*2.3. Fuzzy Measure*

As discussed in our previous publications, the semantics of the intermediate quantifiers assumes the idea of a "size" of a (fuzzy) set, which we describe by the concept of a fuzzy measure. In our theory, we will assume the fuzzy measure below:

**Definition 1.** *Let R* ∈ *Formo*(*oα*)(*oα*) *be a formula.*

• *A formula, μ* ∈ *Formo*(*oα*)(*oα*)*, defined by:*

$$
\lambda \mu\_{o(\alpha a)(\alpha a)} := \lambda z\_{\alpha a} \lambda x\_{\alpha a} \,(Rz\_{\alpha a}) x\_{\alpha a} \,\tag{4}
$$

*represents a* measure on fuzzy sets *in the universe of type α* ∈ *Types if it has the following properties:*


The fuzzy measure introduced above is defined using three axioms—the axiom of normality, the axiom of monotonicity, and the fuzzy measure is closed with respect to the negation.

**Example 1.** *A fuzzy measure on a finite universe can be introduced as follows. Let M be a finite set and A*, *B* ⊂ ∼*M be fuzzy sets. Put:*

$$|A| = \sum\_{m \in \text{Supp}(A)} A(m). \tag{5}$$

*Furthermore, let us define a function F<sup>R</sup>* ∈ (*L*F(*M*))F(*M*) *by:* 

$$F^R(B)(A) = \begin{cases} 1, & \text{if } B = A = \bigcirc, \\ \min\left\{ 1, \frac{|A|}{|B|} \right\}, & \text{if } \text{Supp}(A) \subseteq \text{Supp}(B), \\ 0, & \text{otherwise} \end{cases} \tag{6}$$

*for all A*, *B* ⊂ ∼*M.*

#### *2.4. Formal Definition of Intermediate Quantifiers*

In this subsection, we will recall the modified definition of the fuzzy intermediate quantifier, which is based on a special fuzzy set representing the cut of a fuzzy set (support).

In this article, we will work with the special fuzzy sets; they represent "cuts" of the universe *B*.

Let *y*, *z* ∈ *Formo<sup>α</sup>*. The cut of *y* by *z* is the fuzzy set:

$$y|z \equiv \lambda x\_{\alpha} \cdot zx \clubsuit \mathbf{a} \, \mathbf{a} (\mathbf{Y}(zx) \bullet (yx \equiv zx)).$$

The following result can be proved.

**Proposition 1.** *Let* M *be a model such that B* = M(*y*) ⊂ ∼ *M<sup>α</sup>, Z* = M(*z*) ⊂ ∼ *M<sup>α</sup>. Then, for any m* ∈ *Mα:* 

$$\mathcal{M}(y|z)(m) = (B|Z)(m) = \begin{cases} B(m), & \text{if } B(m) = Z(m), \\ 0 & \text{otherwise.} \end{cases}$$

We can observe that the operation *B*|*Z* picks only those elements *m* ∈ *Mα* from the support *B*, whose membership degree *<sup>B</sup>*(*m*) is equal to the degree *<sup>Z</sup>*(*m*); otherwise, it is equal to zero ((*B*|*Z*)( *m*) = 0). If there is no such element, then the cut is represented by an empty set (*B*|*Z* = ∅).

**Definition 2.** *Let Ev be a formula representing an evaluative expression, x be a variable and A*, *B*, *z be formulas. Then, either of the formulas:*

$$(Q\_{Ev}^{\vee} \mathbf{x})(B, A) \equiv (\exists z)[(\forall \mathbf{x})((B|z) \, \mathbf{x} \Rightarrow A \, \mathbf{x}) \, \mathsf{A} \, \mathrm{Ev}((\mu B)(B|z))],\tag{7}$$

$$(Q\_{Ev}^{\preceq}x)(B,A) \equiv (\exists z)[(\exists x)((B|z)x \land Ax) \land Ev((\mu B)(B|z))],\tag{8}$$

*construe the sentence:*

> *"Quantifier Bs are A".*

Below, we introduce several examples of fuzzy intermediate quantifiers which fulfill the property of the monotonicity.

