- a fuzzy ABox A is a finite set of fuzzy concept assertion **axioms of the form**
ha:C ≥ αi and ha:C ≤ αi, and fuzzy role assertion **axioms of the form**4 h(a, b):R ≥ αi, where a, b are individual names, C is a concept, R **is a role** and α ∈ [0, **1] is a rational number.**
- a fuzzy TBox T is a finite set of fuzzy general concept inclusion **(GCI)**
axioms hC ⊑ D ≥ αi, where C, D are concepts and α ∈ [0, **1] is a rational**
number.
It is common to write τ as a shorthand for hτ ≥ 1i, ha:C = αi as an abbreviation of the pair of axioms ha:C ≥ αi and ha:C ≤ αi, and the **concept equivalence**
C ≡ D as a shorthand of the two axioms hC ⊑ D ≥ 1i and hD ⊑ C ≥ 1i.

For the sake of concrete illustration, let us introduce a couple of examples of fuzzy KBs. The second one will be used later as a counterexample to **the FMP.**
Example 2.1 The pair K1 = hA1, T1i **with** A1 = {hjim:YoungPerson ≥ 0.2i, h(jim, mary):likes ≥ 0.8i} and T1 = {hInn ⊑ Hotel ≥ 0.5i} **is a fuzzy KB.**
Example 2.2 K2 **is the fuzzy KB with the following axioms**
(1) (a : A**) = 0**.5 (2) **⊤ ⊑ ∃**R.⊤ (3) (∀R.A) ≡ (∃R.A) (4) A ≡ (∀R.A) ⊓ (∀R.A)
Semantics **In fuzzy DLs, concepts and roles are interpreted, respectively,** as fuzzy subsets and fuzzy relations over an interpretation domain. **However, the** axioms of a fuzzy KB (i.e., the elements of its ABox or its TBox) are either satisfied (true) or unsatisfied (false) in an interpretation. Hence, the axioms behave as in the crisp case and they are not interpreted as a degree of truth in [0, 1].

Informally speaking, a fuzzy axiom hτ ≥ αi in a ABox constrains the membership degree of τ to be at least α**. And the intended semantics of** hC ⊑ D ≥ αi states that all instances of concept C are instances of concept D **to degree** α, i.e. the subsumption degree (to be clarified later) between C and D **is at least** α**. Next, we give precise definitions for the semantics just explained.**
A fuzzy interpretation (or model) is a pair I **= (∆**I, ·
I**) consisting of a**
nonempty (crisp) set ∆I(the domain) and of a **fuzzy interpretation function** ·
I
that assigns:
4The reader may be surprised about the fact that axioms of the form h(a, b):R ≤ αi **are not**
considered. The reason to not accept this kind of statements is that in the case of crisp ALC
in the ABox it is not allowed to use statements of the form h(a, b):R ≤ 0i **(remember that**
in ALC the negation of a role is not allowed). On the other hand, in crisp ALC **it is indeed**
allowed to use ha:C ≤ 0i since it corresponds to ha:¬C ≥ 1i **(remember that the negation of**
a concept is allowed).