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Commonsense Reasoning and Argumentation 15/16 HC 10: Structured argumentation (3)

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Title: Commonsense Reasoning and Argumentation 15/16 HC 10: Structured argumentation (3)


1
Commonsense Reasoning and Argumentation 15/16HC
10 Structured argumentation (3)
  • Henry Prakken
  • 30 March 2016

2
Overview
  • Recap
  • More about rationality postulates
  • Related research
  • The need for defeasible rules

3
Aspic framework overview
  • Argument structure
  • Directed acyclic graphs where
  • Nodes are wff of a logical language L
  • Links are applications of inference rules
  • Rs Strict rules (?1, ..., ?n ? ?) or
  • Rd Defeasible rules (?1, ..., ?n ? ?)
  • Reasoning starts from a knowledge base K ? L
  • Defeat attack on conclusion, premise or
    inference, preferences
  • Argument acceptability based on Dung (1995)

4
Deriving the strict rules from a monotonic logic
  • For any logic L with (monotonic) consequence
    notion -L define

S ? p ? Rs iff S is finite and S -L p
5
Argument(ation) schemes general form
  • But also critical questions

Premise 1, , Premise n Therefore
(presumably), conclusion
6
Argument schemes in ASPIC
  • Argument schemes are defeasible inference rules
  • Critical questions are pointers to
    counterarguments
  • Some point to undermining attacks
  • Some point to rebutting attacks
  • Some point to undercutting attacks

7
Two example argument orderings
  • (Informal Kp ?, no strict-and-firm arguments)
  • Weakest link ordering
  • Compares the weakest defeasible rule of each
    argument
  • Last-link ordering
  • Compares the last defeasible rules of each
    argument

8
Comparing ordered sets (elitist ordering, strict
version)
  • Ordering lts on sets in terms of an ordering ? (or
    ?) on their elements
  • If S2 ? then not S2 lts S1
  • If S1 ? ? and S2 ? then S1 lts S2
  • Else S1 lts S2 if there exists an s1 ? S1 such
    that for all s2 ? S2 s1 lt s2

9
Consistency in ASPIC(with symmetric 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

9
10
Rationality postulates(Caminada Amgoud 2007)
  • Let E be any Dung-extension and
  • Conc(E) ?? Conc(A) for some A ? E
  • An AT satisfies
  • subargument closure iff B ? E whenever A ? E and
    B ? Sub(A)
  • direct consistency iff Conc(E) is directly
    consistent
  • strict closure iff Cl(Conc(E)) Conc(E)
  • indirect consistency iff Conc(E) is indirectly
    consistent

11
Rationality postulatesfor ASPIC (whether
consistent premises or not)
  • Closure under subarguments always satisfied
  • Strict closure, direct and indirect consistency
  • without preferences satisfied if
  • Rs closed under transposition or AS closed under
    contraposition and
  • Kn is indirectly consistent
  • with preferences satisfied if in addition lta is
    reasonable
  • If A is plausible or defeasible and B is
    strict-and-firm then A lt B
  • If A B ? ? then A and B have the same
    strength
  • (Complicated condition)
  • Weakest- and last link ordering are reasonable

12
Subtleties concerning rebuttals (1)
  • d1 Ring ? Married
  • d2 Party animal ? Bachelor
  • s1 Bachelor ? Married
  • Kn Ring, Party animal

d2 lt d1
13
Subtleties concerning rebuttals (2)
  • d1 Ring ? Married
  • d2 Party animal ? Bachelor
  • s1 Bachelor ? Married
  • s2 Married ? Bachelor
  • Kn Ring, Party animal

d2 lt d1
14
Subtleties concerning rebuttals (3)
  • d1 Ring ? Married
  • d2 Party animal ? Bachelor
  • s1 Bachelor ? Married
  • s2 Married ? Bachelor
  • Kn Ring, Party animal

15
Subtleties concerning rebuttals (4)
  • Rd ?, ? ? ? ? ?
  • Rs all deductively valid inference rules
  • Kn
  • d1 Ring ? Married
  • d2 Party animal ? Bachelor
  • n1 Bachelor ? Married
  • Ring, Party animal

16
The lottery paradox (Kyburg 1960)
  • Assume
  • A lottery with 1 million tickets and 1 prize.
  • The probability that some ticket wins is 1
  • The probability that a given ticket Ti wins is
    0.000001.
  • Suppose
  • a highly probable belief is justified and
  • what can be deduced from a set of justified
    beliefs is justified.
  • Then 1,2,3 is inconsistent

