Henry Prakken February 23, 2018 Commonsense Reasoning and Argumentation 17/18 HC 6 Abstract Argumentation Semantics (2) Henry Prakken February 23, 2018
Overview (Self-defeat) Default logic as argumentation
Self-defeating arguments Grounded semantics: A is justified if A is In in the grounded s.a. A is overruled if is Out in the grounded s.a. A is defensible if A is undecided in the grounded s.a. Preferred semantics: A is justified if A is In in all preferred s.a. A is overruled if A is Out or undecided in all preferred s.a. A is defensible if A is In in some but not all preferred s.a. In grounded and preferred semantics self-defeating arguments can prevent other arguments from being justified or defensible They can make that there are no stable extensions Now give A defeats A and B Then give A defeats A and A <-> B 3
Default-logic argumentation theories Every default theory has an associated abstract argumentation theory AF () as follows: Args = {| is a finite process of } A defeats B iff In(A) for some Out(B) Show that T: A / A yields a self-defeating argument. Show other example: d1: T:a/b, d2: b:-a/-a. Three arguments empty, d1 and d1,d2. d1,d2 defeats d1 and itself. If time, then do exercise: W = {a,b}, d1: a:c/c, d2: b: -c/d. Yields 4 arguments: any argument with d1 defeats any argument with d2. 4
Relation with stable semantics Notation (for a given default theory ): For every set S Args: Concs (S) = { | In (A) for some A S} For every set E of wffs: Args (E) = {A Args| for all k Out (A): E |- k} Proposition: For every default theory : If S is a stable extension of AF () then Concs (S) is a Reiter-extension of If E is a Reiter-extension of then Args (E) is a stable extension of AF () Intuition of first bullet: glue all arguments together into one giant process. Successful since S is conflict-free. Closed since S attacks any argument outside it. Intuition of second bullet: Chop the generating process into pieces. Conflict-free since In and out do not overlap. Stable since process is closed. 5