**Appendix A**

*Appendix A.1. Main Properties of Ł-FTT*

> The following properties are provable in Ł-FTT and will be used in the proofs.

**Lemma A1.** *(Propositional properties [32]) Let A*, *B*, *C*, *D* ∈ *Formo. Then, the following is provable:*

*(a)* ((*A*&*B*) ⇒ *C*) ≡ (*A* ⇒ (*B* ⇒ *<sup>C</sup>*))*; (b)* (*A* ⇒ (*B* ⇒ *C*)) ⇒ (*B* ⇒ (*A* ⇒ *<sup>C</sup>*))*; (c)* (*A*&*B*) ⇒ *<sup>A</sup>*;(*<sup>A</sup>* ∧ *B*) ⇒ *<sup>A</sup>*;(*<sup>A</sup>*&*<sup>B</sup>*) ⇒ (*A* ∧ *<sup>B</sup>*)*; (d)* (*A*&*B*) ≡ (*<sup>B</sup>*&*<sup>A</sup>*)*; (e)* (*B* ⇒ *C*) ⇒ ((*A* ⇒ *B*) ⇒ (*A* ⇒ *<sup>C</sup>*))*; (f)* (*C* ⇒ *A*) ⇒ ((*C* ⇒ *B*) ⇒ (*C* ⇒ (*B* ∧ *<sup>A</sup>*)))*; (g)* (*A* ⇒ *B*) ⇒ ((*A* ∧ *C*) ⇒ (*B* ∧ *<sup>C</sup>*))*; (h)* (*A* ⇒ *B*) ⇒ (¬*<sup>B</sup>* ⇒ ¬*<sup>A</sup>*)*; (i)* ((*A* ⇒ *B*)&(*C* ⇒ *D*)) ⇒ ((*A*&*C*) ⇒ (*<sup>B</sup>*&*<sup>D</sup>*))*.*

**Lemma A2.** *(Properties of quantifiers [32]). Let A*, *B* ∈ *Formo and α* ∈ *Types. Then, the following is provable:*

*(a)* (∀*<sup>x</sup>α*)(*<sup>A</sup>* ⇒ *B*) ⇒ ((∀*<sup>x</sup>α*)*<sup>A</sup>* ⇒ (∀*<sup>x</sup>α*)*B*)*; (b)* (∀*<sup>x</sup>α*)(*<sup>A</sup>* ⇒ *B*) ⇒ ((<sup>∃</sup>*<sup>x</sup>α*)*<sup>A</sup>* ⇒ (<sup>∃</sup>*<sup>x</sup>α*)*B*)*; (c)* (∀*<sup>x</sup>α*)(*<sup>A</sup>* ⇒ *B*) ⇒ (*A* ⇒ (∀*<sup>x</sup>α*)*B*)*, xα is not free in A;(d)* (∀*<sup>x</sup>α*)(*<sup>A</sup>* ⇒ *B*) ⇒ ((<sup>∃</sup>*<sup>x</sup>α*)*<sup>A</sup>* ⇒ *<sup>B</sup>*)*, xαis not free in B.*

**Lemma A3.** *Let T be a theory and A*, *B*, *C*, *D* ∈ *Formo. IfT A*⇒(*B*⇒*<sup>C</sup>*)*,then T A*⇒((*B*∧*D*)⇒(*C*∧*<sup>D</sup>*)).

In the proofs, we also use the rules of modus ponens and generalization, which are derived rules in our theory.

**Theorem A1** ([32])**.** *Let T be a theory, and A*, *B* ∈ *Formo and α* ∈ *Types.*

•*If T A and T A* ⇒ *B, then T B;*

•*If T A, then T* (∀*<sup>x</sup>α*)*A.*

• *Appendix A.2. Graded Peterson's Square of Opposition*

The characteristics and position of the above-mentioned fuzzy intermediate quantifiers were studied using a graded Peterson's square of opposition. In this part of the article, we will not deal with the whole construction of the square in detail (for details, see [19]). We will recall the main definitions that form the before-mentioned square of oppositions, and show the connection between the property of monotonicity and the property of sub-altern.

**Definition A1.** *Let T be a consistent theory of Ł-FTT,* M |= *T be a model, and P*1, *P*2 *be closed formulas.*


The proposed mathematical definitions generalize the classical definitions that form both Aristotle's and Peterson's squares of opposition. At this point, we would like to emphasize that all formally proven syllogisms apply in every model of *T*IQ.

**Figure A1. <sup>7</sup>**-graded Peterson's square of opposition.

Let us remind that the dashed lines denote contraries, the straight lines indicate contradictories, and the dotted lines represent subcontraries. The arrows denote the relation between superaltern–subaltern.

We continue with the theorem which represents the property of the monotonicity of quantifiers which form the **<sup>7</sup>**-graded Peterson's square of opposition.

**Theorem A2.** *[16] Let A,. . . ,O be intermediate quantifiers. Then, the following set of implications is provable in TIQ:*

	- *TIQ K* ⇒ *F*, *TIQ F* ⇒ *S*, *TIQ S* ⇒ *I*;

There are, of course, other studies of the graded cube of opposition. At this point, we also recall the classic cubes of opposition that were proposed by Moretti and Keyne. Gradual extensions to these two structures have been made and deeply studied by Dubois et al. in [36]. In the possibility theory, a graded extension of these cubes of opposition was analyzed by Dubois in [37].

**Figure A2.** Graded Peterson's cube of opposition.
