Pollock s undercutting defeat

Transcription

1 Argumentation in Artificial Intelligence, With Alications in the Law Course at the Institute of Logic and Cognition, Sun Yat-Sen University Ib Abstract Argumentation and Argument Structure Bart Verheij CodeX, Stanford University Artificial Intelligence, University of Groningen IA Introduction Toics: Argumentation in Artificial Intelligence Historical Background Goals: Get an overview of the course and its subject matter Acquire insight about the historical background Literature: Van Eemeren et al. (in rearation). Sections IB Abstract Argumentation, Argument Structure Pollock s undercutting defeat Toics: Abstract Argumentation Argument Structure q q Goals: Acquire knowledge of abstract argumentation and its semantics Acquire insight into the relation between argument structure and abstract argumentation r Literature: Van Eemeren et al. (in rearation). Sections q is warranted by the argument from q is not warranted by the argument from A hilosohical uzzle (Pollock) Pollock s research question q q r r ( q) is a rima facie reason for q q is a rima facie reason for r r is an excetion that undercuts the suort of q by How is argumentative warrant determined by the structure of the available arguments and counterarguments? He roduced a series of roosals, amongst other things driven by hilosohical uzzles. Is q warranted by the argument from? 1

2 ung 1995 The attack relation as a directed grah (ung) On the accetability of arguments and its fundamental role in non-monotonic reasoning, logic rogramming and n-erson games Artificial Intelligence journal Pollock s research question, revisited Pollock s research question, revisited How is argumentative warrant determined by the structure of the available arguments and counterarguments? How is argumentative warrant determined by the structure of the available arguments and counterarguments? of the attack relation between arguments? - Mathematically clean - More abstract, so simler structure ung s basic rincile of argument accetability ung s basic rincile of argument accetability The one who has the last word laughs best. The one who has the last word laughs best. 2

3 ung s basic rincile of argument accetability ung s basic rincile of argument accetability The one who has the last word laughs best. The one who has the last word laughs best. Admissible sets Admissible sets ζ α β δ η A set of arguments A is admissible if 1. it is conflict-free: There are no arguments α and β in A, such that α attacks β. γ ε 2. the arguments in A are accetable with resect to A: For all arguments α in A, such that there is an argument β that attacks α, there is an argument γ in A that attacks β. Admissible, e.g.: {α, γ}, {α, γ, δ, ζ, η} Not admissible, e.g.: {α, β}, {γ} ung s referred and stable extensions Admissible sets An admissible set of arguments is a referred extension if it is an admissible set that is maximal with resect to set inclusion. A conflict-free set of arguments is a stable extension if all arguments that are not in the set are attacked by an argument in the set. ζ α β γ δ η ε Preferred and stable extension: {α, γ, δ, ζ, η} 3

4 Even-length attack cycles Odd-length attack cycles α β α 1 α 3 α 2 Preferred extensions: (the emty set) Preferred and stable extensions: {α}, {β} Stable extensions: none Basic roerties of ung s extensions A stable extension is a referred extension, but not the other way around. An attack relation always has a referred extension. Not all attack relations have a stable extension. An attack relation can have more than one referred/stable extension. A well-founded attack relation has a unique stable extension. ung s grounded and comlete extensions A set of arguments is a comlete extension if it is an admissible set that contains all arguments of which all attackers are attacked by the set. A set of arguments is a (the) grounded extension if it is a minimal comlete extension. ung s four semantics Labelings Preferred Stable Comlete Grounded 4

5 Labelings Stable labeling: Stages α η An argument α is labelled efeated if and only if There is an argument β that attacks α and that is labelled Justified. β δ ζ γ ε Stages, e.g.: β (γ), α (β) γ, α (β) γ δ (ε) ζ η Non-stages, e.g.:β γ, β (δ ε) Labelings 1. Using labelings instead of sets simlifies the formal analysis and increases its transarency. 2. Labelings allow a new natural idea of maximal interretation: maximize the set of labeled nodes. Stage extensions Semi-stable semantics A set of arguments is a semi-stable extension if it is an admissible set, for which the union of the set with the set of arguments attacked by it is maximal. Notion introduced by Verheij (1996) Term coined by Caminada (2006) 3. Some referred extensions are better than others. Semi-stable extensions Verheij (1996). Two Aroaches to ialectical Argumentation: Admissible Sets and Argumentation Stages. Proerties 1. Stable extensions are semi-stable. 2. Semi-stable extensions are referred. 3. Preferred extensions are not always semi-stable. 4. Semi-stable extensions are not always stable. Preferred extensions always exist, but stable extensions do not. o all attack grahs have a semi-stable extension? Answered negatively by Verheij (2000, 2003) Proerties 1. There exist attack grahs without a semi-stable extension. 2. Finite attack grahs always have a semi-stable extension. 3. An attack grah with a finite number of referred extensions has a semi-stable extension. 4. An attack grah with a stable extension has a semi-stable extension. 5. If an attack grah has no semi-stable extension, then there is an infinite sequence of referred extensions with strictly increasing ranges. 5

