Note that in the work of Straccia (1998), no fuzzy assertion of the **form** hα ≤ ni is allowed.

3.3 Fuzzy Terminological **Axiom**
Fuzzy terminological axioms we will consider are a natural extension of classical terminological axioms to the fuzzy case. From a syntax point of view, a fuzzy terminological **axiom**
(denoted by ˜τ ) is either a fuzzy concept specialisation or a fuzzy concept definition. A **fuzzy**
concept specialisation is an expression of the form A≺·C, where A is a primitive **concept**
and C is a concept. On the other hand, a fuzzy concept definition is an expression of the form A:≈ C, where A is a primitive concept and C is a concept. From a semantics **point** of view, we consider the natural extension of classical set inclusion to the fuzzy case **(Zadeh,**
1965). A fuzzy interpretation I satisfies a fuzzy concept specialisation A≺·C iff

$$\forall d\in\Delta^{\mathcal{I}},A^{\mathcal{I}}(d)\leq C^{\mathcal{I}}(d),$$
$$(16)$$
(d), **(16)**
whereas I satisfies a fuzzy concept definition A:≈ C iff

$$\forall d\in\Delta^{\mathcal{I}},A^{\mathcal{I}}(d)=C^{\mathcal{I}}(d).$$
$$(17)$$
(d). **(17)**
Note that in the work of Straccia (1998) a fuzzy specialisation is "non-standard". Indeed, Straccia (1998) considered fuzzy specialisations of the form hA → C ≥ ni **where**
(A → C)
I = mind∈∆I {max{1 − AI(d), C
I
(d)}}. A drawback of this formulation is **that**
it is not clear where the n in hA → C ≥ ni comes from, i.e. who defines the value n and how it is determined. We prefer to rely here on the standard interpretation of fuzzy **subsets.**
3.4 Fuzzy Knowledge Base, Fuzzy Entailment and Fuzzy **Subsumption**
A fuzzy knowledge base is a finite set of fuzzy assertions and fuzzy terminological **axioms.**
As for the crisp case, with ΣA we will denote the set of fuzzy assertions in Σ, **with** ΣT we will denote the set of fuzzy terminological axioms in Σ (the terminology), if ΣT = ∅ **then** Σ
is purely assertional, and we will assume that a terminology ΣT is such that no **concept** A
appears more than once on the left hand side of a fuzzy terminological axiom ˜τ ∈ ΣT and that no cyclic definitions are **present** in ΣT .

An interpretation I satisfies (is a model of ) a set of fuzzy Σ iff I satisfies each **element**
of Σ. A fuzzy KB Σ fuzzy entails a fuzzy assertion ψ (denoted by Σ|≈ψ) iff every **model** of Σ also **satisfies** ψ.

Furthermore, let ΣT be a terminology and let C, D be two concepts. We will say that D fuzzy subsumes C w.r.t. ΣT (denoted by C ¹ΣT D) iff for every **model** I of ΣT ,
∀d ∈ ∆I, C
I
(d) ≤ DI(d) **holds.**
Finally, given a fuzzy KB Σ and an assertion α, it is of interest to compute α's **best**
lower and upper truth-value bounds. To this end we define the greatest lower **bound** of α w.r.t. Σ (denoted by glb(Σ, α)) to be sup{n : Σ|≈hα ≥ ni}. Similarly, we define the least upper bound of α with respect to Σ (denoted by lub(Σ, α)) to be inf{n : Σ|≈hα ≤ ni}
(sup ∅ = 0, inf ∅ = 1). Determining the lub and the glb is called the Best Truth-Value **Bound**
(BTVB) **problem.**