Introduction to Structured Argumentation Anthony Hunter Department of Computer Science, University College London, UK September 6, 2016 1 / 134
Introduction toStructured Argumentation
Anthony Hunter
Department of Computer Science, University College London, UK
September 6, 2016
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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
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Computational models of argument
Abstract argumentation
Structured Argumentation
Dialogical Argumentation
Decisionmaking
Sensemaking
Persuasion NegotiationDistributed
decision making
[Besnard + Hunter 2008]
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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)]
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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.
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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.
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Overview of deductive argumentation
Generative graphsDescriptive graphs
Counterarguments
Arguments
Base logic
[Besnard + Hunter 2014]
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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.
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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)〉
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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)〉
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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)〉
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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.
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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.
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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)〉
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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〉
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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.
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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〉
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Counterarguments based on classical logic
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〉
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Counterarguments based on classical logic
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〉
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Counterarguments based on classical logic
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)〉
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Counterarguments based on classical logic
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).
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Counterarguments based on classical logic
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)〉
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Counterarguments based on classical logic
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.
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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.
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Argument graphs
Abstract graph
Instantiated graph
AttacksArguments
Knowledgebase
desc
riptive
generative
Approaches to constructing logical argument graphs
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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〉
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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
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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)
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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〉
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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〉
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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.
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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.
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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.
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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.
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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]
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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.
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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
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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]
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Extension properties
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}
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Extension properties
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}
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Extension properties
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〉
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Extension properties
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.
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Extension properties
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)
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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 × × × ×
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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.
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Examples w.r.t. consistency postulates
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’.
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Examples w.r.t. consistency postulates
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’.
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Examples w.r.t. consistency postulates
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’.
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Extension properties
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.
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Extension properties
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
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Extension properties
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.
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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 .
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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 ∆.
Structural properties
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]
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Structural properties
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.
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Structural properties
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〉
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Structural properties
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}, . . .〉
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Structural properties
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]
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Structural properties
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)〉
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Structural properties
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.
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Structural properties
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.
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Defeasible reasoning
Monotonicity
An important property for many logics `i including classical logic.
∆ `i α∆ ∪ {β} `i 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)} `i flying-thing(Tweety)∆ ∪ {bird(Tweety), ostrich(Tweety)} 6`i flying-thing(Tweety)
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Defeasible reasoning
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).
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Defeasible reasoning
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)
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Defeasible reasoning
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.
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Defeasible reasoning
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.
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Defeasible reasoning
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}, . . .〉
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Defeasible reasoning
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.
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Defeasible reasoning
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
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Defeasible reasoning
Example (Defeasible reasoning using simple logic with normality predicate)
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〉
Example (Defeasible reasoning using simple logic augmented with rule names)
The first argument A1 captures the general rule that if workDay holds, thenuseMetro(Sid) holds.
A1 = 〈{workDay, r1 : workDay→ useMetro(Sid)}, useMetro(Sid)〉
A2 = 〈{workAtHome(Sid), workAtHome(Sid)→ ¬r1},¬r1〉
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Defeasible reasoning
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).
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Defeasible reasoning
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]
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Defeasible reasoning
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
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Defeasible reasoning
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
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Defeasible reasoning
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〉
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Defeasible reasoning
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
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Defeasible reasoning
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〉
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Defeasible reasoning
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.
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Defeasible reasoning
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]
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Defeasible reasoning
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
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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.
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Defeasible reasoning
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.
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Defeasible reasoning
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]
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Defeasible reasoning
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]
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Defeasible reasoning
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
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Defeasible reasoning
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]
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Defeasible reasoning
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
Defeasible reasoning
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.
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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]
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Meta-level argumentation
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]
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Meta-level argumentation
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.
Critical questions :
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]
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Meta-level argumentation
Scheme for argument from negative consequences
Premise If A is brought about, bad consequences will plausible occur.
Conclusion A should not be brought about.
Critical questions :
Probability: How probable are the consequences?Evidence: What is the evidence?Alternatives: Are there positive consequences thatshould be taken into account?
[See Walton 2006]
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Meta-level argumentation
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
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Meta-level argumentation
Meta-level argumentation
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
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Meta-level argumentation
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))
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Meta-level argumentation
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
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Meta-level argumentation
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}}
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Meta-level argumentation
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]
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Meta-level argumentation
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)
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Meta-level argumentation
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)
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Meta-level argumentation
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)〉
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Meta-level argumentation
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
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Meta-level argumentation
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)
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Meta-level argumentation
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)
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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.
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Meta-level argumentation
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
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Meta-level argumentation
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
Meta-level argumentation
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
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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.
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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
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Understanding arguments in free text
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.
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Understanding arguments in free text
S
poison’(ethel’,the-cat’)
NP V [+PAS ] NP
the-cat’ poison’ ethel’
NP Npr
The cat was poisoned by Ethel
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Understanding arguments in free text
S
¬rain’→ snow’
S S
¬rain’ snow’
If it didn’t rain then it snowed
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Understanding arguments in free text
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
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Understanding arguments in free text
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
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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.
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Modelling enthymemes in logical argumentation
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]
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Modelling enthymemes in logical argumentation
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.
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Modelling enthymemes in logical argumentation
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.
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Modelling enthymemes in logical argumentation
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〉 ×√
× × ×
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Modelling enthymemes in logical argumentation
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 .
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Modelling enthymemes in logical argumentation
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〉
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Modelling enthymemes in logical argumentation
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〉
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Modelling enthymemes in logical argumentation
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
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Modelling enthymemes in logical argumentation
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.
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Modelling enthymemes in logical argumentation
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〉
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Modelling enthymemes in logical argumentation
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.
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Modelling enthymemes in logical argumentation
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
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Modelling enthymemes in logical argumentation
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
Modelling enthymemes in logical argumentation
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.
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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
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Further directions
Selectivity in argumentation (i.e. don’t exhaustively present all arguments,but select appropriate arguments for the purpose and audience).
Defeasible reasoning
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
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