**2. Main Methods**

The main approach of this section is to recall the theory of fuzzy natural logic (FNL) that was designed based on the fuzzy type theory (FTT). FNL is a formal mathematical theory that includes three theories:


#### *2.1. Fuzzy Type Theory*

This section focuses on the reminder of the main symbols and of the fuzzy type theory. We will not repeat all the details here; we refer readers to previous publications [17,32].

Let us recall, at this point, that the mathematical theory of fuzzy quantifiers was proposed over the Łukasiewicz fuzzy type theory (Ł-FTT). The structure of truth values is represented by a linearly ordered MVΔ-algebra that is extended by the delta operation (see [33,34]). A particular case is the standard Łukasiewicz MVΔ-algebra:

$$\mathcal{L} = \langle [0, 1], \vee, \wedge, \otimes, \rightarrow, 0, 1, \Delta \rangle. \tag{1}$$

The fundamental objects that represent the syntax of Ł-FTT are classical (cf. [35]). We assume atomic types as follows: *-* (elements) and *o* (truth values). General types are marked by Greek letters *α*, *β*, . . .. A set of all types is marked by *Types*. The (meta-)symbol ":= " used below means "is defined by".

The *language* contains of variables *xα*, ..., special constants *cα*, ... (*α* ∈ *Types*), symbol *λ*, and parentheses. The connectives are *fuzzy equality/equivalence* <sup>≡</sup>, *conjunction* ∧, *implication* ⇒, *negation* <sup>¬</sup>, *strong conjunction* &, *strong disjunction* ∇, *disjunction* ∨, and *delta* Δ. The fuzzy type theory is *complete*, i.e., a theory *T* is consistent iff it has a (Henkin) model (M |= *T*). We sometimes use the equivalent notion: *T Ao*iff *T* |= *Ao*.

The following special formulas are important in our theory below:

$$\begin{aligned} \Upsilon\_{\text{oo}} & \equiv \lambda z\_{\text{o}} \cdot \neg \mathsf{A}(\neg z\_{\text{o}}), & \text{(nonzero truth value)},\\ \hat{\Upsilon}\_{\text{oo}} & \equiv \lambda z\_{\text{o}} \cdot \neg \mathsf{A}(z\_{\text{o}} \lor \neg z\_{\text{o}}). \end{aligned} \tag{\text{(nonzero truth value)}}$$

Thus, M(Υ(*Ao*)) = 1 iff M(*Ao*) > 0, and M(Υ<sup>ˆ</sup>(*Ao*)) = 1 iff M(*Ao*) ∈ (0, 1) holds in any model M.

#### *2.2. Evaluative Linguistic Expressions*

As we stated in the introduction, the formal definitions of fuzzy intermediate quantifiers are based on evaluation linguistic expressions. In this subsection, we recall the theory of evaluative linguistic expressions.

Evaluative linguistic expressions are special expressions of a natural language, for example, *very small, roughly medium, extremely large, very long, quite roughly, extremely rich*, etc. Their theory is the main part of the fuzzy natural logic. The evaluative language expressions themselves play an important role in everyday human reasoning. We can find them in a wide variety of areas, such as economics, decision making, and more.

The language *J*Ev contains special symbols as follows:


The evaluative expressions are interpreted by special formulas *Sm* ∈ *Formoo*(*oo*) (*small*), *Me* ∈ *Formoo*(*oo*) (*medium*), *Bi* ∈ *Formoo*(*oo*) (*big*), and *Ze* ∈ *Formoo*(*oo*) (*zero*) that can be expanded by several linguistic hedges. Let us remind that a *hedge*, which is often an adverb such as "extremely, significantly, very, roughly", etc., is, in general, represented by a formula *ν* ∈ *Formoo* with specific properties. We proposed a formula *Hedge* ∈ *Formo*(*oo*). Then, *T*Ev *Hedge ν* means that *ν* is a hedge. The other details can be found in [17]. The formula *T*Ev *Hedge ν* is provable for all *ν* ∈ {*Ex*, *Si*, *Ve*, *ML*,*Ro*, *QR*, *VR*}. Furthermore, evaluative linguistic expressions are represented by formulas:

$$\exists sm\,\Psi,\mathsf{M}\varepsilon\,\Psi,\mathsf{B}\,\Psi,\mathsf{Z}\varepsilon\,\Psi\in\mathsf{Form}\_{\mathsf{@}\diamond},\tag{2}$$

where *ν* is a hedge. We will also assume an *empty hedge ν*¯ that is introduced in front of *small, medium*, and *big* if no other hedge is assumed. A special hedge is Δ*oo*, that represents the expression "utmost" and occurs below in the evaluative expression *Bi*Δ.

Let *<sup>ν</sup>*1,*oo*, *<sup>ν</sup>*2,*oo* be two hedges, i.e., *TEv Hedge <sup>ν</sup>*1,*oo* and *TEv Hedge <sup>ν</sup>*2,*oo*. We propose a relation of the partial ordering of hedges by:

$$\lnot\asymp := \lambda \, p\_{\alpha \nu} \lambda q\_{\alpha \nu} \cdot (\forall z\_o) \, (p\_{\alpha \nu} z \Rightarrow q\_{\alpha \nu} z) \,.$$

**Lemma 1** ([17])**.** *The following ordering of the specific hedges can be proved.*

$$T^{\mathsf{L}\overline{\mathsf{u}}} \vdash \mathbf{A} \llcorner \mathbf{Ex} \llcorner \mathbf{Si} \llcorner \mathbf{Ne} \llcorner \mathbf{\bar{\nu}} \llcorner \mathbf{ML} \llcorner \mathbf{Ro} \llcorner \mathbf{QR} \llcorner \mathbf{VR}.\tag{3}$$

**Theorem 1.** *If TIQ ν*1 *ν*2*, then*

$$T^{I\backslash\mathbb{Q}} \vdash (Bi\,\nu\_1)((\mu(\neg B))(\neg B|z)) \Rightarrow (Bi\,\nu\_2)((\mu(\neg B))(\neg B|z)).$$

**Proof.** Analogously to [16], this can be proved using Theorem 1(g), by replacing *B* with its negation.

Evaluative expressions represent certain unspecified positions on a bounded linearly ordered scale. It is important to introduce the *context* in which we characterize them. The context can be characterized by a function *w* : *L* → *N* for some set *N*. We sugges<sup>t</sup> the context as a triple of numbers *vL*, *vS*, *vR* ∈ *N*, such that *vL* < *vS* < *vR* (the ordering on *N* is induced by *w*). Then, *x* ∈ *w* iff *x* ∈ [*vL*, *vS*] ∪ [*vS*, *vR*]. Introducing the concept of context means defining the concept of intension and extension. The intension of the evaluative linguistic expressions (2) is equal to a simple fuzzy set in the support of truth values. For further details, we recommend readers to the publication [17].
