**3. Results**

*3.1. Formal Structure of Peterson's Syllogisms Related to Peterson's Square*

**Definition 4** (Syllogism)**.** *A* syllogism *is a triple <sup>P</sup>*1, *P*2, *C of three statements. P*1, *P*2 *are called* premises *(P*1 *represents* major*, P*2 *is* minor*) and C denotes a* conclusion*. S (*subject*) is somewhere in P*2 *and also as the first formula of the conclusion C, formula P (*predicate*) is somewhere in P*1 *and as the second formula of C; a formula that is not introduced in the conclusion C is called a* middle formula *M.*

Fuzzy syllogisms *are obtained by replacing the classical quantifier (or classical quantifiers) with the fuzzy quantifier (or fuzzy quantifiers).*

**Definition 5.** *Syllogism <sup>P</sup>*1, *P*2, *C is strongly valid in TIQ if TIQ* (*<sup>P</sup>*1&*P*2) ⇒ *C, or equivalently, if TIQ P*1 ⇒ (*<sup>P</sup>*2 ⇒ *<sup>C</sup>*)*.*

Naturally speaking, the syllogism is valid if the Łukasiewicz conjunction of the degrees of both premises is less than or equal to the value of the conclusion.

Since we assume one middle formula, we can therefore consider four possible figures, which will arise according to the position of the middle formula.

**Definition 6.** *Let Q*1, *Q*2, *Q*3 *be fuzzy quantifiers and M*, *P*, *S* ∈ *Formoα be formulas. Let M be a middle formula, S be a subject, and P be a predicate. Then, we distinguish four corresponding figures:*


For demonstration, we present an overview of the valid Peterson's logical syllogisms of Figure I, which were formally proved in [18]. We can observe that the main role plays the property of the monotonicity.


Below, we present an example of valid logical syllogism **AKK**-I with the fuzzy intermediate quantifiers.

> *All cows are herbivores. Many animals on the farm are cows. Many animals on the farm are herbivores.*

We follow with the negative syllogisms of Figure I.

**Theorem 3** ([18])**.** *The following syllogisms are strongly valid in TIQ:*


At this point, we will not present other figures and their valid syllogisms. We refer readers to publications in which we have addressed the formal construction of mathematical proofs of all of Peterson's syllogisms (see [18]).

*3.2. New Forms of Fuzzy Intermediate Quantifiers Related to Graded Peterson's Cube*

Below, we introduce the mathematical definitions of intermediate quantifiers, which form a graded Peterson's cube of opposition (see Figure A2).

**Definition 7** (New forms of fuzzy intermediate quantifiers)**.** *Let Ev be a formula representing an evaluative expression, x be a variable, and A*, *B*, *z be formulas. Then, for either of the formulas:*

$$(Q\_{\mathbb{E}v}^{\vee}x)(\neg B,\neg A) \equiv (\exists z)[(\forall x)((\neg B|z) \, x \Rightarrow \neg A \, x) \, \wedge \, \to v ((\mu(\neg B))(\neg B|z))],\tag{11}$$

$$(Q\_{\mathbb{E}\boldsymbol{\nu}}^{\exists}\mathbf{x})(\neg B,\neg A) \equiv (\exists z)[(\exists \mathbf{x})((\neg B|z)\mathbf{x}\land\neg A\mathbf{x})\land E\mathbf{v}((\mu(\neg B))(\neg B|z))],\tag{12}$$

*either of the quantifiers (11) or (12) construes the sentence "quantifier not Bs are not A".*

Below, we introduce the list of several forms of fuzzy intermediate quantifiers.

