introduction to structured argumentation · abstract argumentation: pros and cons pros an argument...

Introduction toStructured Argumentation

Anthony Hunter

Department of Computer Science, University College London, UK

September 6, 2016

1 / 134

Computational models of argument

1 Introduction

2 Framework for deductive argumentation

Definitions with examples of instancesGeneral properties

3 Defeasible reasoning

4 Meta-level argumentation

Representing argument schema

5 From natural language arguments to formalized arguments

Understanding arguments in free textHandling enthymemes

6 Conclusions and further directions

2 / 134

Computational models of argument

Abstract argumentation

Structured Argumentation

Dialogical Argumentation

Decisionmaking

Sensemaking

Persuasion NegotiationDistributed

decision making

[Besnard + Hunter 2008]

3 / 134

Abstract argumentation: Graphical representation

Graphical representations of argumentation have a long history (see for exampleWigmore, Toulmin, etc. )

A1 = Patient hashypertension so

prescribe diuretics

A2 = Patient hashypertension so pre-scribe betablockers

A3 = Patient hasemphysema whichis a contraindica-

tion for betablockers

[See Dung (AIJ 1995)]

4 / 134

Abstract argumentation: Pros and cons

Pros

An argument graph provides a simple and intuitive representation of acontroversial topic.

Various dialectical semantics by Dung, and others, provide valuableinsights into the nature of argumentation.

Various tools have been developed for analysing arguments in terms ofabstract argumentation.

Cons

Arguments in abstract argumentation are atomic.

They cannot be broken down.They do not have an internal structure.They cannot be combined.

There is no formal definition for arguments or attacks.

5 / 134

Approaches to structured argumentation

Origins of research in structured argumentation

John Pollock pioneered the formalization of arguments and counterargumentsusing logic (See Pollock 1987).

Some frameworks for structured argumentation

Deductive argumentation (Hunter, Besnard, Cayrol, Amgoud, et al)

Defeasible logic programming (Simari, et al)

Assumption-based argumentation (Toni, et al)

ASPIC+ (Prakken, et al)

See the special issue in Argument & Computation, volume 5 (1), 2014, fortutorials on each of these frameworks.

6 / 134

Overview of deductive argumentation

Generative graphsDescriptive graphs

Counterarguments

Arguments

Base logic

[Besnard + Hunter 2014]

7 / 134

Base logic

Choice of base logic

Here we focus on simple logic and classical logic, but other options includenon-monotonic logics, conditional logics, temporal logics, description logics,and paraconsistent logics.

A few definitions for base logic

Let L be a language for a logic, and let `i be the consequence relationfor that logic.

If α is an atom in L, then α is a positive literal in L and ¬α is anegative literal in L.

For a literal β, the complement of β is defined as follows:

If β is a positive literal, i.e. it is of the form α, then thecomplement of β is the negative literal ¬α,if β is a negative literal, i.e. it is of the form ¬α, then thecomplement of β is the positive literal α.

8 / 134

Arguments

Definition for deductive argument

Given a base logic `i , a deductive argument is an ordered pair 〈Φ, α〉where Φ `i α.

Φ is the support, or premises, or assumptions of the argument, and α isthe claim, or conclusion, of the argument.

For an argument A = 〈Φ, α〉, the function Support(A) returns Φ and thefunction Claim(A) returns α.

Example

〈{report(rain), report(rain)→ carry(umbrella)}, carry(umbrella)〉

〈{study(Sid, logic),¬study(Sid, logic)}, KingOfFrance(Sid)〉

9 / 134

Arguments

The consistency constraint

An argument 〈Φ, α〉 satisfies the consistency constraint when Φ is consistent.

Example

If we assume the consistency constraint, then the following are not arguments.

〈{study(Sid, logic),¬study(Sid, logic)},study(Sid, logic)↔ ¬study(Sid, logic)〉

〈{study(Sid, logic),¬study(Sid, logic)}, KingOfFrance(Sid)〉

Consistency constraint is not essential

If we assume the base logic is a paraconsistent logic (such as Belnap’s fourvalued logic), and we do not impose the consistent constraint, then thefollowing are arguments.

〈{study(Sid, logic) ∧ ¬study(Sid, logic)}, study(Sid, logic)〉

〈{study(Sid, logic) ∧ ¬study(Sid, logic)},¬study(Sid, logic)〉

10 / 134

Arguments

The minimality constraint

An argument 〈Φ, α〉 satisfies the minimality constraint when there is no Ψ ⊂ Φsuch that Ψ `i α.

Example

If we assume the minimality constraint, then the following is not an argument.

〈{report(rain), report(rain)→ carry(umbrella), happy(Sid)},carry(umbrella)〉

11 / 134

Arguments based on simple logic

Simple logic

Simple logic is based on a language of literals and simple rules where eachsimple rule is of the form α1 ∧ . . . ∧ αk → β where α1 to αk and β areliterals.

The consequence relation is modus ponens (i.e. implication elimination)as defined next.

∆ `s β iff there is an α1 ∧ · · · ∧ αn → β ∈ ∆and for each αi ∈ {α1, . . . , αn}either αi ∈ ∆ or ∆ `s αi

Example

Let ∆ = {a, b, a ∧ b → c, c → d}. Hence, ∆ `s c and ∆ `s d . However,∆ 6`s a and ∆ 6`s b.

12 / 134

Arguments based on simple logic

Simple argument

Let ∆ be a simple logic knowledgebase. For Φ ⊆ ∆, and a literal α, 〈Φ, α〉 is asimple argument iff Φ `s α and there is no proper subset Φ′ of Φ such thatΦ′ `s α.

Example

Let p1, p2, and p3 be the following formulae.

p1 = oilCompany(BP)p2 = goodPerformer(BP)p3 = oilCompany(BP) ∧ goodPerformer(BP))→ goodInvestment(BP)

Then 〈{p1, p2, p3}, goodInvestment(BP)〉 is a simple argument.

13 / 134

Arguments based on classical logic

Classical logic argument

A classical logic argument from a set of formulae ∆ is a pair 〈Φ, α〉 such that

1 Φ ⊆ ∆

2 Φ 6` ⊥3 Φ ` α4 there is no Φ′ ⊂ Φ such that Φ′ ` α.

Example

The following classical argument uses a universally quantified formula incontrapositive reasoning to obtain the following claim about number 77.

〈{∀X.multipleOfTen(X)→ even(X),¬even(77)},¬multipleOfTen(77)〉

14 / 134

Counterarguments based on simple logic

Rebut and undercut for simple logic

For simple arguments A and B, we consider the following type of simpleattack:

A is a simple undercut of B if there is a simple rule α1 ∧ · · · ∧ αn → β inSupport(B) and there is an αi ∈ {α1, . . . , αn} such that Claim(A) is thecomplement of αi

A is a simple rebut of B if Claim(A) is the complement of Claim(B)

Example

A1 = 〈{efficientMetro, efficientMetro→ useMetro}, useMetro〉A2 = 〈{strikeMetro, strikeMetro→ ¬efficientMetro},¬efficientMetro〉

A3 = 〈{govDeficit, govDeficit→ cutGovSpending}, cutGovSpending〉A4 = 〈{weakEconomy, weakEconomy→ ¬cutGovSpending},¬cutGovSpending〉

15 / 134

Counterarguments based on simple logic

Example (Defeasible reasoning)

The first argument A1 captures the general rule that if workDay holds, thenuseMetro(Sid) holds.

A1 = 〈{workDay, normal, workDay ∧ normal→ useMetro(Sid)},useMetro(Sid)〉

A2 = 〈{workAtHome(Sid), workAtHome(Sid)→ ¬normal},¬normal〉

Here we use normal as an assumption of normality for using the rule.

16 / 134

Counterarguments based on classical logic

Counterarguments for classical logic

If 〈Φ, α〉 and 〈Ψ, β〉 are arguments, then

〈Φ, α〉 rebuts 〈Ψ, β〉 iff α ≡ ¬β〈Φ, α〉 undercuts 〈Ψ, β〉 iff α ≡ ¬ ∧Ψ′ for some Ψ′ ⊆ Ψ

Direct undercut

A direct undercut for an argument 〈Φ, α〉 is an argument of the form 〈Ψ,¬φi 〉where φi ∈ Φ.

Example (Using classical logic)

〈{β, β → α}, α〉 rebuts 〈{γ, γ → ¬α},¬α〉

〈{γ, γ → ¬β},¬(β ∧ (β → α))〉 undercuts 〈{β, β → α}, α〉

〈{δ → ¬β},¬β〉 is a direct undercut for 〈{α, β}, α ∧ β〉

17 / 134


A rebut denotes a disagreement with the claim, whereas an undercut denotes adisagreement with the support (i.e. a disagreement of the explanation orjustification).

Example

a = “garlic is horrible”

b = “this dish contains garlic”

c = “this dish is horrible”

〈{a, b, a ∧ b → c}, c〉〈{¬c},¬c〉

〈{¬a},¬a〉〈{¬a ∧ c},¬a〉

18 / 134


Example (Integrity constraint as a counterargument)

Essentially, the attack says that the flight cannot be both a low cost flight anda luxury flight.

〈{lowCostFly, luxuryFly, lowCostFly ∧ luxuryFly→ goodFly}, goodFly〉

〈{¬lowCostFly ∨ ¬luxuryFly},¬lowCostFly ∨ ¬luxuryFly〉

19 / 134


Example (Using first-order predicate formulae)

Because Tweety is a bird, and birds fly, there is a bird that flies. But, there is abird that doesn’t fly, and so it is not the case that all birds fly.