17
Solutions to the lottery paradox in the literature
  • Reject the conjunction principle for justified
    beliefs (Kyburg)
  • Reject that what is highly probable is justified
    (Pollock?)
  • Reject consistency for justified beliefs
  • But retain restricted forms of consistency and
    deductive closure (Makinson)

18
The lottery paradox in ASPIC
X1 will win and the other tickets will not win
  • Define ? is justified iff some argument for ? is
    in all S-extensions

Kp ?T1,,?T1.000.000 Kn X1 xor xor
X1.000.000 (Rs S ? ? S -PL ? and S is
finite Rd ?
19
Kp ?T1, ?T2, ?T3 Kn X1 xor X2 xor X3
Option 1 C1 A1 But then for all i Ci
Ai So none of A1,A2,A3 are in all extensions
Violates principle that highly probable beliefs
are justified
C1
T1
X1 xor X2 xor X3
?T3
?T2
?T1
B
A3
A2
A1
20
Kp ?T1, ?T2, ?T3 Kn X1 xor X2 xor X3
Excluded by third condition on lt
Option 2 C1 lt A1 But then for all i Ci lt
Ai So A1,A2,A3,B,C1,C2,C3 ? E for any extension
E Violates direct and indirect consistency
C1
T1
X1 xor X2 xor X3
?T3
?T2
?T1
B
A3
A2
A1
21
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

See reader (next four slides not discussed during
lecture)
22
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 ?

23
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 lta ? or
  • A rebuts B on B and either A contrary-rebuts B
    or not A lta B or
  • A undercuts B on B

24
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

24
25
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 reasonable

AT is well-formed if If ? is a contrary of ?
then (1) ? ? Kn and (2) ? is not the
consequent of a strict rule
26
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

27
Reduction of ASPIC defeasible rules to ABA rules
(Dung Thang, JAIR 2014)
1-1 correspondence between grounded, preferred
and stable extensions of ASPIC and ABA
  • Assumptions
  • L consists of literals
  • No preferences
  • No rebuttals of undercutters

p1, , pn ? q becomes di, p1, , pn,notq ?
q where di n(p1, , pn ? q) di, notq are
assumptions ? -(not?), ? -(?), ? -(?)
28
From defeasible to strict rules example
  • r1 Quaker ? Pacifist
  • r2 Republican ? Pacifist

Pacifist
?Pacifist
r1
r2
Quaker
Republican
29
From defeasible to strict rules example
  • s1 Appl(s1), Quaker, notPacifist ? Pacifist
  • s2 Appl(s2), Republican, notPacifist ? Pacifist

Pacifist
Pacifist
Quaker
Appl(s1)
notPacifist
Republican
notPacifist
Appl(s2)
30
Can ASPIC preferences be reduced to ABA
assumptions?
d1 Bird ? Flies d2 Penguin ? Flies d1 lt
d2 Becomes d1 Bird, not Penguin ? Flies d2
Penguin ? Flies
Only works in special cases, e.g. not with
weakest-link ordering
31
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 indirect
    consistency.

32
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

33
Results on classical argumentation (Cayrol 1995
Amgoud Besnard 2013)
Lindebaums lemma Every consistent set is
contained in a maximal consistent set
  • 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

34
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

35
The case of classical argumentation
  • Birds usually fly
  • Penguins usually dont fly
  • All penguins are birds
  • Penguins are abnormal birds w.r.t. flying
  • Tweety is a penguin

36
The case of classical argumentation
  • Birds usually fly
  • Bird Ab1 ? Flies
  • Penguins usually dont 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

37
The case of classical argumentation
  • Bird Ab1 ? Flies
  • Penguin Ab2 ? Flies
  • Penguin ? Bird
  • Penguin ? Ab1
  • Penguin
  • Ab1
  • Ab2

Kn
Kp
Arguments - for Flies using Ab1 - for Flies
using Ab2
38
The case of classical argumentation
  • Bird Ab1 ? Flies
  • Penguin Ab2 ? Flies
  • Penguin ? Bird
  • Penguin ? Ab1
  • Penguin
  • Ab1
  • Ab2

Kn
Kp
Arguments - for Flies using Ab1 - for Flies
using Ab2 - and for Ab1 and Ab2 But Flies
follows
39
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

40
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

41
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 gt Ab2 gt Ab1 makes Flies follow
  • But is this ordering natural?

42
My conclusion
  • Classical logics material implication is too
    strong for representing defeasible
    generalisations in argumentation
  • gt General models of argumentation 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
43
Next lecture
  • Self-defeat
  • Odd defeat loops
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