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

Material for this course Website: Available at website: Lecture notes Argumentation Logics Answers to exercises Powerpointslides Schedule, reading + exercises Prerequisite knowledge: - Propositional logic - First-order predicate logic - Elementary set theory

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, … 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 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 reasons for actions Normally if we have a reason to do something, we should do it, unless we also have good reasons not to do it. We should raise taxes to increase productivity, which is good / but lower taxes increase inequality, which is bad …

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 ( ) 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

AB 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: ABC

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: ABC AB

Two solutions Change conditions so that always a unique status assignment results Use multiple status assignments: and ABC ABAB ABC AB

Unique status assignments: Grounded semantics, extension- based (informal) Given AAT =  Args,Defeat , A  Args and S  Args: A is acceptable wrt S (or S defends A) if all arguments in Args that defeat A are defeated by S S defeats A if an argument in S defeats A Construct a sequence such that: S0: the empty set Si+1: Si + all arguments in Args that are acceptable wrt Si The endpoint is the grounded extension of AAT

AB 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 F 0 AAT =  F i+1 AAT = {A  Args | A is acceptable wrt F i AAT } F ∞ AAT =  ∞ i=0 (F i+1 AAT ) Problem: does not always contain all intuitively justified arguments.

Grounded semantics (formal 2) Let AAT =  Args,Defeat  and S  Args F AAT (S) = {A  Args | A is acceptable wrt S} Since F AAT is monotonic (and since...), F AAT has a least fixed point. Now: The grounded extension of AAT is the least fixed point of F AAT An argument is (w.r.t. grounded semantics) justified on the basis of AAT if it is in the grounded extension of AAT. Proposition (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 (extension-based) A is justified if A is in the grounded extension A is overruled if A is not justifed and A is defeated by an argument that is justified 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(?): AB C D AB C D

A problem(?) with grounded semantics AB C D 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

Multiple labellings AB C D AB C D

A problem(?) with grounded semantics AB C D 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 E = Alice and Bob are unreliable since they contradict each other