**a:** All ¬*B*<sup>s</sup> are not *A* := (*Q*<sup>∀</sup>*Bi*Δ*<sup>x</sup>*)(¬*B*,<sup>¬</sup>*<sup>A</sup>*) ≡ (∀*x*)(¬*Bx* ⇒ ¬*Ax*); **e:** No ¬*B*<sup>s</sup> are not *A* := (*Q*<sup>∀</sup>*Bi*Δ*<sup>x</sup>*)(¬*B*, *A*) ≡ (∀*x*)(¬*Bx* ⇒ *Ax*); **p:** Almost all ¬*B*<sup>s</sup> are not *A* := (*Q*<sup>∀</sup>*Bi Ex<sup>x</sup>*)(¬*B*,<sup>¬</sup>*<sup>A</sup>*); **b:** Almost all ¬*B*<sup>s</sup> are *A* := (*Q*<sup>∀</sup>*Bi Ex<sup>x</sup>*)(¬*B*, *<sup>A</sup>*); **t:** Most ¬*B*<sup>s</sup> are not *A* := (*Q*<sup>∀</sup>*Bi Ve<sup>x</sup>*)(¬*B*,<sup>¬</sup>*<sup>A</sup>*); **d:** Most ¬*B*<sup>s</sup> are *A* := (*Q*<sup>∀</sup>*Bi Ve<sup>x</sup>*)(¬*B*, *<sup>A</sup>*); **k:** Many ¬*B*<sup>s</sup> are not *A* := (*Q*∀¬ *Sm<sup>x</sup>*)(¬*B*,<sup>¬</sup>*<sup>A</sup>*); **g:** Many ¬*B*<sup>s</sup> are *A* := (*Q*∀¬ *Sm<sup>x</sup>*)(¬*B*, *<sup>A</sup>*); **f:** A few (A little) ¬*B*<sup>s</sup> are not *A* := (*Q*<sup>∀</sup>*Sm Si<sup>x</sup>*)(¬*B*,<sup>¬</sup>*<sup>A</sup>*); **v:** A few (A little) ¬*B*<sup>s</sup> are *A* := (*Q*<sup>∀</sup>*Sm Si<sup>x</sup>*)(¬*B*, *<sup>A</sup>*); **s:** Several ¬*B*<sup>s</sup> are not *A* := (*Q*<sup>∀</sup>*Sm Ve<sup>x</sup>*)(¬*B*,<sup>¬</sup>*<sup>A</sup>*); **z:** Several ¬*B*<sup>s</sup> are *A* := (*Q*<sup>∀</sup>*Sm Ve<sup>x</sup>*)(¬*B*, *A*) **i:** Some ¬*B*<sup>s</sup> are not *A* := (*Q*<sup>∃</sup>*Bi*Δ*<sup>x</sup>*)(¬*B*,<sup>¬</sup>*<sup>A</sup>*) ≡ (<sup>∃</sup>*x*)(¬*Bx* ∧ ¬*Ax*); **o:** Some ¬*B*<sup>s</sup> are *A* := (*Q*<sup>∃</sup>*Bi*Δ*<sup>x</sup>*)(¬*B*, *A*) ≡ (<sup>∃</sup>*x*)(¬*Bx* ∧ *Ax*).

If the presupposition is needed, we will denote the corresponding quantifiers by **\*a, \*e, \*p, \*b, \*t, \*d, \*k, \*g, \*f, \*v, \*s, \*z, \*i,** and **\*o**

Just as the theorem represents the monotonicity of the quantifiers that form Peterson's square of opposition, we can also prove the monotonic behavior for new forms of quantifiers.

**Theorem 4.** *The set of implications below is provable in Ł-FTT:*

*1. TIQ a* ⇒ *p*, *TIQ p* ⇒ *t*, *TIQ t* ⇒ *k*, *TIQ k* ⇒ *f*, *TIQ f* ⇒ *s*, *TIQ s* ⇒ *i*; *2. TIQ e* ⇒ *b*, *TIQ b* ⇒ *d*, *TIQ d* ⇒ *g*, *TIQ g*⇒*v*,*TIQ v*⇒*z*,*TIQ z*⇒*o*.

