1. Henry Prakken February 23, 2018
Commonsense Reasoning and Argumentation 17/18 HC 6 Abstract Argumentation Semantics (2)
Henry Prakken
February 23, 2018

2. Overview
(Self-defeat)
Default logic as argumentation

3. 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

4. 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

5. 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