〈{bird(tweety),∀X .bird(X)→ fly(X)}, ∃X .bird(X ) ∧ fly(X)〉

〈{∃X .bird(X ) ∧ ¬fly(X )},¬∀X .bird(X)→ fly(X)〉

Example (Also using first-order predicate formulae)

Some students know nothing. But, they all know their own name

〈{∃X .∀Y .¬knows(X, Y)}, ∃X .∀Y .¬knows(X, Y)〉

〈{∀X .knows(X, name(X))}, ∀X , ∃Y .knows(X, Y)〉

20 / 134


Some kinds of classical attack

Let A and B be two classical arguments.

A is a classical defeater of B if Claim(A) ` ¬∧

Support(B).

A is a classical direct defeater of B if

∃φ ∈ Support(B) s.t. Claim(A) ` ¬φ

A is a classical undercut of B if

∃Ψ ⊆ Support(B) s.t. Claim(A) ≡ ¬∧

Ψ

A is a classical direct undercut of B if

∃φ ∈ Support(B) s.t. Claim(A) ≡ ¬φ

A is a classical canonical undercut of B if Claim(A) ≡ ¬∧

Support(B).

A is a classical rebuttal of B if Claim(A) ≡ ¬Claim(B).

A is a classical defeating rebuttal of B if Claim(A) ` ¬Claim(B).

21 / 134


Example (Attack relations)

〈{a ∨ b, c}, (a ∨ b) ∧ c〉 is a classical defeater of 〈{¬a,¬b},¬a ∧ ¬b〉

〈{a ∨ b, c}, (a ∨ b) ∧ c〉 is a classical direct defeater of 〈{¬a ∧ ¬b},¬a ∧ ¬b〉

〈{¬a ∧ ¬b},¬(a ∧ b)〉 is a classical undercut of 〈{a, b, c}, a ∧ b ∧ c〉

〈{¬a ∧ ¬b},¬a〉 is a classical direct undercut of 〈{a, b, c}, a ∧ b ∧ c〉

〈{¬a ∧ ¬b},¬(a ∧ b ∧ c)〉 is a classical canonical undercut of 〈{a, b, c}, a ∧ b ∧ c〉

〈{a, a → b}, b ∨ c〉 is a classical rebuttal of 〈{¬a ∧ ¬b,¬c},¬(b ∨ c)〉

〈{a, a → b}, b〉 is a classical defeating rebuttal of 〈{¬a ∧ ¬b,¬c},¬(b ∨ c)〉

22 / 134


Hierarchy of arrack relations

Defeater

Direct defeat Undercut Direct rebut

Direct undercut Canonical undercut Rebut

An arrow from D1 to D2 indicates that D1 ⊆ D2.

23 / 134

Argument graphs

Approaches to constructing logical argument graphs

Descriptive graphs Here we assume that the structure of the argumentgraph is given, and the task is to identify the premises and claim of eachargument. Therefore the input is an abstract argument graph, and theoutput is an instantiated argument graph.

Generative graphs Here we assume that we start with a knowledgebase(i.e. a set of logical formula), and the task is to generate the argumentsand counterarguments (and hence the attacks between arguments).Therefore, the input is a knowledgebase, and the output is an instantiatedargument graph.

24 / 134

Argument graphs

Abstract graph

Instantiated graph

AttacksArguments

Knowledgebase

desc

riptive

generative

Approaches to constructing logical argument graphs

25 / 134

Argument graphs

Example (Abstract graph and descriptive graph)

The flight is low cost and luxury, therefore it is a good flight

A flight cannot be both low cost and luxury

A1 = 〈{lowCostFly, luxuryFly, lowCostFly ∧ luxuryFly→ goodFly}, goodFly〉

A2 = 〈{¬(lowCostFly ∧ luxuryFly)},¬lowCostFly ∨ ¬luxuryFly〉

26 / 134

Argument graphs

Example (Abstract graph)

A1 = Patient hashypertension so

prescribe diuretics

A2 = Patient hashypertension so pre-scribe betablockers

A3 = Patient hasemphysema whichis a contraindica-

tion for betablockers

27 / 134

Argument graphs

Example (Descriptive graph using classical logic with integrity constraint)

bp(high)

ok(diuretic)

bp(high) ∧ ok(diuretic)

→ give(diuretic)

¬ok(diuretic) ∨ ¬ok(betablocker)

give(diuretic) ∧ ¬ok(betablocker)

bp(high)

ok(betablocker)

bp(high) ∧ ok(betablocker)

→ give(betablocker)

¬ok(diuretic) ∨ ¬ok(betablocker)

give(betablocker) ∧ ¬ok(diuretic)

symptom(emphysema),symptom(emphysema)→ ¬ok(betablocker)

¬ok(betablocker)

28 / 134

Argument graphs

Example (Generative graph using simple logic)

Let ∆ = {a, b, c, a ∧ c → ¬a, b → ¬c, a ∧ c → ¬b}.

〈{a, c, a ∧ c → ¬a},¬a〉

〈{a, c, a ∧ c → ¬b},¬b〉〈{b, b → ¬c},¬c〉

29 / 134

Argument graphs

Example (Generative graph using classical logic)

Consider ∆ = {a, b, a→ ¬a, b → ¬a, a→ ¬b}, let the arguments be thosethat involves one or more rules.

〈{b, b → ¬c},¬c〉

〈{c, c → ¬a},¬a〉

〈{a, a→ ¬b},¬b〉

〈{c, b → ¬c},¬b〉

〈{a, c → ¬a},¬c〉

〈{b, a→ ¬b},¬a〉

30 / 134

Summary of deductive argumentation

Specification of a deductive argumentation system

Base logic Specification of language and of consequence relation (prooftheoretic) or entailment relation (semantic) for definingarguments.

Arguments Specification of conditions for premises and claims ofarguments.

Attack Specification of claim of attacker and its relationship withpremises or claim of attackee.

Generative graph Specification of which arguments and attacks (that can begenerated from knowledgebase) are to appear in the graph.

Most applications also need meta-level information about formulae (e.g. pref-erences, probability assignments, ethical value judgments, etc.) which areharnessed in the specification of a deductive argumentation system.

31 / 134

Comparison with ASPIC+ and ABA

Controversial statement:The difference betweenABA, ASPIC+, anddeductive argumentationis not substantial.

All provide structure to arguments with some/all premises and claim.

All implicitly/explicitly use a base logic.

All have notions of counterargument.

All can instantiate abstract argument graphs.

All can capture defeasible reasoning.

All can be implemented.

32 / 134

Comparison with ASPIC+ and ABA

There are some differences

ASPIC+ has proof structure in argument.

ASPIC+ use names to block application of rules.

ASPIC+ distinguishes between strict and defeasible information.

ASPIC+ and ABA do not put all the premises in the argument.

ASPIC+ and ABA use rules for inference rules and for object-levelinformation.

ABA is tightly coupled with abstract argumentation.

ASPIC+ and ABA use simple logic (or similar) for most examples.

ASPIC+ is perhaps the most fully developed for rule-based applications.

However, most of what can be done in practice in one approach can be donein the other approaches.

33 / 134

Properties

Overview of properties of deductive argumentation systems

Deductive argumentation offers numerous choices for the specification ofbase logic, argument, attack, and graph construction.

Need to compare and contrast these choices

Need to ensure choices are well-understood and well-behaved.

We will consider the following kinds of property

Attack properties concern classifications of attack relationsExtension properties concern properties of the formulae in thesupport of arguments in extensionsStructural properties concern the types of graph that can beconstructed by given different choices for argument and attackrelation.

34 / 134

Attack properties

Some desirable properties of attack relations

Name Property

D0 if A ≡ A′,B ≡ B ′ then D(A,B) = D(A′,B ′)

D1 if D(A,B) = > then {Claim(A)} ∪ Support(B) ` ⊥D2 if D(A,B) = > and Claim(C) ≡ Claim(A) then D(C ,B) = >D2i if D(A,B) = > and Claim(C) ` Claim(A) then D(C ,B) = >D3 if D(A,B) = > and support(B) = support(C) then D(A,C) = >D3i if D(A,B) = > and support(B) ⊆ support(C) then D(A,C) = >D4 if Arcs(G) = ∅ then MinIncon(∆) = ∅

Postulates for an attack relation D. We denote an attack relation from A to Bas holding when D(A,B) = >.

[Gorogiannis + Hunter 2011]

35 / 134

Attack properties

Postulates satisfied by attack functions

DD DDD DU DDU DCU DR DDR

D0 Y Y Y Y Y Y YD1 Y Y Y Y Y Y YD2 Y Y Y Y Y Y YD2i Y Y N N N N YD3 Y Y Y Y Y N ND3i Y Y Y Y N N ND4 Y Y Y Y Y Y Y

where DD is defeater, DDD is direct defeater, DU is undercut, DDU is directundercut, DCU is cannonical undercut, DR is rebuttal, and DDR is defeatingrebuttal.

36 / 134

Attack properties

Conclusions on attack properties

Since classical logic offers a variety of different options for defining acounterargument (i.e. an attack relation), it is helpful to characterize theoptions in terms of postulates.

Furthermore, these postulates can be used or adapted for classifyingattack relations for a variety of base logics.

