argumentation logics lecture 5: argumentation with structured arguments (1) argument structure

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Argumentation LogicsLecture 5:

Argumentation with structured arguments (1)

argument structure

Henry PrakkenChongqing

June 2, 2010

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Contents Structured argumentation:

Arguments Argument schemes

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Merits of Dung (1995) Framework for nonmonotonic

logics Comparison and properties Guidance for development

From intuitions to theoretical notions

But should not be used for KR

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The structure of arguments: two approaches

Both approaches: arguments are inference trees

Assumption-based approaches (Dung-Kowalski-Toni, Besnard & Hunter, …)

Sound reasoning from uncertain premises Arguments attack each other on their assumptions

(premises)

Rule-based approaches (Pollock, Vreeswijk, …) Risky (‘defeasible’) reasoning from certain premises Arguments attack each other on applications of defeasible

inference rules

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Aspic framework: overviewArgument structure: Trees where

Nodes are wff of a logical language L Links are applications of inference rules

Rs = Strict rules (1, ..., 1 ); or Rd= Defeasible rules (1, ..., 1 )

Reasoning starts from a knowledge base K L Defeat: attack on conclusion, premise or

inference, + preferences Argument acceptability based on Dung

(1995)

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Argumentation systems An argumentation system is a tuple AS = (L,

-,R,) where: L is a logical language - is a contrariness function from L to 2L R = Rs Rd is a set of strict and defeasible inference

rules is a partial preorder on Rd

If -() then: if -() then is a contrary of ; if -() then and are contradictories

= _, = _

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Knowledge bases A knowledge base in AS = (L, -,R,= ’) is

a pair (K, =<’) where K L and ’ is a partial preorder on K/Kn. Here: Kn = (necessary) axioms Kp = ordinary premises Ka = assumptions

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Structure of arguments

An argument A on the basis of (K, ’) in (L, -,R, ) is: if K with

Conc(A) = {} Sub(A) = DefRules(A) =

A1, ..., An if there is a strict inference rule Conc(A1), ..., Conc(An)

Conc(A) = {} Sub(A) = Sub(A1) ... Sub(An) {A} DefRules(A) = DefRules(A1) ... DefRules(An)

A1, ..., An if there is a defeasible inference rule Conc(A1), ..., Conc(An)

Conc(A) = {} Sub(A) = Sub(A1) ... Sub(An) {A} DefRules(A) = DefRules(A1) ... DefRules(An) {A1, ..., An

}

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Q1 Q2

P

R1 R2

R1, R2 Q2

Q1, Q2 P

Q1,R1,R2 K

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ExampleR: r1: p q r2: p,q r r3: s t r4: t ¬r1 r5: u v r6: v,q ¬t r7: p,v ¬s r8: s ¬pKn = {p}, Kp = {s,u}

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Types of arguments An argument A is:

Strict if DefRules(A) = Defeasible if not Firm if Prem(A) Kn Plausible if not firm

S |- means there is a strict argument A s.t.

Conc(A) = Prem(A) S

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Domain-specific vs. inference general inference rules

R1: Bird Flies R2: Penguin Bird Penguin K

Rd = {, } Rs = all deductively valid inference rules Bird Flies K Penguin Bird K Penguin K

Flies

Bird

Penguin

Flies

Bird Bird Flies

Penguin Penguin Bird

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Argument(ation) schemes: general form

Defeasible inference rules! But also critical questions

Negative answers are counterarguments

Premise 1, … , Premise nTherefore (presumably), conclusion

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Expert testimony(Walton 1996)

Critical questions: Is E biased? Is P consistent with what other experts say? Is P consistent with known evidence?

E is expert on DE says that PP is within D Therefore (presumably), P is the case

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Witness testimony

Critical questions: Is W sincere? Does W’s memory function properly? Did W’s senses function properly?

W says PW was in the position to observe PTherefore (presumably), P

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Arguments from consequences

Critical questions: Does A also have bad consequences? Are there other ways to bring about G? ...

Action A brings about G, G is goodTherefore (presumably), A should be done

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Temporal persistence(Forward)

Critical questions: Was P known to be false between T1 and

T2? Is the gap between T1 and T2 too long?

P is true at T1 and T2 > T1Therefore (presumably), P isstill true at T2

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Temporal persistence(Backward)

Critical questions: Was P known to be false between T1 and

T2? Is the gap between T1 and T2 too long?

P is true at T1 and T2 < T1Therefore (presumably), P was already true at T2

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X murdered Y

Y murdered in house at 4:45

X in 4:45

X in 4:45{X in 4:30} X in 4:45{X in 5:00}

X left 5:00

W3: “X left 5:00”W1: “X in 4:30” W2: “X in 4:30”

X in 4:30{W1} X in 4:30{W2}

X in 4:30

accrual

testimony testimony

testimony

forwtemp pers

backwtemp pers

dmp

accrual

V murdered in L at T & S was in L at T

S murdered V