Cerutti--Introduction to Argumentation (seminar @ University of Aberdeen)

Introduction to Argumentation

Theory

Federico Cerutti

DL Lunch

Tuesday 18th December, 2012

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.

<f.cerutti@abdn.ac.uk> Tuesday 18th December, 2012 2

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 )}

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 )}

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

The reason

The conclusion

We are justified in believing that we should run LHC

of the Universe

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

of the Universe

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

of the Universe

destroying Earth

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

of the Universe

destroying Earth

Hawking radiationdoes not exist

Dr Azzeccagarbuglisays so

Now we are again justified in believing that we should not run LHC

of the Universe

destroying Earth

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

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.

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

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

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?

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.

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 ′

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))

A3 A1 A4 A2

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))

Tweety: from Prolog to Arguments (the Dung'sway)

A13A6 A11

A12 A8

A14 A3

A15 A7

A16 A9

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

Undefeated

Defeated

Provisionally Defeated

Extension-based Semantics (Courtesy of M.Giacomin)

Set of extensions ℰS(AF) Argumentation framework AF

Semantics S

Defeat/Justification Status

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)

also includes allacceptable elementsw.r.t. itself

Completeextension

Complete semantics

All traditional semanticsselect complete extensions

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

Labelling Approach: Examples (Courtesy of M.Caminada)

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

Nixon: Labellings

A3 A1 A4 A2

quaker(nixon), republican(nixon), pacifist(nixon)

Nixon: Labellings

A3 A1 A4 A2

quaker(nixon), republican(nixon),¬pacifist(nixon)

Nixon: Labellings

A3 A1 A4 A2

quaker(nixon), republican(nixon)

Tweety: Labellings

A13A6 A11

A12 A8

A14 A3

A15 A7

A16 A9

bird(tweety),¬penguin(tweety),¬abnormal(tweety), f lies(tweety)penguin(tux), bird(tux), abnormal(tux),¬flies(tux)

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).

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.

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.

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.

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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