With a few further postulates we can get characterization results for theattack relation (i.e. for each attack relation there is a combination ofposulates that is equivalent to the attack relation)

Further postulates on attack include

Martinez, Garcia + Simari (IJCAI’07) has a postulate on the attackrelation similar to D3’Amgoud+Besnard (SUM’2009) has a postulate on the attackrelation similar to D1Amgoud+Besnard (SUM’2009) have a number of further postulatesand properties for classical logic instantiations

37 / 134

Extension properties

Overview of extension properties

Given an instantiated argument graph, we can look at the deductive

arguments in the extension.

Are the claims of the arguments in the extension consistent?Are the premises of the arguments in the extension consistent?What is the relationships between the conflicts in the knowledgeand the conflicts between the arguments?

We investigate these kinds of issue in terms of postulates.

[Gorogiannis + Hunter 2011]

38 / 134


Sceptical versus credulous extensions

Let Extensionsσ(G) be the set of extensions obtained according toσ ∈ {pr, gr, st, id}

Scepticalσ(G) =⋂

S∈Extensionσ(G)

S

Credulousσ(G) =⋃

S∈Extensionσ(G)

S

Example

A1 A2

A3

A4Extensionpr(G) = {{A1,A4}, {A2,A4}}

Scepticalpr(G) = {A4}

Credulouspr(G) = {A1,A2,A4}

39 / 134


Free formulae

Let Free(∆) be the set of formulae not in any minimal inconsistent subset of ∆.

Free(∆) = {α ∈ ∆ | α 6∈⋃

Γ∈MinIncon(∆)

Γ}

where

MinIncon(∆) = {Γ ⊆ ∆ | Γ ` ⊥ and for all Γ′ ⊂ Γ, Γ′ 6` ⊥}

Example

Let ∆ = {a ∨ b,¬a,¬b ∨ c,¬c, d , d → ¬b, e, e ∨ d}

MinIncon(∆) = {{a ∨ b,¬a,¬b ∨ c,¬c}{a ∨ b,¬a,¬b ∨ c, d , d → ¬b}}

Hence, Free(∆) = {e, e ∨ d}

40 / 134


Free and non-free arguments

FreeArguments(G) = {A ∈ Nodes(G) | Support(A) ⊆ Free(∆)}NonFreeArguments(G) = {A ∈ Nodes(G) | Support(A) 6⊆ Free(∆)}

Example

Let ∆ = {a ∨ b,¬a,¬b ∨ c,¬c, d , d → ¬b, e, e ∨ d}So Free(∆) = {e, e ∨ d}.An example of a free arguments is

〈{e}, e ∨ f 〉

Some examples of non-free arguments are

〈{a ∨ b,¬a}, b〉〈{¬b ∨ c,¬c},¬b〉

41 / 134


Non-free postulate

For a descriptive or generative argument graph G based on classical logicarguments, the non-free postulate is as follows, where σ ∈ {pr, gr, st, id}.

∃∆ s.t.⋃

A∈Nodes(G) Support(A) ⊆ ∆

and NonFreeArguments(G) 6= ∅and Credulousσ(G) 6= Nodes(G)

Failure means that if Support(A) ⊆ ∆, then A is credulously accepted. So forany argument that can be formed from a knowledgebase, there is a preferredextension that contains that argument.

42 / 134


Consistency postulates

For a descriptive or generative argument graph G based on classical logicarguments, the consistency postulates are defined as follows, whereσ ∈ {pr, gr, st, id}.

(CN1)⋃

A∈scepticalσ(G)

Support(A) 6` ⊥

(CN2)⋃A∈S

Support(A) 6` ⊥, for all S ∈ Extensionσ(G)

(CN1′)⋃

A∈Scepticalσ(G)

Claim(A) 6` ⊥

(CN2′)⋃A∈S

Claim(A) 6` ⊥, for all S ∈ Extensionσ(G)

43 / 134

Results on consistency postulates

For grounded and ideal semantics

Attack CN1 CN1’ CN2 CN2’

Direct undercut X X X XDirect defeat X X X XCanonical undercut X X X XRebut × × × ×

For preferred, semi-stable, stable, and complete semantics

Attack CN1 CN1’ CN2 CN2’

Direct undercut X X X XDirect defeat X X X XCanonical undercut X X × ×Rebut × × × ×

44 / 134

Examples w.r.t. consistency postulates

CN1 Postulate⋃A∈Scepticalσ(G) Support(A) 6|= ⊥

Let ∆ = {a ∧ b,¬a ∧ c}.For the reviewed semantics for rebut, the following are arguments in G.

〈{a ∧ b}, b〉〈{¬a ∧ c}, c〉

Clearly {a ∧ b,¬a ∧ c} ` ⊥.

Hence, CN1 postulate fails for rebut.

45 / 134


CN2 Postulate⋃A∈S Support(A) 6|= ⊥, for all S ∈ Extensionsσ(G)

Let ∆ = {a,¬a ∨ ¬b, b} and let D be direct undercut.

〈{a}, a∗〉〈{b,¬a ∨ ¬b},¬a〉

〈{¬a ∨ ¬b}, (¬a ∨ ¬b)∗〉〈{a, b},¬(¬a ∨ ¬b)〉

〈{a,¬a ∨ ¬b},¬b〉〈{b}, b∗〉

Any credulous extension (e.g. red nodes) satisfies CN2 and CN2’.

46 / 134


CN2 Postulate⋃A∈S Support(A) 6|= ⊥, for all S ∈ Extensionsσ(G)

Consider ∆ = {a, b,¬a ∨ ¬b} with canonical undercuts.

〈{a, b},¬(¬a∨ 6 b)〉〈{¬a ∨ ¬b},¬(a ∧ b)〉

〈{a,¬a ∨ ¬b},¬b〉〈{b},¬(a ∧ (¬a ∨ ¬b))〉

〈{b,¬a ∨ ¬b},¬a〉〈{a},¬(b ∧ (¬a ∨ ¬b))〉

Any credulous extension (e.g. red nodes) violates CN2 and CN2’.

47 / 134


Consider ∆ = {a, b,¬a ∨ ¬b} with undercuts.

〈{a, b},¬(¬a ∨ ¬b)〉

〈{a,¬a ∨ ¬b)},¬b〉

〈{b,¬a ∨ ¬b)},¬a〉

〈{¬a ∨ ¬b},¬(a ∧ b)〉

〈{b},¬(a ∧ (¬a ∨ ¬b))〉

〈{a},¬(b ∧ (¬a ∨ ¬b))〉

〈{b}, b〉

〈{a}, a〉

The stable extension (red nodes) violates CN2 and CN2’.

48 / 134


Related work

Cayrol (IJCAI’95) was the first to systematically consider instantiation ofDung’s proposal with classical logic arguments.

Cayrol (IJCAI’95) has a result similar to satisfaction of CR with directundercuts.

Amgoud+Besnard (SUM’2009) have a number of further postulates andproperties for classical logic instantiations.

Caminada+Amgoud (AIJ 2007) postulates for defeasible logicinstantiations.

Amgoud + Vesic (IJAR 2014) have investigate relationships betweenextensions and maximally consistent subsets of the knowledgebase.

49 / 134


Conclusions on extension properties

Postulates on extensions can be used for any logic-based argumentationsystem

For classical logic,

results for the non-free postulate suggest that

sceptical semantics are too scepticalcredulous semantics are too credulous

results for CN2 & CN2’ suggest that exhaustive constructingargument graphs G is too simplistic

need to better understand how meta-level information is usedto select arguments

50 / 134


More on shortcomings of exhaustive construction of argument graph

There is also a need to be aposite for an audience

Consider an article in a current affairs magazine: Only a small subset ofall possible arguments, that either the writer or the reader could constructfrom their own knowledgebases, are used.

A journalist regards some arguments as having higher impact and/ormore believable for the intended audience and/or ... than others, and somakes a selection.

This need for appositeness is reflected in law, medicine, science, politics,advertising, management, ......, and just ordinary every-day life.

This need for appositeness creates challenges for developing nomative forms ofargumentation for decision making with appropriate extension properties.

51 / 134

Structural properties: Motivation

Utility assumption for Dung’s abstract argumentation

For any directed graph, a user can produce a textual description for eachargument that would be an appropriate reflection of the nodes and arcs in thegraph. So every graph has a use in modelling real-world scenarios ofargumentation.

A1

A2 A3 B1 B2 B3

Can a logical-based argumentation system produce all graphs?

For each abstract argument graph G , can we specify a knowledgebase K suchthat the system induces an instantiated argument graph G ′ such that G ′ isisomorphic to G .

52 / 134

Structural properties

Deductive argumentation system with exhaustive generation of graphs

A deductive argumentation system is a tuple (Argumentsi ,Attacksi ) whereArgumentsi (∆) (respectively Attacksi (∆)) gives the arguments (respectivelyattacks) that can be formed from ∆.


A system (Argumentsi ,Attacksi ) constructively covers Φ iff for allG ∈ X , there is a ∆ and there is an A ∈ Argumentsi (∆), such that(Argumentsi (∆),Attacksi (∆)) = G .

A system (Argumentsi ,Attacksi ) is constructively covered by Φ iff for all∆ and for all A ∈ Argumentsi (∆), if (Argumentsi (∆),Attacksi (∆)) = Gthen G ∈ Φ.

