Foundations of Fuzzy Logic and Semantic Web Languages

178 Foundations of Fuzzy Logic and Semantic Web Languages

Remark 23. Like for the fuzzy FOL case, for which ∀ and ∃ are not comple-

mentary in general (see Remark 19), also for fuzzy DLs we have that ∀R.C

and ¬∃R.¬C are not, unlike the classical case, equivalent. However, they are

equivalent under Lukasiewicz logic and SFL.

Remark 24 (On Number Restrictions). In [49] it has been pointed out that,

unlike the classical case (see Section 4.2.2.1), in the fuzzy case we do not have

in general that (≤ n R.C) ≡ ¬(≥ n + 1 R.C) holds.

For instance, in Lukasiewicz logic assume the following interpretation:

((tom, apple): likes)

I

= ((tom, banana) : likes)

I

= ((tom, orange): likes)

I

= ((tom, peach) : likes)

I

= 0.5

(apple:F ruit)

I

= (banana:F ruit)

I

= (orange : F