the context of the first-order language, by H´ajek in [13]. A fuzzy **interpretation**
I is said to be witnessed iff it holds that for every complex concepts C, D**, every**
role R, and every x ∈ ∆I**there is some**
1. y ∈ ∆Isuch that (∃R.C)
I(x) = RI(**x, y**) ⊗ C
I(y).

2. y ∈ ∆Isuch that (∀R.C)
I(x) = RI(**x, y**) ⇒ C
I(y).

3. y ∈ ∆I**such that (**C ⊑ D)
I = C
I(y) ⇒ DI(y).

The idea behind this definition is that in a witnessed interpretation all arbitrary infima and suprema needed in order to compute C
I **and (**C ⊑ D)
I are indeed minima and maxima. It is obvious that all finite fuzzy interpretations
(this means that ∆I**is a finite set) are indeed witnessed, but the opposite is**
not true. A fuzzy KB K is said to be satisfiable in a witnessed interpretation iff there exist a witnessed fuzzy interpretation I **satisfying every axiom in** K.

## 3 Infinite Models In Fuzzy Alc

In this section we prove that there are fuzzy KBs that are not finitely satisfiable while are satisfiable in a witnessed interpretation. We will prove this statement for two cases: the Lukasiewicz and the Product cases, and in both cases we will use the same fuzzy KB K2 **defined in Example 2.2.**
First of all we focus in the Lukasiewicz case. It is worth pointing out **that**
under Lukasiewicz semantics the first axiom in K2 **can be rewritten as**
ha:((A ⊔ A) ⊓ ¬(A ⊓ A)) ≥ 1i, and so it is simply using as a bound the crisp value 1.

Proposition 3.1 Let I be a witnessed model of K2 **under Lukasiewicz fuzzy**
logic. Then, for every natural number n there are individuals b1, b2, b3**, . . . , b**n such that 0.5 = AI(b1) < AI(b2) < AI(b3) < · · · < AI(bn) < 1.

Proof: For the case n **= 1 this is trivial using axiom (1) in the fuzzy KB.**
Now let us assume that there are individuals b1, b2, b3, . . . , bn such that 0.5 =
AI(b1) < AI(b2) < AI(b3) < · · · < AI(bn) < **1. We want to check that there is** an individual bn+1 such that AI(bn) < AI(bn+1) < 1.

Using axiom (2) we know that (∃R.⊤)
I(bn) = 1. Since the model is witnessed, there is a 1-successor of bn that we call b, i.e., RI(bn, b**) = 1. It is**
obvious that (∀R.A)
I(bn) ≤ 1 ⇒ AI(b) = AI(b) and that AI(b**) = 1** ⊗ AI(b) ≤
(∃R.A)
I(bn). Using axiom (3) we get that (∀R.A)
I(bn) = AI(b) = (∃R.A)
I(bn).

Therefore, axiom (4) is saying that AI(bn) = (∀R.A)
I(bn) ⊗ (∀R.A)
I(bn) =
AI(b) ⊗ AI(b). Using Lukasiewicz t-norm, it follows that AI(bn) = AI(b) +
AI(b) − 1, and hence AI(b) = (AI(bn) + 1)/**2. Thus, it is clear that** A(bn) <
A(b) < 1. Hence, we can define bn+1 as b **to finish the proof. Q.E.D.**
Corollary 3.2 There is no finite model for K2 **under Lukasiewicz fuzzy logic.**