1. if ψ is hα ≥ ni **then** n ≤ 0.5; 2. if ψ is hα ≤ ni **then** n ≥ 0.5.

Note that the definitions for KB-normalisation and query-normalisation are dual. The following proposition follows from (Lee, 1972; Straccia, 2000b) and relies on the fact **that**
hα ≥ ni and hα ≤ mi are together inconsistent, if n > 0.5 and m < 0.5. In **particular,**
Straccia (2000b) shows that if Σ and ψ are a normalised fuzzy propositional KB and a query-normalised fuzzy proposition, then Then Σ|≈ψ iff ♯Σ |= ♯ψ. The proof is **given**
by showing that from a deduction proving Σ|≈ψ a deduction proving ♯Σ |= ♯ψ can be build and vice-versa. The extension to our case is straightforward as e.g. for n, m > 0.5, ha:∀R.C ≥ ni,h(a, b):R ≥ mi|≈hb:C ≥ ni iff a:∀R.C,(a, b):R |= b:C holds (the other firstorder cases involving ∀ and ∃ are **similar).**
Proposition 2 Let Σ be a fuzzy KB in which no h(a, b):R ≤ ni occurs and **each** ψ ∈ Σ is KB-normalised. Let ψ be a query-normalised fuzzy assertion. Then Σ|≈ψ iff ♯Σ |= ♯ψ. ⊣
Example 3 Let Σ be the set Σ = {ha:A ⊓ B ≤ 0.4i,ha:A ≥ 0.6i}. Let ψ be ha:B ≤ 0.7i.

Each fuzzy assertion in Σ is KB-normalised and ψ is query-normalised. It follows **that**
♯Σ = {a:¬(A ⊓ B), a:A} and ♯ψ = a:¬B. It is easily verified that Σ|≈ψ and **that** ♯Σ |= ♯ψ, thereby confirming **Proposition** 2. ✷
4.4 **Subsumption**
At first, as for the classical case and with the same method seen before, **subsumption**
between two concepts C and D w.r.t. a terminology ΣT , i.e. C ¹ΣT D, can be **reduced** to the case of an empty terminology, **i.e.** C
′ ¹∅ D′.

Example 4 Suppose we have two images i1 and i2 each being a snapshot of the car **traffic**
on a major street of an European city. An underlying image analysis tool recognizes, **among**
all the recognised objects, that in image i1 there is a Ferrari, while in image i2 **there** is a Porsche. Furthermore, a semantic image indexing tool establishes that, to some **degree** n image i1 is about a Ferrari, whereas to some degree m image i2 is about a Porsche. **Please**
note that, as a weight of a keyword in text is a quantitative description of the **aboutness** of the text w.r.t. the keyword, a truth-degree gives a quantitative description of the **aboutness**
of an images w.r.t. an object, i.e. the aboutness is handled as an imprecise concept. So, let us **consider**
Σ = {hi1:∃About.Ferrari ≥ 0.6i,hi2:∃About.Porsche ≥ 0.8i,

## Ferrari≺·Car, Porsche≺·Car}.

where the axioms specify that both a Ferrari and a Porsche are a car. According to the expansion process, Σ will be **replaced** by

```
Σ
′ = {hi1:∃About.Ferrari ≥ 0.6i,hi2:∃About.Porsche ≥ 0.8i,
       Ferrari:≈ Car ⊓ Ferrari
                     
                      ∗
                      
                      , Porsche:≈ Car ⊓ Porsche
                                        ∗
                                        },

```

which will be **simplified** to