Propositional Argumentation Systems and Symbolic Evidence Theory
Propositional Argumentation Systems and Symbolic Evidence Theory
Propositional Argumentation Systems and Symbolic Evidence Theory
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2.2 Quasi–Support <strong>and</strong> Support 9<br />
The quasi–support qs is determined by the given knowledge or information.<br />
It defines a mapping qs : H −→ A called allocation of quasi–support for<br />
which the following conditions are assumed (Kohlas, 1995):<br />
(Q1) qs(▽) = ▽ (the first ▽ is in H, the second in A),<br />
(Q2) qs(h 1 ⊓ h 2 ) = qs(h 1 ) ⊓ qs(h 2 ),<br />
(Q3) if h 1 ⊑ h 2 , then qs(h 1 ) ⊑ qs(h 2 ),<br />
(Q4) qs(h 1 ⊔ h 2 ) ⊒ qs(h 1 ) ⊔ qs(h 2 ).<br />
(Q1) says that every argument is an argument for the certain hypothesis.<br />
(Q2) means that if a is an argument for the meet h 1 ⊓ h 2 , then a is also<br />
an argument for both h 1 <strong>and</strong> h 2 <strong>and</strong> vice versa. (Q3) <strong>and</strong> (Q4) result from<br />
(Q1) <strong>and</strong> (Q2).<br />
A triple B = (H, A, qs), where H <strong>and</strong> A are finite Boolean algebras of hypotheses<br />
<strong>and</strong> arguments <strong>and</strong> qs is an allocation of quasi–support satisfying<br />
(Q1) to (Q4), is called a body of arguments. This simple algebraic structure<br />
is the basic formal model of the (symbolic) theory of evidence, from<br />
which all other concepts can be derived. It represents the available evidence<br />
allowing to prove certain hypotheses h relative to a given knowledge<br />
or information.<br />
From the definition of qs it cannot be excluded that qs(△) is different from<br />
△. The argument qs(△), which supports the impossible hypothesis △,<br />
is called contradiction. Arguments a ⊑ qs(△) are in contradiction to<br />
the given knowledge <strong>and</strong> they cannot really be true. Such contradictory<br />
arguments should be eliminated from possible arguments. Thus, we define<br />
a new set of proper arguments<br />
A ′ = {a ⊓ qs ❁ (△) : a ∈ A} = A ⊓ qs ❁ (△), (2.1)<br />
which is still a finite Boolean algebra. The minimal element △ ′ of A ′ is<br />
identical to the minimal element △ of A, <strong>and</strong> the maximal element ▽ ′ of A ′<br />
equals qs ❁ (△). SP (h) ⊆ A denotes the set of proper arguments in favour<br />
of h. SP (h) is determined by<br />
SP (h) = QS(h) − QS ❁ (△), (2.2)<br />
which is no longer a downward–set, <strong>and</strong> can therefore not be represented by<br />
a unique maximal element. In fact, SP (h) is a convex space with an upper
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