justify (or refute) the standpoint before a rationaljudge.
A framework for practical and uncertain reasoning able to copewith partial and inconsistent knowledge.
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The Elements of an Argumentation System[Prakken and Vreeswijk, 2001]
1 The de�nition of an argument (possibly including an underlyinglogical language + a notion of logical consequence)
2 The notion of attack and defeat (successful attack) betweenarguments;
3 An argumentation semantics selecting acceptable (justi�ed)arguments
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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 ` α3 Φ is a minimal subset of ∆ satisfying 2.
A defeater for 〈Φ, α〉 is an argument 〈Ψ, β〉 such thatβ ` ¬(φ1 ∧ . . . ∧ φn) for some {φ1, . . . , φn} ⊆ ΦA rebuttal for 〈Φ, α〉 is an argument 〈Ψ, β〉 where β ` ¬αAn undercut for 〈Φ, α〉 is an argument 〈Ψ,¬(φ1 ∧ . . . ∧ φn)〉 where{φ1, . . . , φn} ⊆ Ψ
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Arguments and Attacks: Argument Schemes[Walton, 1996]
An argument scheme is a reasoning pattern giving us thepresumption in favour of its conclusion.
A critical question is a question that can be posed by an opponentin order to undermine the validity of the stated argument.
There are several argument schemes in literature.
Expert testimony
Premise 1: E is expert on DPremise 2: E says PPremise 3: P is in DConclusion: 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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Abstract argumentation: Nixon Diamond
An abstract argumentation framework AF is a tuple 〈A,R〉, where Ais a set of argument (whose origin and structure is not speci�ed), andR ⊆ 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 paci�st;
A2: since Nixon is a republican, he is not a paci�st.
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Nixon: from Prolog to Arguments (the Dung'sway)
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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Nixon: from Prolog to Arguments (the Dung'sway)
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
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Tweety: from Prolog to Arguments (the Dung'sway)
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)}, f lies(tweety))
A14 , ({penguin(tux)}, bird(tux))
A15 , ({penguin(tux)}, abnormal(tux))
A16 , ({bird(tux),¬abnormal(tux)}, f lies(tux))
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Tweety: from Prolog to Arguments (the Dung'sway)
A5 A4
A13A6 A11
A12 A8
A10
A2 A1
A14 A3
A15 A7
A16 A9
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Argumentation Semantics (Courtesy of M.Giacomin)Argument evaluation: given an argumentation framework, determine the
justi�cation state (defeat status) of arguments. In particular, what
argument emerge undefeated from the con�ict, 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
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Extension-based Semantics (Courtesy of M.Giacomin)
Set of extensions ℰS(AF) Argumentation framework AF
Semantics S
Defeat/Justification Status
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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 allacceptable elementsw.r.t. itself
Completeextension
Complete semantics
All traditional semanticsselect complete extensions
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Labelling Approach [Caminada and Gabbay, 2009]
(Courtesy of M. Caminada)
argument labels: in, out, undec
An argument is iniff all its defeaters are out
An argument is outiff it has a defeater that is in
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Labelling Approach: Examples (Courtesy of M.Caminada)
A
B
C
A B
D
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
A
B
C
A B
D
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
A
B
C
A B
D
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
A
B
C
A B
D
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
A
B
C
A B
D
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
A
B
C
A B
D
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
A
B
C
A B
D
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
A
B
C
A B
D
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
D
BA
A B
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
D
BA
A B
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
D
BA
A B
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
D
BA
A B
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
D
BA
A B
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
D
BA
A B
C
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Labelling Approach: Examples (Courtesy of M.Caminada)
D
BA
A B
C
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Semantics and Labelling (Courtesy of M.Caminada)
restriction on Dung-stylecompl. labeling semanticsno restrictions complete semanticsempty undec stable semanticsmaximal in preferred semanticsmaximal out preferred semanticsmaximal undec grounded semanticsminimal in grounded semanticsminimal out grounded semanticsminimal undec semi-stable semantics
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Nixon: Labellings
A5 A6
A3 A1 A4 A2
A7
quaker(nixon), republican(nixon), pacifist(nixon)
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Nixon: Labellings
A5 A6
A3 A1 A4 A2
A7
quaker(nixon), republican(nixon),¬pacifist(nixon)
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Nixon: Labellings
A5 A6
A3 A1 A4 A2
A7
quaker(nixon), republican(nixon)
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Tweety: Labellings
A5 A4
A13A6 A11
A12 A8
A10
A2 A1
A14 A3
A15 A7
A16 A9
bird(tweety),¬penguin(tweety),¬abnormal(tweety), f lies(tweety)penguin(tux), bird(tux), abnormal(tux),¬flies(tux)
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Conclusions
Argumentation as a way for encompassing common sense
reasoning
Argumentation as a way for encompassing non-monotonicreasoning
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).
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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 (SUM2009).
[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 LogicProgramming and Non-Monotonic Reasoning. MIT Press.
[Brachman and Levesque, 2004a] Brachman, R. and Levesque, H. (2004a).Knowledge Representation and Reasoning.Elsevier.
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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):109�145.
[Dung, 1995] Dung, P. M. (1995).On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logicprogramming, and n-person games.Arti�cial Intelligence, 77(2):321�357.
[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 1295�1300.
[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.
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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. (2005�2010).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, 4:65�86.
[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.Arti�cial 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 andContemporary Developments.Lawrence Erlbaum Associates.
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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.
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