Propositional Argumentation Systems and Symbolic Evidence Theory

4.3 Hypotheses and Arguments in Assumption–Based Systems 41
(Q4) qs[h 1 ∨ h 2 , Σ] =| qs[h 1 , Σ] ∨ qs[h 2 , Σ],
such that a triple ([L N ], [L A ], [qs]) forms a body of arguments.
In many cases qs[⊥, Σ] will be different from [⊥]. If a is an argument such
that [a] |= qs[⊥, Σ], i.e. if a ∧ ξ |= ⊥, then a is in contradiction to the knowledge
base Σ and should be eliminated from quasi–support. The support of
h relative to Σ is defined by
sp[h, Σ] = qs[h, Σ] ∧ ∼qs[⊥, Σ], (4.13)
and an argument a with [a] |= sp[h, Σ] and [a] ̸|= qs[⊥, Σ] is called proper
argument of h. Support satisfies the properties (S1) to (S4) from Chapter 2,
(S1) sp[⊤, Σ] = ∼qs[⊥, Σ],
(S1 ′ ) sp[⊥, Σ] = [⊥],
(S2) sp[h 1 ∧ h 2 , Σ] = sp[h 1 , Σ] ∧ sp[h 2 , Σ],
(S3) if h 1 |= h 2 , then sp[h 1 , Σ] |= sp[h 2 , Σ],
(S4) sp[h 1 ∨ h 2 , Σ] =| sp[h 1 , Σ] ∨ sp[h 2 , Σ],
and a triple ([L N ], [L A ] − {[a] ∈ [L A ] : [a] |= qs[⊥, Σ]}, [qs]) therefore forms
a normalized body of arguments.
It may also be interesting to consider arguments against a hypothesis h ∈
L N , in other words, arguments in favour of ∼h. A formula a ∈ L A is called
a refuting argument of h if a ∧ ξ |= ∼h. If a is a refuting argument of h,
and there is no other refuting argument a ′ of h with a ′ |= a and a ′ ≠ a, then
the equivalence class [a] of a is called doubt in h relative to Σ, denoted by
db[h, Σ]. Doubt is obtained from quasi–support and vice versa:
db[h, Σ] = qs[∼h, Σ], (4.14)
qs[h, Σ] = db[∼h, Σ]. (4.15)
From (Q1) to (Q4) follows that the conditions (D1) to (D4) from he general
evidence theory of Chapter 2 remain valid:
(D1) db[⊥, Σ] = [⊤],
(D2) db[h 1 ∨ h 2 , Σ] = db[h 1 , Σ] ∧ db[h 2 , Σ],
(D3) if h 1 |= h 2 , then db[h 2 , Σ] |= db[h 1 , Σ],