 **Proof.** We will show the proof of *TIQ* **p** ⇒ **t**. We know that:

$$T^{lQ} \vdash (\forall x)((\neg B|z) \ge \Rightarrow \neg Ax) \Rightarrow (\forall x)((\neg B|z) \ge \Rightarrow \neg Ax). \tag{13}$$

By Lemma 1, we know that *Ex Ve* and, therefore, from Theorem 1 and from (13), we obtain:

$$\begin{aligned} \left[ \begin{array}{c} T^{IQ} \vdash \left[ \left( \forall \mathbf{x} \right) \left( \left( \neg B \middle| z \right) \mathbf{x} \Rightarrow \neg Ax \right) \land Bi \text{ix} \left( \left( \mu \left( \neg B \middle| z \right) \right) \left( \neg B \middle| z \right) \right) \right] \Rightarrow\\ \Rightarrow \left[ \left( \forall \mathbf{x} \right) \left( \left( \neg B \middle| z \right) \mathbf{x} \Rightarrow \neg Ax \right) \land Bi \text{i} \text{i} \left( \left( \mu \left( \neg B \middle| z \right) \right) \left( \neg B \middle| z \right) \right) \right] . \end{aligned} \end{aligned}$$

Then, by generalization (∀*z*) and the properties of the quantifiers, we obtain

$$\begin{split} \left[ \begin{array}{c} T^{\mathsf{I}\mathsf{Q}} \vdash (\exists z) [(\forall x)((\neg B|z) \, \mathsf{x} \, \mathsf{@\neg A} x) \, \mathsf{A} \, \mathsf{B} i \, \mathsf{B} i \, \mathsf{Ex}((\mu(\neg B)(\neg B|z)))] \Rightarrow \\ \Rightarrow (\exists z)[(\forall x)((\neg B|z) \, \mathsf{x} \, \mathsf{@\neg A} x) \, \mathsf{A} \, \mathsf{B} i \, \mathsf{V} \epsilon((\mu(\neg B)(\neg B|z)))] \, \mathsf{x} \, \mathsf{B} \, \mathsf{B} \, \mathsf{V}] \end{array} \right] \end{split}$$

This proof is analogous to the proof of *TIQ* **P** ⇒ **T** - we only replace each formula in the proof by its negation. The monotonicity of fuzzy intermediate quantifiers is ensured by the monotonicity of evaluative linguistic expressions. The other proofs of implications are similar to the proof of Theorem A2; we only replace each formula in the proof with its negation.

There are valid forms of examples of logical syllogisms with new forms of fuzzy intermediate quantifiers related to a graded Peterson's cube of opposition (see Appendix A).


g: Many diseases which are not lethal are virus diseases.

E: All virus diseases can not be cured by antibiotics.

i: Some diseases which can not be cured by antibiotics are not lethal diseases.

#### *3.3. Valid Forms Related to Second Face*

First, the reader is reminded of the valid syllogisms of the first figure that are related to the second face of the graded Peterson's cube of opposition. It is not necessary to construct mathematical proofs, because the validity of syllogisms can be verified very easily by replacing individual formulas with their negations.

**Theorem 5.** *The following syllogisms are strongly valid in TIQ:*


**Proof.** Analogously to Theorem 2, this can be proved by replacing each formula with its negation.

**Theorem 6.** *The following syllogisms are strongly valid in TIQ:*


**Proof.** Analogously to Theorem 3, this can be proved by replacing each formula by its negation.

Similarly, we can verify the validity of other forms of logical syllogisms of the other figures. We will not repeat other figures at this point.

#### *3.4. New Forms of Figure I*

In the previous part of this paper, we showed strongly valid syllogisms of the first face and strongly valid syllogisms of the second face. Another goal of the publication is to verify the validity of logical syllogisms that describe the relationship between the first face and the second face in a graded Peterson's cube of opposition.

Firstly we prove a strong validity of the following syllogisms by concrete syntactical proofs.

**Theorem 7.** *Syllogisms aEE-I, aBB-I, aDD-I, aGG-I, aVV-I, aZZ-I, and aOO-I are strongly valid in TIQ.*

**Proof.** Let us assume the syllogism as follows:

$$\begin{array}{c} \begin{array}{c} (\forall x)(\neg Mx \Rightarrow \neg Px) \\ (\forall x)(Sx \Rightarrow \neg Mx) \end{array} \\ \begin{array}{c} (\forall x)(Sx \Rightarrow \neg Mx) \\ (\forall x)(Sx \Rightarrow \neg Px) .\end{array} \end{array}$$

We know that:

$$T^{I\mathbb{Q}} \vdash (\neg Mx \Rightarrow \neg Px) \Rightarrow ((Sx \Rightarrow \neg Mx) \Rightarrow (Sx \Rightarrow \neg Px)).$$

