Commonsense Reasoning and Argumentation 14/15 HC 11: Structured argumentation (4) Henry Prakken 18 March 2015
Jan 01, 2016
Commonsense Reasoning and Argumentation 14/15
HC 11: Structured argumentation (4)
Henry Prakken18 March 2015
Overview The lottery paradox Self-defeat and odd defeat loops The need for dynamics
The lottery paradox (Kyburg 1960)
Assume:1. A lottery with 1 million tickets and 1 prize.2. The probability that some ticket wins is 13. 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
Solutions to the lottery paradox in the literature
Ignore the problem (many in nml and arg)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)
The lottery paradox in ASPIC+
Define: is justified iff some argument for is in all S-extensions
Kp = {T1,…,T1.000.000}Kn = {T1 xor … xor T1.000.000}
Rs = {S | S |-PL and S is finite}
Rd =
T1
T2 T3 T1
Kp = {T1, T2, T3}
Kn = {T1 xor T2 xor T3}
BA2
C1
A1
T1 xor T2 xor T3
A3
Option 1: C1 ≈ A1 But then for all i: Ci ≈ AiSo none of {A1,A2,A3} are in all extensions Violates principle that highly probable beliefs are justified
T1
T2 T3 T1
Kp = {T1, T2, T3}
Kn = {T1 xor T2 xor T3}
BA2
C1
A1
T1 xor T2 xor T3
A3
Option 2: C1 < A1 But then for all i: Ci < AiSo {A1,A2,A3,B,C1,C2,C3} E for any extension EViolates direct and indirect consistency
Excluded by third
condition on <
8
Serial self-defeat
p
n(r)
r: q,r p
A’ A
9
Parallel ‘self-defeat’
p p
q q
10
Requiring antecedents of strict rules to be consistent does not
help
p p
q
qp v q
Contamination: example r1: Quaker Pacifist r2: Republican ¬Pacifist S p Rs iff S |- p in Prop. L and S is finite Kn: Quaker, Republican
Pacifist
Quaker
Pacifist
Republican
r1 r2
12
Contamination: example r1: Quaker Pacifist r2: Republican ¬Pacifist r3: American Likes baseball S p Rs iff S |- p in Prop. L and S is finite Kn: Quaker, Republican, American
Pacifist
Quaker
Pacifist
Republican
r1 r2
Likes baseball
American
Likes baseball
r3
13
Contamination: example r1: Quaker Pacifist r2: Republican ¬Pacifist r3: American Likes baseball S p Rs iff S |- p in Prop. L and S is finite Kn: Quaker, Republican, American
Pacifist
Quaker
Pacifist
Republican
r1 r2
Likes baseball
American
Likes baseball
r3
Pollock (1995): preferred (recursive) labellings
solve the problem
14
Contamination: example r1: Quaker Pacifist r2: Republican ¬Pacifist r3: American Likes baseball S p Rs iff S |- p in Prop. L and S is finite Kn: Quaker, Republican, American
Pacifist
Quaker
Pacifist
Republican
r1 r2
Likes baseball
American
Likes baseball
r3
Pollock (1995): preferred (recursive) labellings
solve the problem
15
Contamination: example r1: Quaker Pacifist r2: Republican ¬Pacifist r3: American Likes baseball S p Rs iff S |- p in Prop. L and S is finite Kn: Quaker, Republican, American
Pacifist
Quaker
Pacifist
Republican
r1 r2
Likes baseball
American
Likes baseball
r3
Caminada (2005): not if arguments for the
conflicting conclusions have no status
16
Contamination: example r1: Quaker Pacifist r2: Republican ¬Pacifist r3: American Likes baseball S p Rs iff S |- p in Prop. L and S is finite Kn: Quaker, Republican, American
Pacifist
Quaker
Pacifist
Republican
r1 r2
Likes baseball
American
Likes baseball
r3
Pacifist v Likes baseball
Requiring that premises of strict inferences are
consistent does not help
Solution Grooters (& Prakken) 2014
Rescher & Manor (1970): S |-W p iff S’ |- p for some
consistent subset S’ of S Grooters (2014):
S p Rs iff S |-W p and S is finite No chaining of strict rules in arguments
Since |-W p does not satisfy Cut
Rationality postulates satisfied under the same assumptions as in Modgil & Prakken (2013)
18
Counterexample to Cut for |-W
Pacifist Pacifist
Likes baseball
Pacifist v Likes baseball
S |-W p, S {p} |-W q, So S |-W q
{Pacifist, Pacifist} |-W Pacifist v Likes baseball{Pacifist, Pacifist, Pacifist v Likes baseball} |-WLikes baseballBut not{Pacifist, Pacifist |-WLikes baseball
19
r1: Quaker Pacifist r2: Republican ¬Pacifist r3: American Likes baseball S p Rs iff S |-W p in Prop. L and S is finite Kn: Quaker, Republican, American
Pacifist
Quaker
Pacifist
Republican
r1 r2
Likes baseball
American
Likes baseball
r3
No contamination (1)
20
r1: Quaker Pacifist r2: Republican ¬Pacifist r3: American Likes baseball S p Rs iff S |-W p in Prop. L and S is finite Kn: Quaker, Republican, American
Pacifist
Quaker
Pacifist
Republican
r1 r2
Likes baseball
American
Likes baseball
r3
Pacifist v Likes baseball
No contamination (2)
21
r1: W says that p p
r2: W is unreliable ¬r1
k1: Alice says that Alice is unreliable
¬r1
A is unreliable
A: “A is unreliable”
22
¬r1
A is unreliable
A: “A is unreliable”
J is the killer
A: “J is the killer”
23
¬r1
A is unreliable
A: “A is unreliable”
J is the killer
