commonsense reasoning and argumentation 14/15 hc 10: structured argumentation (3) henry prakken 16...

Commonsense Reasoning and Argumentation 14/15

HC 10: Structured argumentation (3)

Henry Prakken16 March 2015

Overview More about rationality postulates Related research The need for defeasible rules

3

Subtleties concerning rebuttals (1)

d1: Ring Married d2: Party animal Bachelor s1: Bachelor ¬Married Kn: Ring, Party animal

d2 < d1

4


d1: Ring Married d2: Party animal Bachelor s1: Bachelor ¬Married s2: Married ¬Bachelor Kn: Ring, Party animal

d2 < d1

5


d1: Ring Married d2: Party animal Bachelor s1: Bachelor ¬Married s2: Married ¬Bachelor Kn: Ring, Party animal

6


Rd = {, }Rs = all deductively valid inference rulesKn: d1: Ring Married d2: Party animal Bachelor n1: Bachelor ¬Married Ring, Party animal

7

Argumentation systems (with generalised

negation) An argumentation system is a tuple AS = (L,

-,R,n) 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 n: Rd L is a naming convention for defeasible rules

8

Generalised negation

The – function generalises negation. If -() then:

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

We write - = ¬ if does not start with a negation - = if is of the form ¬

9

Attack and defeat(the general case)

A undermines B (on ) if Conc(A) = - for some Prem(B )/ Kn;

A rebuts B (on B’ ) if Conc(A) = -Conc(B’ ) for some B’ Sub(B ) with a defeasible top

rule; A undercuts B (on B’ ) if

Conc(A) = -n(r) ’for some B’ Sub(B ) with defeasible top rule r A contrary-undermines/rebuts B (on /B’ ) if Conc(A) is a

contrary of / Conc(B ’)

A defeats B iff for some B’ A undermines B on and either A contrary-undermines B’ on

or not A <a ; or A rebuts B on B’ and either A contrary-rebuts B’ or not A <a B’ ;

or A undercuts B on B’

10

Consistency in ASPIC+(with generalised

negation) For any S L

S is directly consistent iff S does not contain two formulas and –()

The strict closure Cl(S) of S is S + everything derivable from S with only Rs.

S is indirectly consistent iff Cl(S) is directly consistent.

Parametrised by choice of strict rules

11

Rationality postulatesfor ASPIC+

(with generalised negation)

Closure under subarguments always satisfied Direct and indirect consistency:

without preferences satisfied if Rs closed under transposition or AS closed under

contraposition; and Kn is indirectly consistent; and AT is `well-formed’

with preferences satisfied if in addition is ‘reasonable’

Weakest- and last link ordering are reasonableAT is well-formed if:If is a contrary of then (1) Kn and (2) is not the consequent of a strict rule

Relation with other work (1)

Assumption-based argumentation (Dung, Kowalski, Toni ...) is special case of ASPIC+ (with generalised negation) with

Only ordinary premises Only strict inference rules All arguments of equal priority

…

Reduction of ASPIC+ defeasible rules to ABA rules (Dung & Thang, JAIR 2014)

Assumptions: L consists of literals No preferences No rebuttals of undercutters

p1, …, pn q

becomes

di, p1, …, pn,not¬q q

where: di = n(p1, …, pn q)

di, not¬q are assumptions = -(not), = -(¬), ¬ = -

()

1-1 correspondence

between grounded, preferred and

stable extensions of ASPIC+ and ABA

14

From defeasible to strict rules: example

r1: Quaker Pacifist r2: Republican ¬Pacifist

Pacifist

Quaker

Pacifist

Republican

r1 r2

15

From defeasible to strict rules: example

s1: Appl(s1), Quaker, not¬Pacifist Pacifist s2: Appl(s2), Republican, notPacifist ¬Pacifist

Pacifist

QuakerAppl(s1) not¬Pacifist

¬Pacifist

RepublicannotPacifist Appl(s2)

Can ASPIC+ preferences be reduced to ABA assumptions?

d1: Bird Fliesd2: Penguin ¬Fliesd1 < d2

Becomes

d1: Bird, not Penguin Fliesd2: Penguin ¬Flies

Only works in special cases, e.g. not with weakest-link ordering

Classical argumentation (Besnard & Hunter, …)

Given L a propositional logical language and |- standard-logical consequence over L:

An argument is a pair (S,p) such that S L and p L S |- p S is consistent No S’ S is such that S’ |- p

Various notions of attack, e.g.: “Direct defeat”: argument (S,p) attacks argument (S’,p’) iff

p |- ¬q for some q S’ “Direct undercut”: argument (S,p) attacks argument (S’,p’)

iff p |- ¬q and ¬q |- p for some q S’ Only these two attacks satisfy consistency.