By the rule of generalization of (∀*x*) and using the properties of the quantifiers, we have:

$$T^{IQ} \vdash (\forall \mathbf{x}) (\neg Mx \Rightarrow \neg P\mathbf{x}) \Rightarrow ((\forall \mathbf{x}) (Sx \Rightarrow \neg Mx) \Rightarrow (\forall \mathbf{x}) (Sx \Rightarrow \neg P\mathbf{x})) .$$

**Proof.** Let us assume the syllogism as follows:

$$\mathbf{a}\mathbf{O}\mathbf{O}\text{-}\mathbf{I} \colon \begin{array}{c} (\forall x)(\neg Mx \Rightarrow \neg Px) \\ (\exists x)(Sx \land \neg Mx) \\ \hline (\exists x)(Sx \land \neg Px) .\end{array}$$

We know that:

$$T^{I\mathbb{Q}} \vdash (\neg M\! \mathbf{x} \Rightarrow \neg P\mathbf{x}) \Rightarrow ((S\mathbf{x} \land \neg M\mathbf{x}) \Rightarrow (S\mathbf{x} \land \neg P\mathbf{x})).$$

By the rule of generalization (∀*x*) and using the properties of the quantifiers, we have:

$$T^{I\mathbb{Q}} \vdash (\forall \mathbf{x})(\neg Mx \Rightarrow \neg P\mathbf{x}) \Rightarrow ((\exists \mathbf{x})(\mathbf{Sx} \land \neg Mx) \Rightarrow (\exists \mathbf{x})(\mathbf{Sx} \land \neg P\mathbf{x})) .$$

**Proof.** Let us assume the syllogism as follows:

$$\begin{array}{c} \begin{array}{c} (\forall x)(\neg Mx \Rightarrow \neg Px) \\ (\exists z)[(\forall x)((S|z)x \Rightarrow \neg Mx) \land (BiEx)((\mu S)(S|z))] \end{array} \\ \hline \begin{array}{c} (\exists z)[(\forall x)((S|z)x \Rightarrow \neg Px) \land (BiEx)((\mu S)(S|z))] \end{array} \end{array}$$

Let us denote, by *Ev* := (*BiEx*)((*μ<sup>S</sup>*)(*S*|*z*)).We know that:

$$T^{I\downarrow\downarrow} \vdash (\neg M \mathbf{x} \Rightarrow \neg P \mathbf{x}) \Rightarrow \left( ((S|z) \mathbf{x} \Rightarrow \neg M \mathbf{x}) \Rightarrow ((S|z) \mathbf{x} \Rightarrow \neg P \mathbf{x}) \right).$$

By the rule of generalization (∀*x*) and using the properties of the quantifiers, we have:

$$T^{l\mathbb{Q}} \vdash (\forall \mathbf{x}) (\neg M\mathbf{x} \Rightarrow \neg P\mathbf{x}) \Rightarrow \left( (\forall \mathbf{x}) ((S|z)\mathbf{x} \Rightarrow \neg M\mathbf{x}) \Rightarrow ((\forall \mathbf{x})(S|z)\mathbf{x} \Rightarrow \neg P\mathbf{x}) \right).$$

By Ł-FTT properties, we have the provable formula as follows:

$$\begin{aligned} \left( \begin{array}{c} T^{lQ} \vdash (\forall \mathbf{x}) (\neg \mathsf{Mx} \Rightarrow \neg \mathsf{Px}) \Rightarrow \\ \Rightarrow \left( [(\forall \mathbf{x}) ((S|z) \mathbf{x} \Rightarrow \neg \mathsf{Mx}) \land Ev] \Rightarrow [(\forall \mathbf{x}) ((S|z) \mathbf{x} \Rightarrow \neg \mathsf{Px}) \land Ev] \right) .\end{array} \right) .\end{aligned}$$

