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 lower taxes
Lower taxes increase productivity
Increased productivity is good
We should not lower taxes
Lower taxes increase inequality
Increased inequality is bad
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
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
Prof. P says that …
USA lowered taxes but productivity decreased
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
Prof. P says that …
Prof. P has political ambitions
People with political ambitions are not objective
Prof. P is not objective
USA lowered taxes but productivity decreased
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
Prof. P says that …
Prof. P has political ambitions
People with political ambitions are not objective
Prof. P is not objective
USA lowered taxes but productivity decreased
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
Prof. P says that …
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
USA lowered taxes but productivity decreased
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
Prof. P says that …
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
USA lowered taxes but productivity decreased
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 problem(?) with grounded semantics
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 problem(?) with grounded semantics
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
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