Commonsense Reasoning and Argumentation 14/15

HC 9Structured argumentation (2)

Henry PrakkenMarch 4, 2015

2

Overview Argument schemes Preferences Rationality postulates

3

Domain-specific vs. inference general inference rules

d1: Bird Flies s1: Penguin Bird Penguin K

Rd = {, } Rs includes {S | S |-PL and

S is finite} Bird Flies K Penguin Bird K Penguin K

Flies

Bird

Penguin

Flies

Bird Bird Flies

Penguin Penguin Bird

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 nTherefore (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

Perception

Critical questions: Are the observer’s senses OK? Are the circumstances such that

reliable observation of P is impossible? …

P is observedTherefore (presumably), P

8

Reasoning with default generalisations

But defaults can have exceptions And there can be conflicting defaults

PIf P then normally/usually/typically QSo (presumably), Q

- What experts say is usually true - People with political ambitions are usually not objective about security- People with names typical from country C usually have nationality C- People who flea from a crime scene when the police arrives are normally involved in the crime- Chinese people usually don’t like coffee

9

How are generalisations justified?

Scientific research (induction) Experts Commonsense Individual opinions Prejudice?

Very reliable

Very unreliable

10

Inducing generalisations

Critical questions: Is the size of the sample large enough? was the sample selection biased?

Almost all observed P’s were Q’sTherefore (presumably), If P then usually Q

In 16 of 17 tests the ballpoint shot with this bow caused this type of

eye injury

A ballpoint shot with this type of bow will usually cause this type of

eye injury

11

Expert testimony

Critical questions: Is E biased? Is P consistent with what other experts say? Is P consistent with known evidence?

E is expert on DE says that PP is within D Therefore (presumably), P is the case

Supporting and using generalisations

V’s injury was caused by a fall

This type of eye injury is usually caused by a fall

V has this type of injury

E says that his type of injury is usually caused

by a fall

E is an expert on this type of injury

Expert testimony scheme

Defeasible modus ponens

13

Witness testimony

Critical questions: Is W sincere? Does W’s memory function properly? Did W’s senses function properly?

W says PW was in the position to observe PTherefore (presumably), P

P is usually of the form“I remember that I observed that ...”

Memory

Critical questions: Is the memory contaminated with

other information? …

P is recalledTherefore (presumably), P

15

Temporal persistence(Forward)

Critical questions: Was P known to be false between T1 and T2? …

P is true at T1 and T2 > T1Therefore (presumably), P isstill true at T2

16

Temporal persistence(Backward)

Critical questions: Was P known to be false between T1 and T2? …

P is true at T1 and T2 < T1Therefore (presumably), P was already true at T2

17

X murdered Y

Y murdered in house at 4:45

X in 4:45

X in 4:45{X in 4:30} X in 4:45{X in 5:00}

X left 5:00

W3: “X left 5:00”W1: “X in 4:30” W2: “X in 4:30”

X in 4:30{W1} X in 4:30{W2}

X in 4:30

accrual

testimony testimony

testimony

forwtemp pers

backwtemp pers

d.m.p.

accrual

V murdered in L at T & S was in L at T

S murdered V

18

Arguments from consequences

Critical questions: Does A also have bad (good) consequences? Are there other ways to bring about G? ...

Action A causes G, G is good (bad)Therefore (presumably), A should (not) be done

19

Example (arguments pro and con an action)

We should lower taxes

Lower taxes increase

productivity

Increased productivity is

good

We should not lower taxes

Lower taxes increase inequality

Increased inequality is bad

20

Example (arguments pro alternative actions)

We should lower taxes

Lower taxes increase

productivity

Increased productivity is

good

We should invest in public

infrastructure

Investing in public infrastructure

increases productivity

Increased productivity is

good

21

Refinement: promoting or demoting legal/societal values

Critical questions: Are there other ways to cause G? Does A also cause something else that

promotes or demotes other values? ...

