Argumentation Logics Lecture 1: Introduction

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Argumentation Logics Lecture 1: Introduction. Henry Prakken Chongqing May 26, 2010. Nonmonotonic logic. Standard logic is monotonic : If S |- and S S then S |- But commonsense reasoning is often non monotonic: - PowerPoint PPT Presentation

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  • Argumentation LogicsLecture 1:IntroductionHenry PrakkenChongqingMay 26, 2010

  • Nonmonotonic logicStandard 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 employedBut suppose also that John is a student and students are usually not employed

    We often reason with rules that have exceptionsWe apply the general rule if we have no evidence of exceptionsBut must retract our conclusion if we learn evidence of an exception

  • Sources of nonmonotonicityEmpirical generalisationsAdults are usually employed, birds can typically fly, Chinese usually do not like coffee, Conflicting information sourcesExperts who disagree, witnesses who contradict each other, conflicting sensory input, Alternative explanationsThe grass is wet so it has rained / but the sprinkler was onConflicting goalsWe should raise taxes to increase productivity, which is good / but lower taxes increase inequality, which is badExceptions to legal rulesWhen a father dies, his son can inherit, except when the son killed the fatherExceptions to moral principlesNormally one should not lie, except when a lie can save lives

  • Some nonmonotonic logicsDefault logic (Ray Reiter)Circumscription (John McCarthy)Logic programming (Robert Kowalski)Argumentation logics

  • Argumentation as a nonmonotonic logicNonmonotonic logic deals with:Rules and exceptionsConflicts and their resolutionBoth can be modelled as argumentation:General rule gives rise to argument, exception gives rise to counterargumentException defeats general ruleConflicts give rise to argument and counterargumentConflicts are resolved with preferences

  • Some historyJohn Pollock (1987-1995)Ron Loui (1987)With Guillermo Simari (1992)Gerard Vreeswijk (1993,1997)Phan Minh Dung (1995)

  • We should lower taxesLower taxes increase productivityIncreased productivity is good

  • We should lower taxesLower taxes increase productivityIncreased productivity is goodWe should not lower taxesLower taxes increase inequalityIncreased inequality is bad

  • We should lower taxesLower taxes increase productivityIncreased productivity is goodWe should not lower taxesLower taxes increase inequalityIncreased inequality is badLower taxes do not increase productivityUSA lowered taxes but productivity decreased

  • We should lower taxesLower taxes increase productivityIncreased productivity is goodWe should not lower taxesLower taxes increase inequalityIncreased inequality is badLower taxes do not increase productivityProf. P says that USA lowered taxes but productivity decreased

  • We should lower taxesLower taxes increase productivityIncreased productivity is goodWe should not lower taxesLower taxes increase inequalityIncreased inequality is badLower taxes do not increase productivityProf. P says that Prof. P has political ambitionsPeople with political ambitions are not objective Prof. P is not objectiveUSA lowered taxes but productivity decreased

  • We should lower taxesLower taxes increase productivityIncreased productivity is goodWe should not lower taxesLower taxes increase inequalityIncreased inequality is badLower taxes do not increase productivityProf. P says that Prof. P has political ambitionsPeople with political ambitions are not objective Prof. P is not objectiveUSA lowered taxes but productivity decreased

  • We should lower taxesLower taxes increase productivityIncreased productivity is goodWe should not lower taxesLower taxes increase inequalityIncreased inequality is badLower taxes do not increase productivityProf. P says that Prof. P has political ambitionsPeople with political ambitions are not objective Prof. P is not objectiveIncreased inequality is goodIncreased inequality stimulates competitionCompetition is goodUSA lowered taxes but productivity decreased

  • We should lower taxesLower taxes increase productivityIncreased productivity is goodWe should not lower taxesLower taxes increase inequalityIncreased inequality is badLower taxes do not increase productivityProf. P says that Prof. P has political ambitionsPeople with political ambitions are not objective Prof. P is not objectiveIncreased inequality is goodIncreased inequality stimulates competitionCompetition is goodUSA lowered taxes but productivity decreased

  • ABCDE

  • Overview of this courseAbstract argumentation (Lectures 1-4)Semantics (Lectures 1-3)Labelling-basedExtension-basedArgument 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,DefeatOUTPUT: 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 conditionsEvery 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:

    ABCAB

  • Two solutionsChange conditions so that always a unique status assignment results

    Use multiple status assignments:

    and

    ABCABABABCAB

  • Unique status assignments: Grounded semantics (informal)The endpoint (or union) of a sequence s.t.:S0: the empty setSi+1: Si + all arguments acceptable wrt Si...

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

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

  • Grounded semantics (formal 1)Let AAT be an abstract argumentation theoryF0AAT = Fi+1AAT = {A Args | A is acceptable wrt FiAAT}FAAT = i=0 (Fi+1AAT) Problem: does not always contain all intuitively justified arguments.

  • Grounded semantics (formal 2)Let AAT = Args,Defeat and S ArgsFAAT(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 FAATAn 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 justifiedIf every argument has at most a finite number of defeaters, then A FAT A is justified

  • Acceptability status with unique status assignmentsA is justified if A is InA is overruled if A is Out and A is defeated by an argument that is InA is defensible otherwise

  • Self-defeating argumentsIntuition: should always be overruled (?)Problem: in grounded semantics they are not always overruledSolution: several possibilities (but intuitions must be refined!)

  • A problem(?) with grounded semanticsWe have: We want(?):

    ABCDABCD

  • A problem(?) with grounded semantics

    ABCDA = 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

    ABCDA = 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 EUEE = Alice and Bob are unreliable since they contradict each other

  • Multiple labellingsABCDABCD

    ******Mention the special case of deductive arguments, which can only be attacked on their premises.***So far we have illustrated:Arguments have structure: premises, a conclusion, and an inference (I will not tell you yet what inference rules licence the inference)So arguments can be attacked in three ways.We can resolve attacks on the basis of our preference, to obtain defeat relationsArguments can be complex (so premise attack often is attack on subconclusions)But what we have not yet seen is how inferences can be drawn from all this. Now Dungs beautiful insight was that inference can be studied by fully abstracting from the structure of arguments and the nature of defeat. *Letters are NOT propositions!Arrows are defeat relations. A typical case of mutual defeat is equally strong arguments with contradictory conclusions.Two typical cases of strict defeat:- arguments with contradictory conclusions, one of which is stronger.An undercutter

    NB: We could leave B and E both white!*NB: Dung does not use a labelling approach.

    Reinstatement: think of Tweety can fly, since it is a bird, Tweety cannot fly since it was observed as a penguin so it is a penguin, the observation was done with faulty instruments.

    **With multiple assignments we could say that an argument is really acceptable if it is in in all assignments.*Examples: after following slide*This example: first ask which are acceptable wrt the empty set.

    Other examples of acceptability: A

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