Foundations of Fuzzy Logic and Semantic Web Languages

Fuzzy Sets and Mathematical Fuzzy Logic Basics 151

TABLE 8.8: Mapping of propositional SFL statements to crisp statements.

x y ρ(x, y)

> c >

⊥ 0 >

⊥ c ⊥ if c > 0

A c A

≥c

¬A c ¬A

>1−c

φ ∧ ψ c ρ(φ, c) ∧ ρ(ψ, c)

φ ∨ ψ c ρ(φ, c) ∨ ρ(ψ, c)

Therefore, by relying on Equation (8.60), for c > 0, we have

K |= hφ, ci iff K ∪ {h¬φ, 1 − c

−

i} is not satisfiable , (8.66)

where c

−

is the next smaller value than c in L

n

and, thus, we may apply

Proposition 59 to solve the entailment decision problem. Eventually, the best

entailment degree problem can be reduced to calls to the entailment problem

for various c ∈ L

n

, similarly as described in Section 8.2.2.3.

We next show how the