Propositional Argumentation Systems and Symbolic Evidence Theory

26 3 SYMBOLIC THEORY OF HINTS
Example 3.4 (Peter’s Secret Love): Let us look again at Peter’s love story from
Example 3.1. Peter wants to judge the hypothesis H = {♥} (i.e. “Susan loves Peter
too”) in the light of the given hint. From (3.3) to (3.8) he gets:
qs({♥}, H) = {ω 1 }, (3.10)
sp({♥}, H) = {ω 1 }, (3.11)
db({♥}, H) = {ω 3 }, (3.12)
pl({♥}, H) = {ω 1 , ω 2 }. (3.13)
In the framework of hints the basic arguments of a hypothesis H are simply
the interpretations ω whose focal sets Γ(ω) equal H. The basic–support
of H relative to a hint H is the set all basic arguments of H:
m(H, H) = {ω ∈ Ω : Γ(ω) = H}. (3.14)
Note that m(H, H) satisfies the properties (M1) to (M4) of Section 2.4,
(M1) m(H 1 , H) ∩ m(H 2 , H) = Ø, for all H 1 , H 2 ⊆ Θ, H 1 ≠ H 2 ,
⋃
(M2) m(H, H) = Ω.
H⊆Θ
and that it can be obtained from quasi–support (see Equation 2.13):
m(H, H) = qs(H, H) −
(⋃ { qs(H ′ , H) : H ′ ⊂ H }) . (3.15)
From (2.16) to (2.19) we know that basic–support determines quasi–support,
support, doubt, and plausibility:
qs(H, H) = ⋃ {m(H ′ , H) : H ′ ⊆ Θ, H ′ ⊆ H}, (3.16)
sp(H, H) = ⋃ {m(H ′ , H) : H ′ ⊆ Θ, H ′ ⊆ H, H ′ ≠ Ø}, (3.17)
db(H, H) = ⋃ {m(H ′ , H) : H ′ ⊆ Θ, H ′ ∩ H = Ø}, (3.18)
pl(H, H) = ⋃ {m(H ′ , H) : H ′ ⊆ Θ, H ′ ∩ H ≠ Ø}. (3.19)
As basic–support can immediately be derived from the given focal sets,
m(H, H) will be the most convenient represention for the information contained
in the hint. Here, it is therefore not necessary to define the pseudo–
support for hints as an alternative representation.