A system (Argumentsi ,Attacksi ) is constructively complete for Φ iff(Argumentsi ,Attacksi ) constructively covers Φ and is constructivelycovered by Φ

[See Hunter + Woltran 2013]

53 / 134


Argumentation based on simple logic

Let (Argumentsi ,Attacksi ) be the argument system based on simple logicarguments and simple logic undercuts.

(Argumentsi ,Attacksi ) is constructively complete for all graphs.

Showing constructive completeness for simple logic

For each node x in the graph, create a rule where the atom αn+1 is aunique identifier for this rule

α1 ∧ . . . ∧ αn → ¬αn+1

For each attack on the node x , there is a condition αk in the above rulewhere αk is the unique identified for the attacker.

The knowledgebase ∆ is the set of these rules plus the set of atomsformed from the conditions in these rules.

Each argument contains exactly one of these rules.

54 / 134


From ∆ = {a, b, c, a ∧ c → ¬a, a ∧ c → ¬b, b → ¬c}, we can construct thefollowing exhaustive argument graph which is isomorphic to the above directedgraph.

〈{a, c, a ∧ c → ¬a},¬a〉〈{a, c, a ∧ c → ¬b},¬b〉

〈{b, b → ¬c},¬c〉

55 / 134


There is no knowledgebase such that we can instantiate the following abstractgraph with classical logic arguments and undercuts.

From ∆ = {a,¬a}, we can only produce a more complicated graph below.

A1 = 〈{a}, a〉 A2 = 〈{¬a},¬a〉

A3 = 〈{¬a}, . . .〉 A4 = 〈{a}, . . .〉

56 / 134


Preference-based argumentation frameworks

PAF introduces a preference relation over arguments that in effect causesan attack to be ignored when the attacked argument is preferred over theattacker.

From this, we need to define a defeat relation D as follows, and then(A,D) is used as the argument graph, instead of (A,R)

D = {(Ai ,Aj) ∈ R | (Aj ,Ai ) 6∈�}

Example

For the left graph, if A2 � A1 and A2 � A3, then (A,D) is the right graph.

A1 A2 A3 A1 A2 A3

[Amgoud and Cayrol 2002]

57 / 134


Example (Using preference-based argumentation)

p1 is the fact that the patient has glaucoma.

p2 is the assumption that it is ok to give PGA treatment.

p3 is the assumption that it is ok to give BB treatment.

p6 is an integrity constraint that ensures that only one treatment is given.

p1 = glaucoma p4 = glaucoma ∧ ok(PGA)→ give(PGA)p2 = ok(PGA) p5 = glaucoma ∧ ok(BB)→ give(BB)p3 = ok(BB) p6 = ¬ok(PGA) ∨ ¬ok(BB)

There are numerous arguments that can be constructed from this set offormulae such as the following.

A1 = 〈{p1, p2, p4, p6}, give(PGA) ∧ ¬ok(BB)〉 A5 = 〈{p1, p2, p4}, give(PGA)〉A2 = 〈{p1, p3, p5, p6}, give(BB) ∧ ¬ok(PGA)〉 A6 = 〈{p1, p3, p5}, give(BB)〉A3 = 〈{p2, p3}, ok(PGA) ∧ ok(BB)〉 A7 = 〈{p2, p6},¬ok(BB)〉A4 = 〈{p6},¬ok(PGA) ∨ ¬ok(BB)〉 A8 = 〈{p3, p6},¬ok(PGA)〉

58 / 134


Example (Using preference-based argumentation (continued))

Let Argumentsc(∆) be the set of all classical arguments that can beconstructed from {p1, . . . , p6}.Let the preference relation � be such that Ai � A1 and Ai � A2 for all isuch that i 6= 1 and i 6= 2.

Following is the subgraph involving A1 and A2 plus any arguments thatdefeat them.

A1 = 〈{p1, p2, p4, p6}, give(PGA) ∧ ¬ok(BB)〉

A2 = 〈{p1, p3, p5, p6}, give(BB) ∧ ¬ok(PGA)〉

By taking this focal graph, we have ignored arguments such as A3 to A8 whichdo not affect the dialectical status of A1 or A2 given this preference relation.

59 / 134


Some results

1 For simple logic arguments, with simple rebuts and simple undercuts, theexhaustive graphs are constructively complete for the set of all graphs.

2 For classical logic arguments, with any choice of attack, the exhaustive

graphs are not constructively complete for the set of all graphs.

For classical logic arguments, with any choice of attack, it is anopen question as to which set of graphs for which the exhaustivegraphs are complete.For classical logic arguments, with direct rebuttal, the exhaustiveconnective graphs are constructively complete for the set ofbipartite graphs.

3 Every graph can be regarded as an argument graph, and we should beable to construct every argument graph from a logical knowledgebaseusing a generative mechanism.

4 To meet this requirement, we need generative mechanisms that harnessmeta-level information.

60 / 134

Defeasible reasoning

Monotonicity

An important property for many logics ì including classical logic.

∆ ì α∆ ∪ {β} ì α

Example (Failure of monotonicity for defeasible reasoning)

Consider the following set of formulae ∆.

bird(x)→ flying-thing(x)ostrich(x)→ ¬flying-thing(x)ostrich(x)→ bird(x)

Therefore,

∆ ∪ {bird(Tweety)} ì flying-thing(Tweety)∆ ∪ {bird(Tweety), ostrich(Tweety)} 6ì flying-thing(Tweety)

61 / 134


Default logic (by Ray Reiter in 1980)

A default theory is a set of first-order formulae and a set of default rulesof the following form, where α, β and γ are classical formulae,

α : β

γ

The inference rules are those of classical logic plus a special mechanismto deal with default rules

If α is inferred, and ¬β cannot be inferred, then infer γ.

All classical inferences from the classical information in a default theory isderivable (if there is an extension).

62 / 134


Default logic (by Ray Reiter in 1980)

The operator Γ indicates the conclusions (i.e. the extension) to beassociated with a set of classical formulae E .

Let (D,W ) be a default theory, where D is a set of default rules and W isa set of classical formulae.

Let Cn return the set of classical consequences of a set of formulae.

Γ(E) is the smallest set of classical formulae such that the following threeconditions are satisfied.

1 W ⊆ Γ(E)2 Γ(E) = Cn(Γ(E))3 For each default in α : β/γ ∈ D, the following holds:

if α ∈ Γ(E) and ¬β 6∈ E then γ ∈ Γ(E)

63 / 134


Attempt E Γ(E) Extension?

1 bird(Tweety) bird(Tweety) E ⊂ Γ(E)fly(Tweety)

2 fly(Tweety) bird(Tweety) E ⊂ Γ(E)fly(Tweety)

3 bird(Tweety) bird(Tweety) E = Γ(E)fly(Tweety) fly(Tweety)

4 bird(Tweety), bird(Tweety) Γ(E) ⊂ E¬fly(Tweety)

5 ¬bird(Tweety), bird(Tweety) Γ(E) 6⊆ E & E 6⊆ Γ(E)¬fly(Tweety)

A non-exhaustive number of attempts are made for determining an extension.In each attempt, a guess is made for E , and then Γ(E) is calculated.

64 / 134


Example (Default logic)

Let D be the following set of defaults:

bird(X) : fly(X)

fly(X)

penguin(X) : bird(X)

bird(X)

penguin(X) : ¬fly(X)

¬fly(X)

and W is {bird(Tweety)}. For (D,W ), we obtain one extension

Cn({bird(Tweety), fly(Tweety)})

Now, consider W ′ = {bird(Tweety), penguin(Nigel)}. For (D,W ′), weobtain one extension

Cn({bird(Tweety), fly(Tweety), penguin(Nigel), bird(Nigel),¬fly(Nigel)})

Default logic suppresses inconsistency

Note how potential inconsistency is suppressed by the definition of default logic.

65 / 134


Argumentation is non-monotonic

Adding formulae to the knowledgebase can cause arguments to be withdrawn.

Example

Let ∆ = {a}, and so the following is a generated argument graph. Hence, {A1}is the grounded extension of the graph.

A1 = 〈{a}, a〉 A4 = 〈{a}, . . .〉

If we add ¬a to ∆, we get the following generated argument graph. Hence, A1

is no longer in the grounded extension.

A1 = 〈{a}, a〉 A2 = 〈{¬a},¬a〉

A3 = 〈{¬a}, . . .〉 A4 = 〈{a}, . . .〉

66 / 134


Non-monotonic logics versus argumentation systems

Both are non-monotonic processes

However

Non-monotonic logics suppress inconsistency: The aim is toprovide a consistent set of inferences.Argumentation systems highlight inconsistency via the use ofarguments and counterarguments.

67 / 134


Overview

Argumentation is a non-monotonic process.

This reflects the fact that argumentation involves uncertain information,and so new information can cause a change in the conclusions drawn.

However, the base logic does not need to be non-monotonic.

Indeed, most proposals for structured argumentation use a monotonicbase logic (e.g. simple logic, or classical logic).

Nonetheless, there are issues in capturing defeasible reasoning in

argumentation.

Choice of base logicModelling of defeasible knowledge

And there are insights and tools to be harnessed for research innon-monontonic logics

68 / 134


Example (Defeasible reasoning using simple logic with normality predicate)


A1 = 〈{workDay, normal, workDay ∧ normal→ useMetro(Sid)},useMetro(Sid)〉

A2 = 〈{workAtHome(Sid), workAtHome(Sid)→ ¬normal},¬normal〉

Example (Defeasible reasoning using simple logic augmented with rule names)


A1 = 〈{workDay, r1 : workDay→ useMetro(Sid)}, useMetro(Sid)〉

A2 = 〈{workAtHome(Sid), workAtHome(Sid)→ ¬r1},¬r1〉

69 / 134


Going beyond simple logic

Simple logic is monotonic.

