commonsense reasoning and argumentation 14/15 hc 10: structured argumentation (3) henry prakken 16...
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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
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Subtleties concerning rebuttals (1)
d1: Ring Married d2: Party animal Bachelor s1: Bachelor ¬Married Kn: Ring, Party animal
d2 < d1
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Subtleties concerning rebuttals (2)
d1: Ring Married d2: Party animal Bachelor s1: Bachelor ¬Married s2: Married ¬Bachelor Kn: Ring, Party animal
d2 < d1
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Subtleties concerning rebuttals (3)
d1: Ring Married d2: Party animal Bachelor s1: Bachelor ¬Married s2: Married ¬Bachelor Kn: Ring, Party animal
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Subtleties concerning rebuttals (4)
Rd = {, }Rs = all deductively valid inference rulesKn: d1: Ring Married d2: Party animal Bachelor n1: Bachelor ¬Married Ring, Party animal
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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
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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 ¬
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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’
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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
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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
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From defeasible to strict rules: example
r1: Quaker Pacifist r2: Republican ¬Pacifist
Pacifist
Quaker
Pacifist
Republican
r1 r2
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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
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Modelling default reasoning in classical argumentation
Quakers are usually pacifist Republicans are usually not pacifist Nixon was a quaker and a republican
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A modelling in classical logic
Kp: Quaker Pacifist Republican ¬Pacifist Quaker, Republican
Pacifist
Quaker Quaker Pacifist
¬Pacifist
Republican Republican ¬Pacifist
¬(Quaker Pacifist)
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A modelling in classical logic
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
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A modelling in classical logic
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
The case of classical argumentation
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
The case of classical argumentation
Bird & ¬Ab1 Flies Penguin & ¬Ab2 ¬Flies Penguin Bird Penguin Ab1 Penguin ¬Ab1 ¬Ab2 Arguments:
- for Flies using ¬Ab1 - for ¬Flies using ¬Ab2
Kp
Kn
The case of classical argumentation
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
The case of classical argumentation
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
The case of classical argumentation
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- and for Ab1 and Ab2 and Ab3
The case of classical argumentation
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- and for Ab1 and Ab2 and Ab3 ¬Ab3 > ¬Ab2 > ¬Ab1 makes ¬Flies followBut is this ordering natural?
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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
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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