Action A causes G, G promotes (demotes) legal/societal value VTherefore (presumably), A should (not) be done

22

Example (arguments pro and con an action)

We should save DNA of all citizens

Saving DNA of all citizens leads to

solving more crimes

Solving more crimes promotes

security

We should not save DNA of all

citizens

Saving DNA of all citizens makes

more private data publicly accessible

Making more private data

publicly available

demotes privacy

23

Example (arguments pro alternative actions)

We should save DNA of all citizens

Saving DNA of all citizens leads to

solving more crimes

Solving more crimes promotes

security

We should have more police

Having more police leads to solving more

crimes

Solving more crimes promotes

security

Argument schemes about action(generalised)

Action A results in C1…Action A results in CnWe should achieve C1…We should achieve CnTherefore, We should do A

Action A results in C1…Action A results in CnWe should avoid C1…We should avoid CnTherefore, We should not do A

25

Argument preference In general its origin is undefined General constraint: A <a B if B is strict-

and-firm and A is defeasible or plausible.

Could otherwise be defined in terms of partial preorders (on Rd) and ’ (on Kp) Origins of and ’: domain-specific!

26

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

27

Example Rd: r1: p q r2: p r r3: s t

Rs: q, r ¬t

K: p,s

28

Comparing ordered sets (elitist ordering, weak version)

Ordering s on sets in terms of an ordering (or ’) on their elements: If S1 = then not S1 s S2 If S1 ≠ and S2 = then S1 <s S2 Else S1 s S2 if there exists an s1 S1 such

that for all s2 S2: s1 s2

29

Comparing ordered sets (elitist ordering, strict version)

Ordering <s on sets in terms of an ordering (or ’) on their elements: If S1 = then not S1 <s S2 If S1 ≠ and S2 = then S1 <s S2 Else S1 <s S2 if there exists an s1 S1 such

that for all s2 S2: s1 < s2

Weakest-link ordering (formal)

A <a B if B is strict-and-firm and A is defeasible or plausible. Otherwise:

A a B iff If both A and B are strict, then Premp(A) s

Premp(A2) If both A and B are firm, then DefRules(A) s

DefRules(B); else Premp(A) s Premp(A2) and DefRules(A) s

DefRules(B)

30

Last-link ordering (formal) A <a B if B is strict-and-firm and A is

defeasible or plausible. Otherwise: A a B iff

LDR(A) s LDR(B); or A and B are strict and Premp(A) s

Premp(B)

31

32

Last link vs. weakest link (1)

r1: Born in Scotland Scottish r2: Scottish Likes Whisky r3: Fitness Lover ¬Likes Whisky

Kn: Born in Scotland, Fitness Lover r1 < r2, r1 < r3, r2 ≈ r3

Likes Whisky

Scottish

Born in Scotland

Likes Whisky

Fitness lover

r1

r2 r3

33

Weakest link

r1: Born in Scotland Scottish r2: Scottish Likes Whisky r3: Fitness Lover ¬Likes Whisky

Kn: Born in Scotland, Fitness Lover r1 < r2, r1 < r3, r2 ≈r3

Likes Whisky

Scottish

Born in Scotland

Likes Whisky

Fitness lover

r1

r2 r3

34

Last link

r1: Born in Scotland Scottish r2: Scottish Likes Whisky r3: Fitness Lover ¬Likes Whisky

Kn: Born in Scotland, Fitness Lover r1 < r2, r1 < r3, r2 ≈r3

Likes Whisky

Scottish

Born in Scotland

Likes Whisky

Fitness lover

r1

r2 r3

35

Last link vs. weakest link (2)

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

Kn: Snores, Professor r1 < r2, r1 < r3, r2 ≈r3

May be removed

Misbehaves

Snores

May be removed

Professor

r1

r2 r3

36

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

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

38

Violation of direct and indirect consistency in

ASPIC+

s1: r ¬q Kn = ; Kp = {q,r} r <’ q

q

q

r

s1>

B1A1

B2

B1A1

B2

39

Violation of direct and indirect consistency in

ASPIC+

s1: r ¬q s2: q ¬r Kn = ; Kp = {q,r} r <’ q

q

q

r

s1

r

Constraint on a:If A = B then A ≈ a B

>B1A1

B2A2

A2

s2

Trans- and contraposition Transposition:

If S p Rs then S/{s} U {–p} –s Rs

Contraposition: If S |- p and s S then S/{s} U {– p}

|- –s

41

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 is ‘reasonable’ If A is plausible or defeasible and B is strict-and-firm then A

< B If A = B then A ≈ B (Complicated condition)

Weakest- and last link ordering are reasonable