Using the generalization rule for (∀*z*) and by the properties of the quantifiers we know that:

$$\begin{aligned} T^{l\mathbb{Q}} &\vdash (\forall \mathbf{x})(\neg Mx \Rightarrow \neg P\mathbf{x}) \Rightarrow\\ &\clubsuit \quad ((\exists \mathbf{z})[(\forall \mathbf{x})((S|\mathbf{z}) \texttt{x} \mathbin{\sf m} \neg Mx) \land Ev] \Rightarrow (\exists \mathbf{z})[(\forall \mathbf{x})((S|\mathbf{z}) \texttt{x} \mathbin{\sf m} \neg P\mathbf{x}) \land Ev]) .\end{aligned}$$

By putting *Ev* := (*BiEx*)((*μ<sup>S</sup>*)(*S*|*z*)), we obtain the strong validity of **aBB**-I. If we denote *Ev* := (*BiVe*)((*μ<sup>S</sup>*)(*S*|*z*)), we obtain the strong validity of **aDD**-I. If we put *Ev* := (¬*Sm*)((*μ<sup>S</sup>*)(*S*|*z*)), we have the strong validity of syllogism **aGG**-I. By denoting *Ev* := (*SmSi*)((*μ<sup>S</sup>*)(*S*|*z*)), we obtain the strong validity of syllogism **aVV**-I. Finally, if we denote *Ev* := (*SmVe*)((*μ<sup>S</sup>*)(*S*|*z*)), we conclude that the syllogism **aZZ**-I is strongly valid.

From these strongly valid syllogisms, we can obtain other strongly valid syllogisms by using monotonicity. Below, we continue with other forms of valid syllogisms of Figure I.

**Theorem 8.** *Let syllogisms* **aEE***-I,* **aBB***-I,* **aDD***-I,* **aGG***-I,* **aVV***-I,* **aZZ***-I,* **aOO***-I be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** From strongly valid syllogism **aEE-I** (Theorem 7) and from monotonicity (Theorem A2) by transitivity, we prove the strong validity of syllogisms in the first column. We prove the strong validity of syllogisms in other columns analogously.

**Theorem 9.** *Syllogisms* **Aee***-I,* **Abb***-I,* **Add***-I,* **Agg***-I* **Avv***-I,* **Azz***-I,* **Aoo***-I are strongly valid in TIQ.*

**Proof.** Analogously to the proof of Theorem 7, this can be proved by replacing each formula by its negation.

**Theorem 10.** *Let syllogisms* **Aee***-I,* **Abb***-I,* **Add***-I,* **Agg***-I,* **Avv***-I,* **Azz***-I, and* **Aoo***-I be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** This can be proved by monotonicity (Theorem 4) similarly to Theorem 8.

In other constructions, we will assume a selected group of valid syllogisms (these proofs can be obtained similarly as in Theorem 7), from which we will verify the validity of

other forms of syllogisms, especially with the help of monotonicity. In Theorem 11 and in Theorem 12, we present other strongly valid syllogisms of Figure I.

**Theorem 11.** *Let syllogisms* **Eea***-I,* **Ebp***-I,* **Edt***-I,* **Egk***-I,* **Evf***-I,* **Ezs***-I, and* **Eoi***-I be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** This can be proved by monotonicity (Theorem 4), similarly to Theorem 8.

**Theorem 12.** *Let syllogisms* **eEA***-I,* **eBP***-I,* **eDT***-I,* **eGK***-I,* **eVF***-I,* **eZS***-I, and* **eOI***-I be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** This can be proved by monotonicity (Theorem A2), analogously to Theorem 8.

We can prove other strongly valid syllogisms using the following proposition, which shows the relationship of the sub-alterns between the first and second squares of opposition.

**Proposition 2** ([19])**.** *The following is provable:*


**Theorem 13.** *Let syllogisms* **AAA***-I,* **aaa***-I,* **EAE***-I,* **eae***-I,* **Aee***-I,* **aEE***-I,* **Eea***-I, and* **eEA***-I be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:* **(\*A)Ai***-I* **(\*a)aI***-I,* **(\*E)Ao***-I,* **(\*e)aO***-I,* **(\*A)eO***-I,* **(\*a)Eo***-I,* **(\*E)eI***-I, and* **(\*e)Ei***-I.*

**Proof.** This can be proved by transitivity and by Proposition 2.

**Theorem 14.** *Syllogisms* **oAo***-I,* **iAi***-I,* **oeI***-I,* **ieO***-I,* **IEo***-I,* **OEi***-I,* **IaI***-I, and* **OaO***-I are strongly valid in TIQ.*