Simple logic is very weak — it only has modus ponens.

There is a range of conditional logics

that are monotonicthat are richer than classical logicthat capture interesting aspects of defeasible reasoning

There is also a range of non-monotonic logics that may useful inargumentation (e.g. default logic).

70 / 134


Proof rules for system P

(REF )A⇒ A

(CUT )A⇒ B A ∧ B ⇒ C

A⇒ C

(LLE)A ≡ B A⇒ C

B ⇒ C(RW )

` A→ B C ⇒ A

C ⇒ B

(AND)A⇒ B A⇒ C

A⇒ B ∧ C(OR)

A⇒ C B ⇒ C

A ∨ B ⇒ C

(CM)A⇒ B A⇒ C

A ∧ B ⇒ C

(LOOP)A0 ⇒ A1 A1 ⇒ A2 . . .Ak−1 ⇒ Ak Ak ⇒ A0

A0 ⇒ Ak

[See Kraus Lehmann and Magidor 1990]

71 / 134


Example

From the following statements

penguin⇒ bird

penguin⇒ ¬flybird⇒ fly

we get the following inferences

penguin ∧ bird⇒ ¬flyfly⇒ ¬penguinbird⇒ ¬penguinbird ∨ penguin⇒ fly

bird ∨ penguin⇒ ¬penguin

72 / 134


Example

From the following statements,

teenager⇒ poor

teenager⇒ student

poor⇒ employed

student⇒ ¬employedwe get the following inferences

> ⇒ ¬teenager> ⇒ ¬(poor ∧ student)

but we don’t get the following.

poor⇒ ¬studentstudent⇒ ¬poor

73 / 134


Conditional logic argument

For a set of conditional statements Φ and a set of propositional formulae Ψ,

if Φ `P α⇒ β and ∧Ψ ≡ α, then a preferential argument is

〈Φ ∪Ψ, α⇒ β〉

Example

For Φ = {penguin⇒ bird, penguin⇒ ¬fly} and Ψ = {penguin, bird}, thefollowing is a preferential argument.

〈Φ ∪Ψ, penguin ∧ bird⇒ ¬fly〉

74 / 134


Some preferential attack relations

Let A1 = 〈Φ ∪Ψ, α⇒ β〉 and A2 = 〈Φ′ ∪Ψ′, γ ⇒ δ〉.A2 is a rebuttal of A1 iff δ ` ¬β and γ ` αA2 is a direct rebuttal of A1 iff δ ≡ ¬β and γ ` αA2 is a undercut of A1 iff δ ` ¬αA2 is a canonical undercut of A1 iff δ ≡ ¬αA2 is a direct undercut of A1 iff there is σ ∈ Ψ such that δ ≡ ¬σ

75 / 134


Example

For A1 and A2 below, A2 is a direct rebuttal of A1, but not vice versa,

A1 = 〈Φ1 ∪Ψ1, bird⇒ fly〉A2 = 〈Φ2 ∪Ψ2, penguin ∧ bird⇒ ¬fly〉

where

Φ1 = {bird⇒ fly},Ψ1 = {bird},Φ2 = {penguin⇒ bird, penguin⇒ ¬fly},Ψ2 = {penguin, bird}.

Example

For A1 and A2 below, A2 is a direct rebuttal of A1, but not vice versa,

A1 = 〈Φ1 ∪Ψ1, matchIsStruck⇒ matchLights〉A2 = 〈Φ2 ∪Ψ2, matchIsStruck ∧ matchIsWet⇒ ¬matchLights〉

76 / 134


MP system of conditional logic (Chellas 1975)

Conditional logics can be used for hypothetical statements of the form “If αwere true, then β would be true”.

RCEA`c α↔ β

`c (α⇒ γ)↔ (β ⇒ γ)

RCEC`c α↔ β

`c (γ ⇒ α)↔ (γ ⇒ β)

CC `c ((α⇒ β) ∧ (α⇒ γ))→ (α⇒ (β ∧ γ))

CM `c (α⇒ (β ∧ γ))→ ((α⇒ β) ∧ (α⇒ γ))

CN `c (α→ >)

MP `c (α⇒ β)→ (α→ β)

Informally, α⇒ β is valid when β is true in the possible worlds where α is true.

77 / 134


Example (Using MP system of conditional logic)

Let ∆ = {matchIsStruck⇒ matchLights, matchIsStruck}. From thisknowledgebase, we get the following argument.

〈{matchIsStruck⇒ matchLights, matchIsStruck}, matchLights〉

Note, from ∆, we cannot get the following argument.

〈{matchIsStruck∧matchIsWet⇒ matchLights, matchIsStruck}, matchLights〉

[See Besnard et al 2013]

78 / 134


matchTonightmatchTonight⇒ JohnGoesToTheStadiumJohnGoesToTheStadium

matchTonight ∧ JohnHasEnoughMoney ⇒ JohnGoesToTheStadiumtmatchTonight ∧ JohnHasEnoughMoney ⇒ JohnGoesToTheStadium

A descriptive graph using MP system of conditional logic.

The first argument says John will go to the stadium because there is amatch tonight.

The second corrects the first argument by correcting the circumstancesunder which John will go to watch the match at the stadium.

For more details of definition of counterargument, see Besnard et al 201379 / 134

Defeasible reasoning: What is a defeasible rule?

birds fly (∗)If we take all the normal situations (or observations, or worlds or days), and inthe majority of these birds fly, then (*) may be equal to the following

birds normally fly

Or if we take the set of all birds (or all the birds you have seen, or read about,or watched on TV) and the majority of this set fly, then (*) may be equal tothe following

most birds fly

Or perhaps if we take the set of all birds, we know the majority have thecapability to fly, then (*) could equate with the following. — though this inturn raises the question of what it is to know something has a capability

most birds have the capability to fly

Or if we take an idealized notion of a bird that we in a society are happy toagree upon, then we could equate (*) with the following

a prototypical bird flies

80 / 134

Defeasible reasoning: What is a defeasible rule?

birds lay eggs (∗∗)We may take (**) as meaning the following — but on a given day, (or givensituation, or observation) a given bird bird will probably not lay an egg.

birds normally lay eggs

Or we could say (**) means the following — but half the bird population ismale and therefore don’t lay eggs.

most birds have the capability lay eggs

Or perhaps we could say (**) means the following — where

most species of bird reproduce by laying eggs

However, the above involves a lot of missing information to be filled in, and ingeneral it is challenging to know how to formalize a defeasible rule.

81 / 134


The choice of proof rules depends on the interpretation of defeasible rules

Example (Failure of contraposition — from Brewka)

Men usually do not have beards

But, this does not imply that if someone does have a beard, then thatperson is not a man.

Example (Another failure of contraposition)

If I buy a lottery ticket, then I will normally not win a prize.

But, this does not imply that if I do win a prize, then I did not buy aticket.

82 / 134


Dichotomy of situations for whether contrapositive reasoning is appropriate

Epistemic Defeasible rules describe how certain facts hold in relation toeach other in the world. So the world exists independently ofthe rules. Contrapositive reasoning may be appropriate.

Constitutive Defeasible rules (in part) describe how the world is constructed(e.g. regulations). So the world does not exist independently ofthe rules. Contrapositive reasoning is not appropriate.

[Caminada 2008]

83 / 134


Note that these examples are identical syntactically.

Example (Epistemic situation)

O⇒ A - goods ordered three months ago will probably have arrived now.

A⇒ C - arrived good will probably have a customs document.

L⇒ ¬C - goods listed as unfulfilled will probably lack customs document.

From these rules, it would be reasonable to infer ¬A from L.

Example (Constitutive situation)

S⇒ M - snoring in the library is form of misbehaviour.

M⇒ R - misbehaviour in the library can result in removal from the library.

P⇒ ¬R - professors cannot be removed from the library.

From these rules, it would not be reasonable to infer ¬M from P.

[Caminada 2008]

84 / 134


Appropriateness of contraposition also depends on the kind of epistemicinformation and how we model it

1 Rule 1: bird⇒ flyingbird

2 Rule 2: flyingbird⇒ bird

3 Rule 3: penguin⇒ ¬flyingbird4 Rule 4: ¬flyingbird⇒ bird

5 From 1: ¬flyingbird⇒ ¬bird6 But step 5 conflicts with Rules 3+4

bird

flyingbird

penguin

85 / 134


Brewka’s preferred subtheories

A default theory Γ is a tuple (Γ1, ..., Γn), where each Γi is a set offormulae in a classical first-order language.

The formulae in Γi are more preferred to those in Γi+1

A preferred subtheory is a set Σ = Σ1 ∪ ∪ Σn such that for i = 1...n,Σ1 ∪ ∪ Σi is a maximal (under set inclusion) consistent subset ofΓ1, ..., Γi .

Example

Γi Σi

1 {a, a→ b,¬a ∨ ¬b} {a, a→ b}2 {¬a ∧ c,¬c} {¬c}3 {c,¬b, d} {d}

[Brewka 1989]

86 / 134


Relationship between preferred subtheories and deductive arguments

Let ∆ = Γ1 ∪ · · · ∪ Γn.

