By David Seabury
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O = CN(KS ). t. sc(O ) < sc(O). t. O = CN(KS ). Since the objective function of LP(K ) corresponds to sc, then S does not minimize the objective function, which is a contradiction. 2. Suppose that O ∈ Opt P (K , WP ). t. O = CN(KS ). Suppose by contradiction that S is not a solution of LP(K ). This means that it does not minimize the objective function. Then, there is a solution S of LP(K ) which has a lower value of the objective function. As shown before, O = CN(KS ) is an option and has a score lower than O, which is a contradiction.
1. Let T = (Arg(K ), Ru ) be an AS over the base K . • ∀E ∈ Ext(T ), ∃O ∈ Opt (K ) such that O = CN( • ∀O ∈ Opt (K ), ∃E ∈ Ext(T ) such that O = CN( Supp(a)). Supp(a)). a∈E a∈E Proof. Straightforward. 1 An AS is finite iff each argument is attacked by a finite number of arguments. It is infinite otherwise. Chapter 6 Conclusions Past works on reasoning about inconsistency in AI have suffered from multiple flaws: (i) they apply to one logic at a time and are often invented for one logic after another.
3. 5. Then, • ∀S ∈ P2 (K ), ∃O ∈ Opt (K ) such that O = CN(S). • ∀O ∈ Opt (K ), ∃S ∈ P2 (K ) such that O = CN(S). Proof. Straightforward. Brewka (1989) provides a weak and strong notion of provability for both the generalizations described above. t. ψ ∈ CN(S); ψ is strongly provable from K iff for every preferred subbase S of K we have ψ ∈ CN(S). 8), whereas the former is not a valid inference mechanism, since the set of weakly provable formulas might be inconsistent. Observe that Brewka’s approach is committed to a specific logic, weakening mechanism and preference criterion, whereas our framework is applicable to different logics and gives the flexibility to choose the weakening mechanism and the preference relation that the end-user believes more suitable for his purposes.
The Art of Selfishness by David Seabury