**Proof.** Let us assume the syllogism as follows:

$$\begin{array}{c} \begin{array}{c} (\exists x)(\neg Mx \land Px) \\ (\forall x)(Sx \Rightarrow Mx) \end{array} \\ \begin{array}{c} (\forall x)(Sx \Rightarrow Mx) \\ (\exists x)(\neg Sx \land Px) .\end{array} \end{array}$$

We know that:

$$T^{IQ} \vdash (\mathcal{S} \mathbf{x} \Rightarrow \mathcal{M} \mathbf{x}) \Rightarrow (\neg \mathcal{M} \mathbf{x} \Rightarrow \neg \mathcal{S} \mathbf{x}) .$$

We also know that:

$$T^{I\mathbb{Q}} \vdash (\neg Mx \Rightarrow \neg Sx) \Rightarrow ((\neg Mx \land Px) \Rightarrow (\neg Sx \land Px)) .$$

By transitivity, we obtain:

> *TIQ* (*Sx* ⇒ *Mx*) ⇒ ((¬*Mx* ∧ *Px*) ⇒ (¬*Sx* ∧ *Px*)).

By the rule of generalization for (∀*x*) and using the properties of the quantifiers, we have:

$$T^{I[\mathbb{Q}]} \vdash (\forall \mathbf{x})(Sx \Rightarrow Mx) \Rightarrow ((\exists \mathbf{x})(\neg Mx \land P\mathbf{x}) \Rightarrow (\exists \mathbf{x})(\neg Sx \land Px)).$$

By adjunction, we obtain:

$$T^{I\mathbb{Q}} \vdash ((\exists x)(\neg Mx \land Px) \& (\forall x)(Sx \Rightarrow Mx)) \Rightarrow (\exists x)(\neg Sx \land Px).$$

The strong validity of syllogism **OaO**-I can be proven analogously, by replacing each formula with its negation. The strong validity of other syllogisms can be proven similarly.

**Theorem 15.** *Let syllogisms* **oAo***-I,* **iAi***-I,* **oeI***-I,* **ieO***-I,* **IEo***-I,* **OEi***-I,* **IaI***-I, and* **OaO***-I be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** By monotonicity (Theorem 4) and the strongly valid syllogism **oAo**-I, we prove, by transitivity, the strong validity of the syllogisms in the first column. We prove the other syllogisms in the other columns analogously by monotonicity (Theorems A2 and 4).

#### *3.5. New Forms of Figure II*

Figure II is similar to Figure I, so we will not present concrete syntactical proofs of syllogisms. The syllogisms can be proved similarly to Theorems 7 and 14.

**Theorem 16.** *Let syllogisms* **aAA***-II,* **aPP***-II,* **aTT***-II,* **aKK***-II,* **aFF***-II,* **aSS***-II, and* **aII***-II be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** This can be proved by monotonicity (Theorem A2), similarly to Theorem 8.

**Theorem 17.** *Let syllogisms* **Aaa***-II,* **App***-II,* **Att***-II,* **Akk***-II,* **Aff***-II,* **Ass***-II, and* **Aii***-II be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** This can be proved by monotonicity (Theorem 4), similarly to Theorem 8.

**Theorem 18.** *Let syllogisms* **eEA***-II,* **eBP***-II,* **eDT***-II,* **eGK***-II,* **eVF***-II,* **eZS***-II, and* **eOI***-II be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** This can be proved by monotonicity (Theorem A2), similarly to Theorem 8.

**Theorem 19.** *Let syllogisms* **Eea***-II,* **Ebp***-II,* **Edt***-II,* **Egk***-II,* **Evf***-II,* **Ezs***-II, and* **Eoi***-II be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** This can be proved by monotonicity (Theorem 4), similarly to Theorem 8.