If Σ is a preferred subtheory of Γ, then Args(Σ) is a stable extension ofArgs(∆) with direct undercut and �.

If E is a stable extension of Args(∆) with direct undercut and �, then⋃A∈E Support(A) is a preferred subtheory of Γ.

where A � B iff ∃α ∈ Support(A) s.t. ∀β ∈ Support(B), α ≤ β.

Example

For Γ = ({a, b}{¬a ∨ ¬b}), the nodes in the stable extension are yellow

〈{a}, a∗〉〈{b,¬a ∨ ¬b},¬a〉

〈{¬a ∨ ¬b}, (¬a ∨ ¬b)∗〉〈{a, b},¬(¬a ∨ ¬b)〉

〈{a,¬a ∨ ¬b},¬b〉〈{b}, b∗〉

[Adapated from Modgil and Prakken 2013] 87 / 134


Conclusions

Defeasible reasoning is a central featureof argumentation

Structured argumentation may benefitfrom logics developed for defeasiblereasoning

Computational models of argument maybenefit from insights into defeasiblereasoning developed by research intonon-monontonic logics.

We need better ways of deciding whatkinds of defeasible rules we have for anapplication, and for deciding which proofrules are appropriate.

88 / 134

Meta-level argumentation

Argument schema

An argument schema is an informal specification of a general pattern of

reasoning used in argumentation, and defined in terms of

Types of premiseConclusionCritical questions that when answered may bring conclusion intodoubt

A variety of argument schema have been proposed by Walton and byothers

They are popular for capturing the complexities of applications

They can easily be adapted for specific applications

They can be formalized (e.g. Prakken, Hunter, etc).

[See Walton 2006]

89 / 134


Scheme for appeal to expert opinion

Major premise Source E is an expert in subject domain S containingproposition P.

Minor premise E asserts that proposition A in domain S is true (false).

Conclusion A may plausibly be taken to be true (false).

Critical questions :

Expertise: How credible is E as an expert source?Field: Is E an expert in the field that A is in?Opinion: What did E assert that implies A?Trustworthiness: Is E personally reliable as a source?Consistency: Is A consistent with what other expertsassert?Evidence: Is E ’s assertion based on evidence?

[See Walton 2006]

90 / 134


Scheme for appeal to popular opinion

General acceptance premise A is generally accepted as true.

Presumption premise If A is generally accepted as true, there exists apresumption in favour of A.

Conclusion There exists a presumption in favour of A.


Evidence: What evidence, such as poll or appeal tocommon knowledge, supports the claim that A isgenerally accepted as true?Alternatives: Even if A is generally accepted as true, arethere any reasons for doubting it is true?

[See Walton 2006]

91 / 134


Scheme for argument from negative consequences

Premise If A is brought about, bad consequences will plausible occur.

Conclusion A should not be brought about.


Probability: How probable are the consequences?Evidence: What is the evidence?Alternatives: Are there positive consequences thatshould be taken into account?

[See Walton 2006]

92 / 134


Further argument schema from Walton

The slippery slope argument

Argument from commitment

The circumstantial ad hominem

Argument from position to know

Argument from analogy

Argument from sign

Argument from example

Argument from popular practice

Argument from precedent

Argument from consequences

Argument from waste

Argument from bias

Ethotic argument

Argument from cause to effect

Argument from established rule

Argument from gradualism

Argument from verbal classification

Argument from correlation to cause

Plausible argument from ignorance

The direct ad hominem argument

Argument from an exceptional case

Argument from inconsistent commitment

Argument from positive consequences

Circumstantial argument against the person

Argument from vagueness of verbal classification

Argument from arbitrariness of a verbal classification

Argument from evidence to a hypothesis

Argument from falsification of a hypothesis

93 / 134



Meta-level argumentation is argumentation about argumentation.

Use logic as a meta-language for meta-level argumentation.

Properties and constraints can be specified in the meta-language.

Notions such as acceptability of an argument can be the subject of

argumentation

For example, arguments and counterarguments in the meta-level foran object-level argument being acceptable.Hence, argument schema can be captured in meta-levelargumentation

94 / 134


Example (Use meta-language of Wooldridge, McBurney + Parsons (2006))

For example, suppose 〈{b, b → c}, c〉 is an argument then the following is ameta-level formula.

〈{b, b → c}, c〉Hence, the following are meta-level arguments.

〈{〈{b, b → c}, c〉}, 〈{b, b → c}, c〉〉〈{〈{b}, b ∨ c〉, a7 = 〈{b}, b ∨ c〉}, a7〉

Following is a meta-level axiom for preference-based reasoning where a1 and a2

are object level arguments, and q is an individual.

Defeats(a1, a2, q)↔ (Attacks(a1, a2) ∧ Val(a1, v1) ∧ Val(a2, v2)→ ¬(v2 >q v1))

95 / 134


Besnard+Doutre 2004 encoding of argument semantics uses a meta-language

Conflictfreeness, admissibility, and extensions under different semantics,are specified by axioms in classical logic

Conflictfree is a constraint ensuring no attacker and attackee are in a setof arguments X

Conflictfree(X ,G) =∧ai∈X

[ai ∧ (∧

aj s.t. (ai ,aj )∈Attacks(G)

¬aj)]

Example (Conflictfree)

a1 a2a3a4

For X1 = {a1, a3}, [a1 ∧ (¬a2 ∧ ¬a3)] ∧ [a3 ∧ (¬a4)] Unsatisfiable

For X2 = {a1, a4}, [a1 ∧ (¬a2 ∧ ¬a3)] ∧ [a4] Satisfiable

96 / 134


Besnard+Doutre 2004 encoding of argument semantics uses a meta-language

The following constraint ensures that X is a stable extension

Stable(G) =∧

ai∈Nodes(G)

[ai ↔ (∧

aj s.t. (ai ,aj )∈Attacks(G)

¬aj)]

Example (Stable)

a1 a2a3a4

Hence, Stable(G) is

[a1 ↔ (¬a2 ∧ ¬a3)] ∧ [a2 ↔ (¬a1)] ∧ [a3 ↔ (¬a4)] ∧ [a4 ↔ (>)]

The models of this constraint are

{{a1, a4}, {a2, a4}}

97 / 134


Some axioms for meta-argumentation

(M1) argument(x)→ acceptable(x)(M2) acceptable(x)→ warranted(x)(M3) acceptable(y) ∧ undercut(y , x)→ ¬acceptable(x)

Example

Let (F1) be argument(〈{b, b → c}, c〉),

〈{F1,M1}, acceptable(〈{b, b → c}, c〉)〉

[see Hunter 2008]

98 / 134


Simulating object-level argumentation

Object level

〈{∀x .p(x) → ∀x .q(x),¬∃x .¬p(x)}, ∀x .q(x)〉

〈{∃x .(¬p(x)∧ r(x))},¬(. . .)〉

〈{∀x .¬r(x)},¬(. . .)〉

Meta-level

〈{F2,M1,M2}, warranted(a1)〉

〈{F3,F5,M1,M3},¬(. . .)〉

〈{F4,F6,M1,M3},¬(. . .)〉

(F2) argument(a1) (M1) argument(x)→ acceptable(x)(F3) argument(a2) (M2) acceptable(x)→ warranted(x)(F4) argument(a3) (M3) acceptable(y) ∧ undercut(y , x)(F5) undercut(a2, a1) → ¬acceptable(x)(F6) undercut(a3, a2)

99 / 134


Logical Formalization of Appropriateness

In order to formalise appropriateness we drop the axiom

(M1) argument(x)→ acceptable(x)

and replace it with the axiom

(M4) assert(x , y) ∧ appropriate(x , y)→ acceptable(y)

where

assert is for x asserts y

appropriate is for x is an appropriate proponent for y

So now our core axioms for appropriateness are

(M2) acceptable(x)→ warranted(x)(M3) acceptable(y) ∧ undercut(y , x)→ ¬acceptable(x)(M4) assert(x , y) ∧ appropriate(x , y)→ acceptable(y)

100 / 134


Logical Formalization of Appropriateness

Our core axioms for appropriateness

(M2) acceptable(x)→ warranted(x)(M3) acceptable(y) ∧ undercut(y , x)→ ¬acceptable(x)(M4) assert(x , y) ∧ appropriate(x , y)→ acceptable(y)

Further axioms for appropriateness may include

(M5) liar(x)→ ¬appropriate(x , y)(M6) celebrity(x)→ appropriate(x , y)(M6′) celebrity(x) ∧ topic(y , showbiz)→ appropriate(x , y)

Example

Let (F6) be liar(Pinnocchio),

〈{F6,M5},¬appropriate(Pinocchio, a)〉

Let (F8) be celebrity(HarrisonFord), and (F9) be topic(a, showbiz),

〈{F8,F9,M6}, appropriate(HarrisonFord , a)〉

101 / 134


Logical Formalization of Expert Argumentation

Our core axioms for expert argumentation

(M2) acceptable(x)→ warranted(x)(M3) acceptable(y) ∧ undercut(y , x)→ ¬acceptable(x)(M4) assert(x , y) ∧ appropriate(x , y)→ acceptable(y)(M7) topic(z ,w) ∧ role(x , y) ∧ scope(y ,w)→ appropriate(x , z)(M8) assert(x2, y2) ∧ acceptable(y2) ∧ competing(y1, y2) ∧ outrank(x2, x1)