6 Abstract argumentation semantics (1995) Stable extension Abstract argumentation semantics (1996) Stable extension Semi-stable extension Stage extension Grounded extension Preferred extension Grounded extension Preferred extension Comlete extension ung 1995 Comlete extension ung 1995 Verheij 1996 Pollock s research question, revisited How is argumentative warrant determined by the structure of the available arguments and counterarguments? of the attack relation between arguments? What haens if we add structure? Not just attack, also suort - Mathematically clean - More abstract, so simler structure - Philosohically still comlex Secificity Conclusive force 1. Conflict by inconsistency 2. efeat by secificity 1.Conflict by inconsistency 2. efeat by conclusive force Simari & Loui 1992 Vreeswijk

7 Combining suort and attack Aroach 1: ung s abstract arguments have internal structure Combining suort and attack Aroach 2: Arguments can attack or suort Abstract version: ASPIC+ Prakken 2010 Nute 1994, eflog Verheij 2003 Arguing about suort and attack eflog Verheij 2003 ArguMed software Verheij 2003 eflog A conditional ~> that validates Modus onens A connective that exresses negation as defeat (dialectical negation) ro: ϕ ~> ψ con: ϕ ~> ψ warrant: ϕ ~> (ψ ~> χ) undercutter: ϕ ~> (ψ ~> χ) rebutter: ((ϕ ~> ψ) ϕ) ~> (χ ~> not-ψ) or ψ ~> (χ ~> not-ψ) eflog Attack I (no warrants) J conflict-free attacked by J naturalized American A defeasible theory is divided in a justified art J and a defeated art. 7

8 Attack I (no warrants) Attack I (no warrants) C ~> C C R naturalized American R ~> x( ~> C) R naturalized American Attack I (no warrants) Attack I (no warrants) ~> C C ~> C C R ~> x( ~> C) R naturalized American R ~> x( ~> C) R naturalized American = { ~> C, R ~> x( ~> C), R, } = { ~> C, R ~> x( ~> C), R, } eflog Undercutting & rebutting 's closure under Modus onens J xϕ ϕ Some assumtions are attacked Some artitions are stable Undercutting-1: Attacking the connection between a reason and its conclusion Undercutting-2: Attacking an assumtion of an argument Rebutting: Attacking by giving a reason against a conclusion 8

9 Undercutting-1 in eflog Attacking a conditional assumtion Undercutting-2 in eflog Passim : ~> q e ~> x( ~> q) e J J J q e Rebutting in eflog Side-ste: No stable extension in eflog Kevin testifies that he saw Peter assaulting Jack Rebutting in eflog Modelled using undercutting-1 Kevin testifies that he saw Peter assaulting Jack 9

10 Wait a minute: we didn't get the oosite conclusion! not Kevin testifies that he saw Peter assaulting Jack Kevin testifies that he saw Peter assaulting Jack [Negation-as-contradiction vs. negation-as-defeat] Attack II (with warrants) in Bermuda naturalized American A man born will generally be a British subject Harry has become a naturalized American A man born will generally be a British subject Rebutting in eflog Also the weighing of reasons (cf. Reason-Based Logic) can be modeled using undercutting-1. Miriam testifies that s Kevin testifies that he saw Peter assaulting Jack 10

11 Abstract argumentation semantics (1996) not Stable extension Miriam testifies that s did not assault Jack Kevin testifies that he saw Peter assaulting Jack Grounded extension Semi-stable extension Preferred extension Comlete extension Stage extension ung 1995 Verheij 1996 Argumentation semantics (2003) Argumentation semantics (2003) Stage Stable Stable Semi-stable Preferred eflog Verheij 2003 eflog Verheij 2003 Pollock s research question, revisited We return to the original version (attack + suort) : How is argumentative warrant determined by the structure of the available arguments and counterarguments? - Mathematically clean (still) - Less abstract, less simle structure - Philosohically very comlex Argumentation in Artificial Intelligence, With Alications in the Law Course at the Institute of Logic and Cognition, Sun Yat-Sen University Ib Abstract Argumentation and Argument Structure Bart Verheij CodeX, Stanford University Artificial Intelligence, University of Groningen