**Theorem 20.** *Let syllogisms* **EAE***-II,* **eae***-II,* **AEE***-II,* **aee***-II,* **aAA***-II,* **Aaa***-II,* **Eea***-II, and* **eEA***-II be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:* **(\*E)Ao***-II,* **(\*e)aO***-II,* **(\*A)Eo***-II,* **(\*a)eO***-II,* **(\*a)Ai***-II,* **(\*A)aI***-II,* **(\*E)eI***-II, and* **(\*e)Ei***-II.*

**Proof.** This can be proved by transitivity and by Proposition 2.

**Theorem 21.** *Let syllogisms* **OAo***-II,* **oaO***-II,* **IEo***-II,* **ieO***-II,* **OeI***-II,* **oEi***-II,* **IaI***-II, and* **iAi***-II be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** This can be proved by monotonicity (Theorems A2 and 4), analogously to Theorem 15.

**Theorem 22.** *Syllogisms* **(\*A)Ai***-II,* **(\*a)aI***-II,* **(\*E)Ei***-II,* **(\*e)eI***-II,* **(\*e)Ao***-II,* **(\*E)aO***-II,* **(\*A)eO***-II, and* **(\*a)Eo***-II are strongly valid in TIQ.*

**Proof.** Let us assume the syllogism as follows:

> **(\*A)Ai**-II: (∀*x*)(*Px* ⇒ *Mx*)&(<sup>∃</sup>*x*)(¬*Mx*) (∀*x*)(*Sx* ⇒ *Mx*) (<sup>∃</sup>*x*)(¬*Sx* ∧ <sup>¬</sup>*Px*).

We know that:

$$T^{IQ} \vdash (Sx \Rightarrow Mx) \Rightarrow (\neg Mx \Rightarrow \neg Sx). \tag{14}$$

We know that:

$$\vdash T^{I\downarrow} \vdash (P\mathbf{x} \Rightarrow M\mathbf{x}) \Rightarrow (\neg M\mathbf{x} \Rightarrow \neg P\mathbf{x}).\tag{15}$$

We also know that:

$$T^{lQ} \vdash (\neg M \mathbf{x} \Rightarrow \neg \mathbf{S} \mathbf{x}) \Rightarrow \left( (\neg M \mathbf{x} \Rightarrow \neg \mathbf{P} \mathbf{x}) \Rightarrow (\neg M \mathbf{x} \Rightarrow (\neg \mathbf{S} \mathbf{x} \land \neg \mathbf{P} \mathbf{x})) \right). \tag{16}$$

By the application of transitivity on (14) and (16), we obtain:

$$T^{l\mathbb{Q}} \vdash (Sx \Rightarrow Mx) \Rightarrow \left( (\neg Mx \Rightarrow \neg Px) \Rightarrow (\neg Mx \Rightarrow (\neg Sx \land \neg Px)) \right).$$

By adjunction, we obtain:

$$(\neg T^{lQ} \vdash (\neg M \mathbf{x} \Rightarrow \neg P \mathbf{x}) \Rightarrow ((\mathbf{Sx} \Rightarrow M \mathbf{x}) \Rightarrow (\neg M \mathbf{x} \Rightarrow (\neg \mathbf{Sx} \land \neg P \mathbf{x}))).\tag{17}$$

By the application of transitivity on (15) and (17), we obtain:

$$(\neg T^{l\mathbb{Q}} \vdash (P\mathbf{x} \Rightarrow M\mathbf{x}) \Rightarrow \left((\mathbf{Sx} \Rightarrow M\mathbf{x}) \Rightarrow (\neg M\mathbf{x} \Rightarrow (\neg S\mathbf{x} \land \neg P\mathbf{x}))\right).$$

By generalization (∀*x*) and the properties of the quantifiers, we obtain:

$$T^{l\mathbb{Q}} \vdash (\forall \mathbf{x})(P\mathbf{x} \Rightarrow M\mathbf{x}) \Rightarrow \left((\forall \mathbf{x})(S\mathbf{x} \Rightarrow M\mathbf{x}) \Rightarrow ((\exists \mathbf{x})\neg M\mathbf{x} \Rightarrow (\exists \mathbf{x})(\neg S\mathbf{x} \land \neg P\mathbf{x}))\right).$$

The strong validity of other fuzzy logical syllogisms can be verified similarly.

#### *3.6. New Forms of Figure III*

Firstly, we will show some syntactical proofs of syllogisms on this Figure.