→ ¬appropriate(x1, y1)

where

topic(z ,w) — argument z is on topic w

role(x , y) — proponent x has role y

scope(y ,w) — role y covers topic w

competing(y1, y2) — argument y1 competes with argument y2

outranks(x1, x2) — proponent x1 outranks proponent x1

102 / 134


Example (Expert argumentation)

(M2) acceptable(x)→ warranted(x)(M4) assert(x, y) ∧ appropriate(x, y)→ acceptable(y)(M7) topic(z, w) ∧ role(x, y) ∧ scope(y, w)→ appropriate(x, z)

(H1) assert(DrJones, diagnosis1)(H2) topic(diagnosis1, infarction)(H3) role(DrJones, GP)(H4) scope(GP, infarction)

appropriate(DrJones, diagnosis1) ∧ warranted(diagnosis1)

103 / 134


Example (Expert argumentation)

(M4) assert(x, y) ∧ appropriate(x, y)→ acceptable(y)(M7) topic(z, w) ∧ role(x, y) ∧ scope(y, w)→ appropriate(x, z)(M8) assert(x2, y2) ∧ acceptable(y2) ∧ competing(y1, y2) ∧ outrank(x2, x1)

→ ¬appropriate(x1, y1)

(H5) assert(DrSmith, diagnosis2)(H6) topic(diagnosis2, infarction)(H7) role(DrSmith, cardiologist)(H8) scope(cardiologist, infarction)(H9) competing(DrJones, DrSmith)(H10) outrank(DrSmith, DrJones)

appropriate(DrSmith, diagnosis2)∧ ¬appropriate(DrJones, diagnosis1)∧ warranted(diagnosis2)

104 / 134

Representing argument schema

Criteria for good expert argumentation

We consider four key criteria for delineating good expert argumentation.

Qualified proponent The expert is suitably qualified in the field of theargument being proposed.

Confident proponent The expert offers sufficient confidence in the argumentbeing proposed.

Best argument The argument by the expert is better than any competingargument by any expert.

Safe argument No counterargument to the expert’s argument has beenoverlooked.

These adapt the conditions and some of the critical questions of Walton’sscheme for appeal to expert opinion.

105 / 134


Criteria for good expert argumentation

Argument a is a good expert argument iff there is a proponent pand an undefeated argument 〈Φ, α〉 such that

Φ ` appropriate(a, p) Qualified proponent

Φ ` assert(a, p) Confident proponent

and there is no undefeated argument 〈Ψ, β〉 such that

Ψ ` ¬appropriate(a, p) Best argument

Ψ ` ¬acceptable(a, p) Safe argument

106 / 134


Meta-level (yellow) and object-level arguments (orange) can be arranged in abimodal argument graph where the object-level arguments and attacks aresupported (dotted line) by meta-level arguments.

〈{. . .},¬appropriate(p1, a1〉

〈{. . .}, acceptable(a1)〉

a1

〈{. . .}, acceptable(a2)〉

a2

〈{. . .}, attack(a1, a2)〉

〈{. . .}, attack(a2, a1)〉

[Muller + Hunter 2013]107 / 134


Advantages of meta-level argumentation

Meta-level argumentation provides a separation of concerns

Object level is for the issue being analyzed.Meta-level is for the quality of participants, evidence, etc.

Meta-level logics can be used directly in deductive argumentation.

Meta-level argumentation can be used for formalize argument schema.

Bimodal argument graphs can be used to arrange and evaluate meta-leveland object-level arguments

108 / 134

From natural language arguments to formalized arguments

Overview

Natural language is complex and so formalization calls for expressivelogics such as classical logic.

Computational linguistics offers mechanisms to translate fragments ofnatural language into logic.

However, most arguments in natural language are enthymemes, and thisraises further complications.

109 / 134

Understanding arguments in free text

Red denotes outer reason-claim coupling, and blue denotes inner reason-claimcoupling. Note, outer reason-claim coupling has two reasons for the claim.

〈claim〉Heathrow needs more capacity〈\claim〉〈reason〉 Heathrow runs at close to 100% capacity. With demandfor air travel predicted to double in a generation, Heathrow will notbe able to cope without a third runway, say those in favour of theplan. 〈\reason〉〈reason〉〈reason〉 Because the airport is over-stretched, anyproblems which arise cause knock-on delays. 〈\reason〉〈claim〉Heathrow, the argument goes, needs extra capacity if it is to reachthe levels of service found at competitors elsewhere in Europe, or itwill be overtaken by its rivals. 〈\claim〉〈\reason〉

http://news.bbc.co.uk/1/hi/uk/7828694.stm

110 / 134

http://news.bbc.co.uk/1/hi/uk/7828694.stm


Beyond argument mining

Argument mining only allows us to find parts of the text that pertain topremises and/or claim.

Argument mining does not give us the logical structure of the arguments.

If we are to use logical reasoning with arguments arising in naturallanguage, we need natural language understanding.

111 / 134


S

poison’(ethel’,the-cat’)

NP V [+PAS ] NP

the-cat’ poison’ ethel’

NP Npr

The cat was poisoned by Ethel

112 / 134


S

¬rain’→ snow’

S S

¬rain’ snow’

If it didn’t rain then it snowed

113 / 134


S

(λQ[λP[∀x [Q(x)→ P(x)]]](student’))(like’(jo’))

NP VP

λQ[λP[∀x [Q(x)→ P(x)]]](student’) like’(jo’)

Det N

λQ[λP[∀x [Q(x)→ P(x)]]] student’

Vt [+FIN] NP

like’ jo’

every student liked Npr

Jane’

Jane

114 / 134


Deep understanding within sentence

Part of speech tagging

Syntactic parsing

Semantic parsing

Identifying the predicate-argument structure

Identifying the structure of the logical formulae

Deep understanding across sentences

Resolving pronouns

Resolving ellipses

Co-ordinating phrases

Identifying implicit information

Identifying rhetorical structure

115 / 134

Modelling enthymemes in logical argumentation

A husband is clearing up breakfast as his wife ispreparing to go to work.

Husband thinks The weather report predicts rain and ifthe weather report predicts rain, thenyou should take an umbrella, so youshould take an umbrella (intendedargument)

Husband speaks The weather report predicts rain, soyou should take an umbrella(enthymeme)

Wife thinks The weather report predicts rain and ifthe weather report predicts rain, thenyou should take an umbrella, so youshould take an umbrella (receivedargument)

Since if the weather report predicts rain, then you should take an umbrella iscommon knowledge, it is not communicated.

116 / 134


Enthymeme

Let 〈Ψ, α〉 be an argument.

〈Φ, α〉 is an enthymeme for 〈Ψ, α〉 iff Φ ⊂ Ψ.

Example

Let α be “you need an umbrella today”, and β be “the weather report predictsrain”.

Intended argument is 〈{β, β → α}, α〉Enthymeme is 〈{β}, α〉.

So β → α is treated as common knowledge.

[Hunter 2007]

117 / 134


Assumption that there is a set of background knowledge

Intended argument Enthymeme

Encode by deletion of premises that are in background knowledge

Decode by abduction of premises from background knowledge

Abduction

Potentially many decodings can be abduced for an enthymemes.

Meta-information (preferences, probabilities, etc) can help to choose thebest decoding.

Risk that wrong argument is chosen for the decoding.

118 / 134


Approximate arguments

An approximate argument is a pair 〈Φ, α〉 where Φ ⊆ L and α ∈ L.

If Φ ` α, then 〈Φ, α〉 is valid.

If Φ 6` ⊥, then 〈Φ, α〉 is consistent.

If Φ ` α, and there is no Φ′ ⊂ Φ s.t. Φ′ ` α, then 〈Φ, α〉 is minimal.

If Φ ` α, and Φ 6` ⊥, then 〈Φ, α〉 is expansive.

If Φ 6` α, and Φ 6` ¬α, then 〈Φ, α〉 is a precursor.

An enthymeme is a precursor

Therefore, if 〈Φ, α〉 is a precursor, then there exists some Ψ ⊂ L such thatΦ ∪Ψ ` α and Φ ∪Ψ 6` ⊥, and hence 〈Φ ∪Ψ, α〉 is expansive.

119 / 134


Example (Approximate arguments)

Let ∆ = {α,¬α ∨ β, γ,¬β, β,¬γ,¬β ∨ γ},

Argument Val

id

Co

nsi

sten

t

Min

imal

Exp

ansi

ve

Pre

curs

or

〈{α,¬α ∨ β, γ, β}, β〉√ √

×√

×〈{γ,¬γ}, β〉

√×

√× ×

〈{α,¬α ∨ β, γ}, β〉√ √

×√

×〈{α,¬α ∨ β, γ,¬γ}, β〉

√× × × ×

〈{α,¬α ∨ β}, β〉√ √ √ √

×〈{¬α ∨ β}, β〉 ×

√× ×

√

〈{¬α ∨ β,¬β ∨ γ,¬γ}, β〉 ×√

× × ×

120 / 134


Real arguments

Each agent i has the following two types of resource.

Perbase A personal knowledgebase ∆i that when i is proponent, is usedfor the support of the intended argument.

Cobase A function µi,j : L 7→ [0, 1] that represents what an agent ibelieves is common knowledge for i and j .

For α ∈ L, the higher the value of µi,j(α), the more that i regards α as beingcommon knowledge for i and j .

Note, we do not assume that µi,j = µj,i for any agents.

