Foundations of Fuzzy Logic and Semantic Web Languages

294 Foundations of Fuzzy Logic and Semantic Web Languages

the following relationships hold:

P oss(φ ∧ ψ) ≤ min(P oss(φ), P oss(ψ)) ;

P oss(φ ∨ ψ) = max(P oss(φ), P oss(ψ)) ;

P oss(¬φ) = 1 − N ec(φ) ;

P oss(⊥) = 0 ;

P oss(>) = 1 (in the normalized case);

Nec(φ ∧ ψ) = min(Nec(φ), Nec(ψ)) ;

Nec(φ ∨ ψ) ≥ max(Nec(φ), Nec(ψ)) ;

Nec(¬φ) = 1 − P oss(φ) ;

Nec(⊥) = 0 (in the normalized case);

Nec(>) = 1 .

(E.6)

A possibilistic interpretation π satisfies a possibilistic formula hPφ, ni (resp.,

hNφ, ni), or π is a model of hPφ, ni (resp., hNφ, ni), denoted π |= hPφ, ni (resp.,

π |= hNφ, ni), iff P oss(φ) ≥n (resp., Nec(φ) ≥n). The notions of satisfiability,

logical consequence, tight logical consequence and best entailment degree for

possibilistic knowledge bases are then ...