Computational Argumentation 20202021 HC 3 b Abstract Argumentation

Computational Argumentation 2020/2021, HC 3 b Abstract Argumentation Frameworks: semantics (1) Henry Prakken September 16 th, 2020

Agents use argumentation … n for their internal reasoning n n n Their information may be inconsistent Reasoning about beliefs, goals, intentions etc often is defeasible for their interaction with other agents n Information exchange, negotiation, collaboration, …

We should lower taxes Lower taxes increase productivity Increased productivity is good

We should lower taxes Lower taxes increase productivity We should not lower taxes Increased productivity is good Attack on conclusion Lower taxes increase inequality Increased inequality is bad

We should lower taxes Lower taxes increase productivity We should not lower taxes Increased productivity is good Lower taxes do not increase productivity Attack on premise … 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 … … often becomes attack on intermediate conclusion 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 should not lower taxes Increased productivity is good Prof. P is not objective Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Attack on inference People with political ambitions are not objective Prof. P has political ambitions USA lowered taxes but productivity decreased

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 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 Indirect defence

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

1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. Stable semantics labels all nodes Grounded semantics minimises In labelling Preferred semantics maximises In labelling A C B D E

Overview HC 3 b-14 n Abstract argumentation frameworks HC 3 b-5 n n n Semantics Proof theory (argument games) Structured argumentation frameworks HC 6 -9 a Dynamics and dialogue HC 9 b-11 Legal argumentation HC 12 -13

Status of arguments: abstract semantics (Dung 1995) n INPUT: an abstract argumentation framework AF = Args, Defeat n n n Args is a set (of arguments) Defeat is a binary relation on Args OUTPUT: the justification status of all members of Args

Status assignments (or ‘labellings’) Given an AF = Args, Defeat : n A status assignment of AF assigns to zero or more members of Args either the status In or Out (but not both) such that: 1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. n n Let Undecided = Args / (In Out): A status assignment is stable iff Undecided = (stable semantics) A status assignment is preferred iff In is -maximal (preferred semantics). A status assignment is grounded iff In is -minimal (grounded semantics)

Properties n n n There always exists exactly one grounded labelling There exists at least one preferred labelling Every stable labelling is preferred (but not v. v. ) The set In of the grounded labelling is a subset of the set In of any preferred or stable labelling Every finite AF without defeat cycles has a unique labelling (which is the same in all semantics). . .

1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. Stable s. a. label all nodes Grounded s. a. minimise In labelling Preferred s. a maximise In labelling A B C

1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. Stable s. a. label all nodes Grounded s. a. minimise In labelling Preferred s. a maximise In labelling A B C

1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. Stable s. a. label all nodes Grounded s. a. minimise In labelling Preferred s. a maximise In labelling A B C

1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. Stable s. a. label all nodes Grounded s. a. minimise In labelling Preferred s. a maximise In labelling A B C

1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. Stable s. a. label all nodes Grounded s. a. minimise In labelling Preferred s. a maximise In labelling A B D C

1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. Stable s. a. label all nodes Grounded s. a. minimise In labelling Preferred s. a maximise In labelling A B D C

Difference between grounded and preferred labellings 1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. A B C D A = Merkel is German since she has a German name B = Merkel is Belgian since she is often seen in Brussels C = Merkel is a fan of Oranje since she wears an orange shirt (unless she is German or Belgian) D = Merkel is not a fan of Oranje since she looks like someone who does not like football (Generalisations are left implicit)

The grounded labelling 1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. A B C D

The preferred labellings 1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. A B C C D D

Justification status of arguments n Grounded semantics: n n A is justified iff A is In in the grounded s. a. A is overruled iff is Out in the grounded s. a. A is defensible iff A is undecided in the grounded s. a. Preferred/stable semantics: n n n A is justified iff A is In in all preferred s. a. A is overruled iff A is Out in all preferred s. a. A is defensible iff A is In in some but not all preferred s. a.

Argument status in grounded and preferred semantics Preferred semantics: A and B defensible C overruled D justified Grounded semantics: all arguments defensible A B A B C C C D D D

Next lecture n n n Alternative formulation of AF semantics Self-defeat Default logic as argumentation