**A:** All *B*s are *A* := (*Q*<sup>∀</sup>*Bi*Δ*<sup>x</sup>*)(*<sup>B</sup>*, *A*) ≡ (∀*x*)(*Bx* ⇒ *Ax*); **E:** No *B*s are *A* := (*Q*<sup>∀</sup>*Bi*Δ*<sup>x</sup>*)(*<sup>B</sup>*,<sup>¬</sup>*<sup>A</sup>*) ≡ (∀*x*)(*Bx* ⇒ ¬*Ax*); **P:** Almost all *B*s are *A* := (*Q*<sup>∀</sup>*Bi Ex<sup>x</sup>*)(*<sup>B</sup>*, *<sup>A</sup>*); **B:** Almost all *B*s are not *A* := (*Q*<sup>∀</sup>*Bi Ex<sup>x</sup>*)(*<sup>B</sup>*,<sup>¬</sup>*<sup>A</sup>*); **T:** Most *B*s are *A* := (*Q*<sup>∀</sup>*Bi Ve<sup>x</sup>*)(*<sup>B</sup>*, *<sup>A</sup>*); **D:** Most *B*s are not *A* := (*Q*<sup>∀</sup>*Bi Ve<sup>x</sup>*)(*<sup>B</sup>*,<sup>¬</sup>*<sup>A</sup>*); **K:** Many *B*s are *A* := (*Q*∀¬ *Sm<sup>x</sup>*)(*<sup>B</sup>*, *<sup>A</sup>*); **G:** Many *B*s are not *A* := (*Q*∀¬ *Sm<sup>x</sup>*)(*<sup>B</sup>*,<sup>¬</sup>*<sup>A</sup>*); **F:** A few (A little) *B*s are *A* := (*Q*<sup>∀</sup>*Sm Si<sup>x</sup>*)(*<sup>B</sup>*, *<sup>A</sup>*); **V:** A few (A little) *B*s are not *A* := (*Q*<sup>∀</sup>*Sm Si<sup>x</sup>*)(*<sup>B</sup>*,<sup>¬</sup>*<sup>A</sup>*); **S:** Several *B*s are *A* := (*Q*<sup>∀</sup>*Sm Ve<sup>x</sup>*)(*<sup>B</sup>*, *<sup>A</sup>*); **Z:** Several *B*s are not *A* := (*Q*<sup>∀</sup>*Sm Ve<sup>x</sup>*)(*<sup>B</sup>*,<sup>¬</sup>*<sup>A</sup>*); **I:** Some *B*s are *A* := (*Q*<sup>∃</sup>*Bi*Δ*<sup>x</sup>*)(*<sup>B</sup>*, *A*) ≡ (<sup>∃</sup>*x*)(*Bx* ∧ *Ax*); **O:** Some *B*s are not *A* := (*Q*<sup>∃</sup>*Bi*Δ*<sup>x</sup>*)(*<sup>B</sup>*,<sup>¬</sup>*<sup>A</sup>*) ≡ (<sup>∃</sup>*x*)(*Bx* ∧ <sup>¬</sup>*Ax*).

The mathematical definition of fuzzy intermediate quantifiers is extended by a formula that ensures the non-emptiness of the fuzzy set representing the antecedent.

**Definition 3** ([19])**.** *Let Ev be a formula representing an evaluative expression, x be a variable, and A*, *B*, *z be formulas. Then, either of the formulas:*

$$(\ (^\*Q\_{Ev}^\forall)(B,A) \equiv (\exists z)[(\forall x)((B|z) \,\texttt{x}\Rightarrow Ax)\&(\exists x)(B|z) \,\texttt{x}\,\,Ev((\mu B)(B|z))],\tag{9}$$

$$(^\*Q\_{\mathbb{E}\nu}^{\exists}x)(B,A) \equiv \quad (\exists z)[(\exists x)((B|z) \text{ x } \land \text{ A}x)\nabla \neg(\exists x)(B|z) \text{ x } \land \text{ E}v((\mu B)(B|z))], \tag{10}$$

*construe the sentence:*

> *"\*Quantifier Bs are A".*

The corresponding quantifiers with presuppositions are denoted by **\*A, \*E, \*P, \*B, \*T, \*D, \*K, \*G, \*F, \*V, \*S, \*Z, \*I,** and **\*O**.