A: “J is the killer”
24
¬r1
A is unreliable
A: “A is unreliable”
J is the killer
A: “J is the killer”
J is the not killer
B: “J is not the killer”
Grounded versus preferred semantics
A B
C
DE
A: Alice says that Bob is unreliable, so Bob is unreliable
B: Bob says that Carole is unreliable, so Carole is unreliable
C: Carole says that Alice is unreliable, so Alice is unreliable
D: Bob says that John was the killer,so John was the killer
E: Eric says that John was not the killer,so John was not the killer
R: W says that p p
Exception: W is unreliable
A: Alice says that Bob is unreliable, so Bob is unreliable
B: Bob says that Carole is unreliable, so Carole is unreliable
C: Carole says that Fred is unreliable, so Fred is unreliable
F: Fred says that Alice is unreliable,so Alice is unreliable
D: Bob says that John was the killer,so John was the killer
R: W says that p p
A B
DE
CFE: Eric says that John was not the killer,so John was not the killer
Exception: W is unreliable
A: Alice says that Bob is unreliable, so Bob is unreliable
B: Bob says that Carole is unreliable, so Carole is unreliable
C: Carole says that Fred is unreliable, so Fred is unreliable
F: Fred says that Alice is unreliable,so Alice is unreliable
D: Bob says that John was the killer,so John was the killer
R: W says that p p
A B
DE
CFE: Eric says that John was not the killer,so John was not the killer
Exception: W is unreliable
A B
C
DE
A B
DE
CF
1. An argument is In iff all arguments defeating it are Out.2. An argument is Out iff it is defeated by an argument that is In.
A B
C
DE
A B
DE
CF
1. An argument is In iff all arguments defeating it are Out.2. An argument is Out iff it is defeated by an argument that is In.
A B
C
DE
A B
DE
CF
E is not justifiedE is justified
3. An argument is justified if it is In in all labellings
1. An argument is In iff all arguments defeating it are Out.2. An argument is Out iff it is defeated by an argument that is In.
A B
DE
CF
S defends A if all defeaters of A are defeated by a member of S
S is admissible if it is conflict-free and defends all its members
{A,C,E} is admissible …
A B
DE
CF
S defends A if all defeaters of A are defeated by a member of S
S is admissible if it is conflict-free and defends all its members
{A,C,E} is admissible …
{B,D,F} is admissible …
A B
C
DE
S defends A if all defeaters of A are defeated by a member of S
S is admissible if it is conflict-free and defends all its members
{E} is admissible …
A B
C
DE
S defends A if all defeaters of A are defeated by a member of S
S is admissible if it is conflict-free and defends all its members
{E} is admissible …
but {B,D} is not …
A B
C
DE
S defends A if all defeaters of A are defeated by a member of S
S is admissible if it is conflict-free and defends all its members
{E} is admissible …
but {B,D} is not …
and {B,C,D} is not
Choosing between semantics (or not?)
37
Bright Rykkje is Norwegian
Brigt Rykkje has a Norwegian
name
Brigt Rykkje is Dutch
Brigt Rykkje was born in Holland
P is justified iff all labellings make an argument with conclusion P in(but it does not have to be the same argument)
Brigt Rykkje likes ice skating
Brigt Rykkje likes ice skating
In preferred semantics P is justified, in grounded semantics P is not justified
38
the suspect stabbed the victim to death
Witness Bob says: the suspect stabbed the victim to death
the suspect shot the victim to death
Witness John says: the suspect shot
the victim to death
The suspect killed the victim
The suspect killed the victim
Floating conclusions:still invalid? (John Horty)
39
the suspect stabbed the victim to death
Witness Bob says: the suspect stabbed the victim to death
the suspect shot the victim to death
Witness John says: the suspect shot
the victim to death
The suspect killed the victim
The suspect killed the victim
John/Bob is unreliable
One solution: add an undercutter “if two witnesses contradict each other, then they are
both unreliable”
Undercutter formalised
d(w,p): Witness w says that p p,ud(w,w’,p,-p): Witness w says that p, Witness w’ says that -p
-d(w,p)
d(w,p): Witness w says that p p,ud(w,w’,p,-p): Witness w says that p, Witness w’ says that –p,
d(w,p) ≤ d(w’,-p) -d(w,p)
Requires reasoning about preferences
Floating conclusions:Don’t ignore dynamics
Any judge would ask further questions Did you hear anything? Where did you stand? How dark was it?
The law’s way of dealing with dynamics: Procedures for fair and effective
dispute resolution
A simpler (imaginary) example
American civil law: evidence has to prove claim “on the balance of probabilities”
(Imaginary) statistic: 51% of American husbands commits adultery within 10 years.
Mary has been married to John for 10 years: can she sue John for divorce?