Multiagent Systems

6. Propositional Logic 4. Calculi for SL
We bear in mind that Cn(W ) is defined semantically but
Cn D (W ) is defined syntactically (using the notion of
proof). Both sets are equal according to the
completeness-theorem in the special case D = ∅.
Lemma 6.16 (Properties of Cn D )
D be a set of general inference rules and W ⊆ F ml. Then:
1 Cn(W ) ⊆ Cn D (W ).
2 Cn D (Cn D (W )) = Cn D (W ).
3 Cn D (W ) is the smallest set which is closed in respect to
D and contains W .
Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 454/731