Argumentation Logics Lecture 1 Introduction Henry Prakken Chongqing

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Argumentation Logics Lecture 1: Introduction Henry Prakken Chongqing May 26, 2010

Argumentation Logics Lecture 1: Introduction Henry Prakken Chongqing May 26, 2010

Nonmonotonic logic n Standard logic is monotonic: n n But commonsense reasoning is often

Nonmonotonic logic n Standard logic is monotonic: n n But commonsense reasoning is often nonmonotonic: n n n If S |- and S S’ then S’ |- 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 n Empirical generalisations n n Conflicting information sources n n When

Sources of nonmonotonicity n Empirical generalisations n n Conflicting information sources n n When a father dies, his son can inherit, except when the son killed the father Exceptions to moral principles n n We should raise taxes to increase productivity, which is good / but lower taxes increase inequality, which is bad Exceptions to legal rules n n The grass is wet so it has rained / but the sprinkler was on Conflicting goals n n Experts who disagree, witnesses who contradict each other, conflicting sensory input, … Alternative explanations n n Adults are usually employed, birds can typically fly, Chinese usually do not like coffee, … … Normally one should not lie, except when a lie can save lives

Some nonmonotonic logics n n n Default logic (Ray Reiter) Circumscription (John Mc. Carthy)

Some nonmonotonic logics n n n Default logic (Ray Reiter) Circumscription (John Mc. Carthy) Logic programming (Robert Kowalski) … Argumentation logics

Argumentation as a nonmonotonic logic n Nonmonotonic logic deals with: n n n Rules

Argumentation as a nonmonotonic logic n Nonmonotonic logic deals with: n n n Rules and exceptions Conflicts and their resolution Both can be modelled as argumentation: n General rule gives rise to argument, exception gives rise to counterargument n n Exception defeats general rule Conflicts give rise to argument and counterargument n Conflicts are resolved with preferences

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

Some history n n John Pollock (1987 -1995) Ron Loui (1987) n n 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 lower taxes Lower taxes increase productivity Increased productivity is good We should

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 We should not lower taxes Increased

We should lower taxes Lower taxes increase productivity We should not lower taxes Increased productivity is good Lower taxes do not increase productivity USA lowered taxes but productivity decreased Lower taxes increase inequality Increased inequality is bad

We should lower taxes Lower taxes increase productivity Prof. P says that … We

We should lower taxes Lower taxes increase productivity Prof. P says that … We should not lower taxes Increased productivity is good Lower taxes do not increase productivity USA lowered taxes but productivity decreased Lower taxes increase inequality Increased inequality is bad

We should lower taxes Lower taxes increase productivity Prof. P says that … People

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

We should lower taxes Lower taxes increase productivity Prof. P says that … People

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

We should lower taxes Lower taxes increase productivity Prof. P says that … People

We should lower taxes Lower taxes increase productivity Prof. P says that … People with political ambitions are not objective We should not lower taxes Increased productivity is good Prof. P is not objective Prof. P has political ambitions Lower taxes increase inequality Increased inequality is good Lower taxes do not increase productivity USA lowered taxes but productivity decreased Increased inequality is bad Increased inequality stimulates competition Competition is good

We should lower taxes Lower taxes increase productivity Prof. P says that … People

We should lower taxes Lower taxes increase productivity Prof. P says that … People with political ambitions are not objective We should not lower taxes Increased productivity is good Prof. P is not objective Prof. P has political ambitions Lower taxes increase inequality Increased inequality is good Lower taxes do not increase productivity USA lowered taxes but productivity decreased Increased inequality is bad Increased inequality stimulates competition Competition is good

A C B D E

A C B D E

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

Overview of this course n Abstract argumentation (Lectures 1 -4) n Semantics (Lectures 1 -3) n n Labelling-based Extension-based Argument games (Lecture 4) Rule-based argumentation (Lectures 5 -7) n n n Structure of arguments, (Lecture 5) Attack, defeat, preferences (Lecture 6) Self-defeat, rationality postulates (Lecture 7)

Status of arguments: abstract semantics (Dung 1995) n n INPUT: an abstract argumentation theory

Status of arguments: abstract semantics (Dung 1995) n n INPUT: an abstract argumentation theory AAT = Args, Defeat OUTPUT: An assignment of the status ‘in’ or ‘out’ to all members of Args n n So: semantics specifies conditions for labeling the ‘argument graph’. Should capture reinstatement: A B C

Possible labeling conditions n Every argument is either ‘in’ or ‘out’. 1. An argument

Possible labeling conditions n 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’. n Works fine with: n But not with: A B C

Two solutions n Change conditions so that always a unique status assignment results A

Two solutions n Change conditions so that always a unique status assignment results A B A n A B Use multiple status assignments: A n C B B and C A B

Unique status assignments: Grounded semantics (informal) n The endpoint (or union) of a sequence

Unique status assignments: Grounded semantics (informal) n The endpoint (or union) of a sequence s. t. : n n S 0: 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 n S defeats A if an argument in S defeats A

A C B D E Is B, D or E defended by S 1?

A C B D E Is B, D or E defended by S 1? Is B or E defended by S 2?

Grounded semantics (formal 1) n Let AAT be an abstract argumentation theory n n

Grounded semantics (formal 1) n Let AAT be an abstract argumentation theory n n F 0 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) n Let AAT = Args, Defeat and S Args n

Grounded semantics (formal 2) n Let AAT = Args, Defeat and S Args n n n FAAT(S) = {A Args | A is acceptable wrt S} Since FAAT is monotonic (and since. . . ), FAAT has a least fixed point. Now: n The grounded extension of AAT is the least fixed point of n An argument is (w. r. t. grounded semantics) justified on the basis of AAT if it is in the grounded extension of AAT. FAAT Proposition 4. 2. 4 (AAT implicit): n n 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 n n n A is justified if A

Acceptability status with unique status assignments n n n 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 n n n Intuition: should always be overruled (? ) Problem: in

Self-defeating arguments n n n 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: A B We want(? ): A

A problem(? ) with grounded semantics We have: A B We want(? ): A B C C D D

A problem(? ) with grounded semantics A A = Frederic Michaud is French since

A problem(? ) with grounded semantics A A = Frederic Michaud is French since he has a French name B = Frederic Michaud is Dutch since he is a marathon skater C = 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 B C D

A problem(? ) with grounded semantics E A A = Frederic Michaud is French

A problem(? ) with grounded semantics E A A = Frederic Michaud is French since Alice says so B = Frederic Michaud is Dutch since Bob says so C = 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 = Alice and Bob are unreliable since they contradict each other B C D

Multiple labellings A B C C D D

Multiple labellings A B C C D D