Propositional Argumentation Systems and Symbolic Evidence Theory

42 4 ASSUMPTION–BASED SYSTEMS
(D4) db[h 1 ∧ h 2 , Σ] =| db[h 1 , Σ] ∨ db[h 2 , Σ].
A formula a ∈ L A is called a possible argument of h if it is not a refuting
argument of h, i.e. if a ∧ ξ ̸|= ∼h. If a is a possible argument of h, and if
there is no other possible argument a ′ of h with a |= a ′ and a ′ ≠ a, then the
equivalence class [a] of a is called plausibility of h relative to Σ, denoted
by pl[h, Σ]. It corresponds to the negated quasi–support of the negated
hypothesis,
pl[h, Σ] = ∼qs[∼h, Σ], (4.16)
qs[h, Σ] = ∼pl[∼h, Σ], (4.17)
and it satisfies the conditions (P1) to (P4) of Chapter 2:
(P1) pl[⊥, Σ] = [⊥],
(P2) pl[h 1 ∨ h 2 , Σ] = pl[h 1 , Σ] ∨ pl[h 2 , Σ],
(P3) if h 1 |= h 2 , then pl[h 1 , Σ] |= pl[h 2 , Σ],
(P4) pl[h 1 ∧ h 2 , Σ] |= pl[h 1 , Σ] ∧ pl[h 2 , Σ].
If a formula a ∈ L A is either a supporting argument, a proper argument, a
refuting argument, or a possible argument of a hypotheses h, then it is called
a symbolic argument of h. Thus, L A is the set of all possible symbolic
arguments.
Example 4.3 (Michael’s Murderer): Consider the example of the previous chapter,
which treats the question: “Who is Michael’s murderer?”. There are three possible
answers: the two suspects Mr. Tanner and J.O. Simpson, or somebody else.
The available information about the crime is encoded in an assumption–based system,
and it will be interesting to find arguments in favour or against the suspect’s
innocence.
Table 4.2 summarizes the arguments for the hypotheses s (Simpson is the murderer),
t (Tanner is the murderer), e (somebody else is the murderer), s ∧ t (Simpson and
Tanner are both murderers), and s ∨ t (one of the two suspects is the murderer).
An argument [a] is represented by ψ(a), that is by the disjunction of its prime
implicants.
In general, supports are the most interesting arguments to be considered. For
example, the support of s is [a 1 ∧ ∼a 3 ∧ ∼a 4 ], i.e. Simpson is responsible for the
crime exactly if Mr. Tanner says the truth, J.O. Simpson lies, and the footprints are