Part 1: Computing acceptability degrees

Let us consider the following ranking-based semantics. Let $\mathcal{F}= (A,R)$ be an AF:

1. $\forall a \in A, Deg(a) = \frac{1}{1+ \sum_{x \in Att(a)} Deg(x)}$
2. $\forall a \in A, Deg(a) = \frac{1}{1+ \max_{x \in Att(a)} Deg(x)}$
3. $\forall a \in A, Deg(a) = \frac{1}{1+ |Att(a)| + \frac{\sum_{x \in Att(a)} Deg(x)}{|Att(a)|}}$

For the graph below, compute the acceptability degrees of all arguments. Then, rank the arguments from the most acceptable to the less.

• Consider the AF $\mathcal{F}=(A,R)$ with $A = \{a,p\}$ and $R= \{(p,a)\}$.

  

  H-categoriser/Max-based

  $p>a$

  

  Card-based

  $p>a$

  

• Consider the AF $\mathcal{F}=(A,R)$ with $A = \{q,r,s,b\}$ and $R= \{(q,b),(r,b),(s,b)\}$.

  

  H-categoriser

  $q,s,r, > b$

  

  Max-based

  $q,s,r, > b$

  

  Card-based

  $q,s,r, > b$

  

• Consider the AF $\mathcal{F}=(A,R)$ with $A = \{p,q,a,b\}$ and $R= \{(p,q), (q,a)\}$.

  

  H-categoriser/Max-based

  $b,p > a > q$

  

  Card-based

  $b,p > a > q$

  

• Consider the AF $\mathcal{F}=(A,R)$ with $A = \{t,p,q,a\}$ and $R= \{(t,q), (t,p), (p,a), (q,a)\}$.

  

  H-categoriser

  $t > p,q,a$

  

  Max-based semantics

  $t > a>p,q$

  

  Card-based

  $t > p,q > a$

  

• Consider the AF $\mathcal{F}=(A,R)$ with $A = \{v,x,y,z,r,s,b\}$ and $R= \{(v,r),(x,r),(x,s), (y,s), (z,s), (r,b), (s,b)\}$.

  

  H-categoriser

  $v,y,x,z > b > r > s$

  

  Max-based

  $v,y,x,z > b > r , s$

  

  Card-based

  $v,y,x,z > b > r > s$

  

• Consider the AF $\mathcal{F}=(A,R)$ with $A = \{a_1, a_2, \dots, a_{10}\}$ and $R= \{(a_{10},a_9), (a_9, a_{10}), (a_9, a_8), (a_8, a_7), (a_8, a_4), (a_6, a_5), (a_5, a_4), (a_2, a_1), (a_1, a_2), (a_2, a_3), (a_1,a_3), (a_3, a_4)\}$

  

  H-categoriser

  $a_6 > a_1, a_2, a_{10}, a_9, a_8, a_7 > a_5 > a_3 > a_4$

  

  Max-based

  $a_6 > a_1, a_2, a_{10}, a_9, a_8, a_7 , a_3 , a_4>a_5$

  

  Card-based

  $a_6 > a_1, a_2, a_{10}, a_9, a_8, a_7 > a_5 > a_3 > a_4$

  

Part 2: Identifying fallacies

Let us consider the following situations. In each of them, identify whether or not there are logical fallacies, and if yes, which ones?