Relation with other work (2)

Two variants of classical argumentation with premise attack (Amgoud & Cayrol, Besnard & Hunter) are special case of ASPIC+ with

Only ordinary premises Only strict inference rules (all valid propositional or first-order

inferences from finite sets) - = ¬ No preferences Arguments must have classically consistent premises

…

Results on classical argumentation (Cayrol 1995; Amgoud & Besnard

2013)

In classical argumentation with premise attack, only ordinary premises and no preferences:

Preferred and stable extensions and maximal conflict-free sets coincide with maximal consistent subsets of the knowledge base

So p is defensible iff there exists an argument for p The grounded extension coincides with the

intersection of all maximal consistent subsets of the knowledge base

So p is justified iff there exists an argument for p without counterargument

Lindebaum’s lemma:

Every consistent set is contained in

a maximal consistent set

20

Modelling default reasoning in classical argumentation

Quakers are usually pacifist Republicans are usually not pacifist Nixon was a quaker and a republican

21

A modelling in classical logic

Kp: Quaker Pacifist Republican ¬Pacifist Quaker, Republican

Pacifist

Quaker Quaker Pacifist

¬Pacifist

Republican Republican ¬Pacifist

¬(Quaker Pacifist)

22


Kn: Quaker & ¬Ab1 Pacifist Republican & ¬Ab2 ¬Pacifist Quaker, Republican

Kp: ¬Ab1, ¬Ab2 (attackable)

Pacifist

Quaker ¬Ab1

¬Pacifist

¬Ab2 RepublicanQuaker & ¬Ab1 Pacifist

Republican & ¬Ab2 ¬Pacifist

Ab1

23


Pacifist

Quaker ¬Ab1

¬Pacifist

¬Ab2 RepublicanQuaker & ¬Ab1 Pacifist

Republican & ¬Ab2 ¬Pacifist

Ab1Ab2

Can defeasible reasoning be reduced to plausible

reasoning? Is it natural to reduce all forms of attack

to premise attack? My answer: no

In classical argumentation: can the material implication represent defaults? My answer: no

Default contraposition in classical argumentation

Heterosexuals are normally married .

John is not married Assume when possible that things are

normal

What can we conclude about John’s sexual orientation?

Default contraposition in classical argumentation

Heterosexuals are normally married H & ¬Ab M

John is not married (¬M) Assume when possible that things are

normal ¬Ab

The first default implies that non-married people are normally not heterosexual

¬M & ¬Ab ¬H So John is not heterosexual

Default contraposition in classical argumentation (2)

Men normally have no beard => Creatures with a beard are normally not

men This type of sensor usually does not

give false alarms => False alarms are usually not given by this

type of sensor Witnesses interrogated by the police

usually tell the truth => People interrogated by the police who do

not speak the truth are usually not a witness Statisticians call these

inferences “base rate fallacies”

The case of classical argumentation

Birds usually fly

Penguins usually don’t fly

All penguins are birds

Penguins are abnormal birds w.r.t. flying

Tweety is a penguin


Birds usually fly Bird & ¬Ab1 Flies Penguins usually don’t fly Penguin & ¬Ab2 ¬Flies All penguins are birds Penguin Bird Penguins are abnormal birds w.r.t. flying Penguin Ab1 Tweety is a penguin Penguin ¬Ab1 ¬Ab2


Bird & ¬Ab1 Flies Penguin & ¬Ab2 ¬Flies Penguin Bird Penguin Ab1 Penguin ¬Ab1 ¬Ab2 Arguments:

- for Flies using ¬Ab1 - for ¬Flies using ¬Ab2

Kp

Kn


Bird & ¬Ab1 Flies Penguin & ¬Ab2 ¬Flies Penguin Bird Penguin Ab1 Penguin ¬Ab1 ¬Ab2 Arguments:

- for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 - and for Ab1 and Ab2 But ¬Flies follows

Kp

Kn


Bird & ¬Ab1 Flies Penguin & ¬Ab2 ¬Flies Penguin Bird Penguin Ab1 ObservedAsPenguin & ¬Ab3 Penguin ObservedAsPenguin ¬Ab1 ¬Ab2 ¬Ab3

Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 and ¬Ab3 - for Penguin using ¬Ab3



Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 and ¬Ab3 - for Penguin using ¬Ab3- and for Ab1 and Ab2 and Ab3



Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 and ¬Ab3 - for Penguin using ¬Ab3- and for Ab1 and Ab2 and Ab3 ¬Ab3 > ¬Ab2 > ¬Ab1 makes ¬Flies followBut is this ordering natural?

35

Contraposition of legal rules

r1: Snores Misbehaves r2: Misbehaves May be removed r3: Professor ¬May be removed

K: Snores, Professor r1 < r2, r1 < r3, r3 < r2

May be removed

Misbehaves

Snores

May be removed

Professor

r1

r2 r3

This is the intuitive outcome

R3 < R2

36

Contraposition of legal rules

r1: Snores Misbehaves r2: Misbehaves May be removed r3: Professor ¬May be removed

K: Snores, Professor r1 < r2, r1 < r3, r3 < r2

May be removed

Misbehaves

Snores

May be removed

Professor

r1

r2 r3

But with contraposition (and last or weakest

link) we have this outcome

My conclusion Classical logic’s material implication is

too strong for representing defeasible generalisations or legal rules

=> Models of legal argument (and many other kinds of argument) need defeasible inference rules Defeasible reasoning cannot be modelled as

inconsistency handling in deductive logic

John Pollock:Defeasible reasoning is the rule, deductive reasoning is the exception

Next lecture The lottery paradox Self-defeat and odd defeat loops Floating conclusions The need for dynamics

commonsense reasoning and argumentation 14/15 hc 10: structured argumentation (3) henry prakken 16...

Documents

b subb

undercuts b

party animal bachelors1

ring marriedd2

defeats b iff

party animal bachelorn1

choice of strict rules

consistent iff s