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?