Download - Argumentation Logics Lecture 1: Introduction

Argumentation LogicsLecture 1:

Introduction

Henry PrakkenChongqing

May 26, 2010

Nonmonotonic logic Standard logic is monotonic:

If S |- and S S’ then S’ |- But commonsense reasoning is often nonmonotonic:

John is an adult, Adults are usually employed, so John is presumably employed

But suppose also that John is a student and students are usually not employed …

We often reason with rules that have exceptions We apply the general rule if we have no evidence of

exceptions But must retract our conclusion if we learn evidence of

an exception

Sources of nonmonotonicity Empirical generalisations

Adults are usually employed, birds can typically fly, Chinese usually do not like coffee, …

Conflicting information sources Experts who disagree, witnesses who contradict each other,

conflicting sensory input, … Alternative explanations

The grass is wet so it has rained / but the sprinkler was on Conflicting goals

We should raise taxes to increase productivity, which is good / but lower taxes increase inequality, which is bad

Exceptions to legal rules When a father dies, his son can inherit, except when the son killed

the father Exceptions to moral principles

Normally one should not lie, except when a lie can save lives …

Some nonmonotonic logics Default logic (Ray Reiter) Circumscription (John McCarthy) Logic programming (Robert

Kowalski) … Argumentation logics

Argumentation as a nonmonotonic logic

Nonmonotonic logic deals with: Rules and exceptions Conflicts and their resolution

Both can be modelled as argumentation: General rule gives rise to argument,

exception gives rise to counterargument Exception defeats general rule

Conflicts give rise to argument and counterargument

Conflicts are resolved with preferences

Some history John Pollock (1987-1995) Ron Loui (1987)

With Guillermo Simari (1992) Gerard Vreeswijk (1993,1997) Phan Minh Dung (1995) …

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







Lower taxes do not increase productivity

USA lowered taxes but productivity decreased








Prof. P says that …










Prof. P has political ambitions

People with political ambitions are not objective

Prof. P is not objective










Prof. P has political ambitions

People with political ambitions are not objective

Prof. P is not objective

Increased inequality is good

Increased inequality stimulates competition

Competition is good


A B

C D E

Overview of this course Abstract argumentation (Lectures 1-4)

Semantics (Lectures 1-3) Labelling-based Extension-based

Argument games (Lecture 4) Rule-based argumentation (Lectures 5-7)

Structure of arguments, (Lecture 5) Attack, defeat, preferences (Lecture 6) Self-defeat, rationality postulates (Lecture 7)

Status of arguments: abstract semantics (Dung 1995)

INPUT: an abstract argumentation theory AAT = Args,Defeat

OUTPUT: An assignment of the status ‘in’ or ‘out’ to all members of Args So: semantics specifies conditions for

labeling the ‘argument graph’. Should capture reinstatement:

A B C

Possible labeling conditions

Every argument is either ‘in’ or ‘out’.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’.

Works fine with:

But not with:

A B C

A B

Two solutions

Change conditions so that always a unique status assignment results

Use multiple status assignments:

and

A B C

A BA B

A B C

A B

Unique status assignments: Grounded semantics (informal)

The endpoint (or union) of a sequence s.t.: S0: the empty set Si+1: Si + all arguments acceptable wrt Si ...

A is acceptable wrt S (or S defends A) if all defeaters of A are defeated by S S defeats A if an argument in S defeats A

A B

C D E

Is B, D or E defended by S1?Is B or E defended by S2?

Grounded semantics (formal 1)

Let AAT be an abstract argumentation theory F0

AAT = Fi+1

AAT = {A Args | A is acceptable wrt Fi

AAT} F∞

AAT = ∞i=0 (Fi+1

AAT)

Problem: does not always contain all intuitively justified arguments.

Grounded semantics (formal 2)

Let AAT = Args,Defeat and S Args FAAT(S) = {A Args | A is acceptable wrt S}

Since FAAT is monotonic (and since ...), FAAT has a least fixed point. Now:

The grounded extension of AAT is the least fixed point of FAAT

An argument is (w.r.t. grounded semantics) justified on the basis of AAT if it is in the grounded extension of AAT.

Proposition 4.2.4 (AAT implicit): A F∞ A is justified If every argument has at most a finite number of

defeaters, then A F∞AT A is justified

Acceptability status with unique status assignments

A is justified if A is In A is overruled if A is Out and A is defeated by

an argument that is In A is defensible otherwise

Self-defeating arguments

Intuition: should always be overruled (?) Problem: in grounded semantics they are not

always overruled Solution: several possibilities (but intuitions

must be refined!)

A problem(?) with grounded semantics

We have: We want(?):

A B

C

D

A B

C

D


A B

C

D

A = Frederic Michaud is French since he has a French nameB = Frederic Michaud is Dutch since he is a marathon skaterC = F.M. likes the EU since he is European (assuming he is not Dutch or French)D = F.M. does not like the EU since he looks like a person who does not like the EU


A B

C

D

A = Frederic Michaud is French since Alice says soB = Frederic Michaud is Dutch since Bob says soC = F.M. likes the EU since he is European (assuming he is not Dutch or French)D = F.M. does not like the EU since he looks like a person who does not like the EU

E

E = Alice and Bob are unreliable since they contradict each other

Multiple labellings

A B

C

D

A B

C

D

Download - Argumentation Logics Lecture 1: Introduction

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