Example

When µi,j(β → α) = 1 then the premise β → α is superfluous in any realargument consigned by proponent i to recipient j .

121 / 134


Encoding arguments

For an argument 〈Φ, α〉, and for a threshold τ ∈ [0, 1], the encodation of〈Φ, α〉 from a proponent i for a recipient j , is the approximate argument 〈Ψ, α〉where

Ψ = {φ ∈ Φ | µi,j(φ) ≤ τ}

Example

When µi,j(β → α) = 1, and µi,j(β) = 0.5, and τ = 0.7.

the encodation of 〈{β, β → α}, α〉 is 〈{β}, α〉

Example

When µi,j(β → α) = 1, and µi,j(β) = 1, and τ = 0.7,

the encodation of 〈{β, β → α}, α〉 is 〈{}, α〉

122 / 134


Decoding arguments

For an encodation 〈Ψ, α〉 from a proponent i for a recipient j , a decodation isof the form 〈Ψ ∪Ψ′, α〉 such that

1 Ψ′ ⊆ L2 〈Ψ ∪Ψ′, α〉 is expansive

3 there is no Ψ′′ s.t. Ψ′′ >j,i Ψ′ and 〈Ψ ∪Ψ′′, α〉 is expansive.

Example

Let µj,i be defined by the following and for all other φ, µj,i (φ) = 0.

µj,i (α→ β) = 1, µj,i (α→ ε) = 1, µj,i (ε→ β) = 1

So for the enthymeme 〈{α}, β〉, the decodations are

〈{α, α→ β}, β〉〈{α, α→ ε, ε→ β}, β〉

123 / 134


Background knowledge

The RNLI is a charity providingmost lifeboats for the UK.

Birmingham is in the centre ofEngland and so it is the furthest

point from the sea.

Conversation a fund-raiser and a potential donor

Would you like to make a donation to the RNLI?

No, I always spend my holidays in Birmingham

124 / 134


E1 and E3 are by a proponent for expanding Heathrow airport with a thirdrunway, and E2 is by an opponent, and so it appears that E3 attacks E2 and E2

attacks E1.

E1 = We should build a third runway at Heathrow because everyone willbenefit from the increased capacity.

E2 = It is not true that everyone will benefit in the community.

E3 = Local residents won’t have problems with traffic because we willincrease public transport to the airport.

Heathrow airport in thesuburbs of London.

125 / 134


h = “we should build a third runway at Heathrow”,

e = “everyone will benefit from the increased capacity”,

t = “local residents will have problems from increased traffic”,

s = “we will increase public transport to the airport”.

〈{e, e → h}, h〉

〈{},¬e〉

〈{s, s → ¬t},¬t〉

126 / 134


E1 and E3 are by a proponent for expanding Heathrow airport with a thirdrunway, and E2 is by an opponent, and so it appears that E3 attacks E2 and E2

attacks E1.

E1 = We should build a third runway at Heathrow because everyone willbenefit from the increased capacity.

E2 = It is not true that everyone will benefit in the community.

E3 = Local residents won’t have problems with traffic because we willincrease public transport to the airport.

Let us suppose that with the common knowledge we have available, we judgethat E2 decodes to either E ′2 or E ′′2 .

E ′2 = It is not true that everyone will benefit in the community. There arelocal residents who will suffer from increased noise from the increasednumber of aircraft.

E ′′2 = It is not true that everyone will benefit in the community. Thereare local residents who will have problems from increased traffic on theroads to the airport.

Given these decodings,E1 is attacked by both interpretations of E2 whereas onlyone interpretation of E2 is attacked by E3.

127 / 134


h = “we should build a third runway at Heathrow”,

e = “everyone will benefit from the increased capacity”,

t = “some local residents will have problems from increased traffic”,

n = “some local residents will suffer from noise from the increasednumber of aircraft”.

s = “we will increase public transport to the airport”.

〈{e, e → h}, h〉

〈{n, n→ ¬e},¬e〉

〈{s, s → ¬t},¬t〉

G1 〈{e, e → h}, h〉

〈{t, t → ¬e},¬e〉

〈{s, s → ¬t},¬t〉

G2

128 / 134


Enthymemes where the claim is implicit

An enthymeme has some/all support being implicit

An enthymeme may have some/all of its claim being implicit

Conversation late at night

Would you like a coffee?

Coffee will keep me awake.

[Black and Hunter 2012]129 / 134


Conclusions on modelling enthymemes

Enthymemes are an important feature of natural language arguments.

Enthymemes are a major source of dispute in argumentation.

We have some understanding of how to model enthymemes in logicalargumentation systems.

But acquiring the commonsense knowledge required for handlingenthymemes is challenging.

130 / 134

Conclusions

Structured argumentation provides explicit reasons and explicit claims

A range of formal definitions for arguments and for attacks can beconsidered

Properties concerning the structure, extension, and attack, ensure thatthe approach is well-understood and well-behaved

Deductive argumentation is useful for representing the formal semanticsof arguments in natural language

Deductive argumentation can be combined with rich backgroundknowledge for handling enthymemes

131 / 134

Further directions

Selectivity in argumentation (i.e. don’t exhaustively present all arguments,but select appropriate arguments for the purpose and audience).


Using different base logicsDraw on insights from research in non-monotonic reasoning.

Uncertainty in argumentation

Translating natural language arguments into logical arguments

Drawing on theories and technologies from computational linguisticsIdentifying background & commonsense knowledge for decodingenthymemes

132 / 134

Bibliography

L Amgoud and Ph Besnard (2009) Bridging the Gap between Abstract Argumentation Systems and Logic.SUM’09. 12-27.

L Amgoud and C Cayrol (2002) A Reasoning Model Based on the Production of Acceptable Arguments.Ann. Math. Artif. Intell. 34(1-3): 197-215.

L Amgoud and S Vesic (2014) Rich preference-based argumentation frameworks. International Journal ofApproximate Reasoning 55(2): 585-606.

Ph Besnard and S Doutre (2004) Characterization of Semantics for Argument Systems. KR 2004: 183-193.

Ph Besnard, E. Gregoire and B. Raddaoui: A Conditional Logic-Based Argumentation Framework. Proc.International Conference on Scalable Uncertainty Management (SUM’13), LNCS, Springer. 44-56

Ph Besnard and A Hunter (2008) Elements of Argumentation, MIT Press.

Ph Besnard and A Hunter (2014) Constructing argument graphs with deductive arguments: A tutorial.Argument and Computation. 5(1): 5-30.

E Black and A Hunter (2008) Using enthymemes in an inquiry dialogue system, Proceedings of theSeventh International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS’08),pages 437-444, ACM Press.

E Black and A Hunter (2012) A relevance-theoretic framework for constructing and deconstructingenthymemes, Journal of Logic and Computation, 22(1):55-78.

G Brewka (1989) Preferred subtheories: An extended logical framework for default reasoning, IJCAI’89, pp.10431048.

M Caminada and L Amgoud (2007) On the evaluation of argumentation formalisms. Artificial Intelligence171(5-6): 286-310.

M Caminada (2008) On the Issue of Contraposition of Defeasible Rules. COMMA 2008: 109-115.

C Cayrol (1995) On the Relation between Argumentation and Non-monotonic Coherence-BasedEntailment. IJCAI’95. 1443-1448.

B Chellas, Basic conditional logic. Journal of Philosophical Logic 4(2), 133153.

133 / 134

Bibliography

P. Dung (1995) On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning,Logic Programming and n-Person Games. Artificial Intelligence 77(2): 321-358.

A Garcia and G Simari (2014) Defeasible logic programming: DeLP servers, contextual queries, andexplanations for answers. Argument and Computation. 5(1): 89-117.

N Gorogiannis and A Hunter (2011) Instantiating Abstract Argumentation with Classical Logic Arguments:Postulates and Properties, Artificial Intelligence 175(9-10):1479-1497.

A Hunter (2007) Real arguments are approximate arguments, Proceedings of the 22nd AAAI Conferenceon Artificial Intelligence (AAAI’07), pages 66-71, MIT Press.

A Hunter (2014) Probabilistic qualification of attack in abstract argumentation, International Journal ofApproximate Reasoning, 55:607-638.

S Kraus, D Lehmann and M Magidor (1990) Nonmonotonic Reasoning, Preferential Models andCumulative Logics. Artificial Intelligence 44(1-2): 167-207.

D Martnez, A. Garcia, G Simari (2007) Modelling Well-Structured Argumentation Lines. IJCAI 2007:465-470.

S Modgil and H. Prakken (2013) A general account of argumentation with preferences. ArtificialIntelligence. 195: 361-397.

S Modgil and H. Prakken (2014) The ASPIC+ framework for structured argumentation: A tutorial.Argument and Computation. 5(1): 31-62.

J Muller, A Hunter and P Taylor (2013) Bimodal graphs for meta-argumentation with argument schemes,Proc. International Conference on Scalable Uncertainty Management (SUM’13), LNCS, Springer.

J Pollock (1987) Defeasible reasoning, Cognitive Science, 11,. 481-518.

R Reiter (1980) A Logic for Default Reasoning. Artificial Intelligence 13(1-2): 81-132.

F Toni (2014) A tutorial on assumption-based argumentation. Argument and Computation. 5(1): 89-117.

D Walton (2006) Fundamentals of Critical Argumentation. Cambridge University Press.

134 / 134

introduction to structured argumentation · abstract argumentation: pros and cons pros an argument...

Documents