**Theorem 23.** *Syllogisms* **AOo***-III,* **\*(PG)o***-III,* **\*(TD)o***-III,* **\*(KB)o***-III, and* **IEo***-III are strongly valid in TIQ.*

**Proof.** Let us assume the syllogism as follows:

$$\begin{array}{c} \left( \forall x \right) (Mx \Rightarrow Px) \\ \left( \exists x \right) (Mx \land \neg Sx) \\ \hline \left( \exists x \right) (\neg Sx \land Px) . \end{array}$$

We know that:

$$T^{IQ} \vdash (\mathsf{Mx} \Rightarrow \mathsf{Px}) \Rightarrow ((\mathsf{Mx} \land \neg \mathsf{Sx}) \Rightarrow (\neg \mathsf{Sx} \land \mathsf{Px})) .$$

Analogously, as in previous proofs, using the properties of the quantifiers, we obtain:

$$T^{I\mathbb{Q}} \vdash (\forall \mathbf{x})(\mathbf{Mx} \Rightarrow \mathbf{Px}) \Rightarrow ((\exists \mathbf{x})(\mathbf{Mx} \land \neg \mathbf{Sx}) \Rightarrow (\exists \mathbf{x})(\neg \mathbf{Sx} \land \mathbf{Px})).$$

**Proof.** Let us assume the syllogism as follows:

$$\begin{array}{c} \begin{array}{c} (\exists x)(Mx \land Px) \\ (\forall x)(Mx \Rightarrow \neg Sx) \end{array} \\ \begin{array}{c} (\exists x)(\exists x \land Px) .\end{array} \end{array}$$

We know that:

$$T^{IQ} \vdash (Mx \Rightarrow \neg Sx) \Rightarrow ((Mx \land Px) \Rightarrow (\neg Sx \land Px)).$$

Using the generalization of (∀*x*) and by the logical properties of the quantifiers, we have:

$$(\neg T^{I\mathbb{Q}} \vdash (\forall x)(Mx \Rightarrow \neg Sx) \Rightarrow ((\exists x)(Mx \land Px) \Rightarrow (\exists x)(\neg Sx \land Px)).$$

By adjunction, we obtain:

$$T^{\mathcal{U}\mathcal{Q}} \vdash (\exists x)(\mathcal{Mx} \land Px) \Rightarrow ((\forall x)(\mathcal{Mx} \Rightarrow \neg Sx) \Rightarrow (\exists x)(\neg Sx \land Px)).$$

**Proof.** Let us assume the syllogism as follows:

$$\begin{array}{c} \left( \exists z \big) \big[ \big( \forall x \big) \big( \big( M|z \big) x \Rightarrow Px \big) \big\hdash \left( \big( BiEx \big) \big( \big( \mu M \big) \big( M|z \big) \big) \big] \big] \\ \mathbf{PGo-III:} \quad \big( \exists z' \big) \big[ \big( \forall x \big) \big( \big( M|z' \big) x \Rightarrow \neg Sx \big) \big( \big( \neg Sm \big) \big( \big( \mu M \big) \big( M|z' \big) \big) \big] \big} \\ \cline{2-4} \\ \hline \left( \exists x \big) \big( \neg Sx \big \land Px \big) . \end{array}$$

We know that:

$$\begin{aligned} \left( \begin{array}{c} T^{\mathsf{Q}} \vdash \left( (M|z) \mathbf{x} \Rightarrow P\mathbf{x} \right) \Rightarrow \\ \qquad \qquad \qquad \qquad \mapsto \left( ((M|z') \mathbf{x} \Rightarrow \neg S \mathbf{x}) \Rightarrow \left( ((M|z) \mathbf{x} \&(M|z') \mathbf{x}) \Rightarrow (\neg S \mathbf{x} \land P\mathbf{x}) \right) \right) \end{array} \right) \end{aligned}$$

By generalization (∀*x*) and the properties of the quantifiers, we obtain:

$$\begin{aligned} \left( \begin{array}{c} T^{lQ} \vdash (\forall \mathbf{x}) ((M|z) \mathbf{x} \Rightarrow P\mathbf{x}) \Rightarrow \\ \Rightarrow \left( (\forall \mathbf{x}) ((M|z') \mathbf{x} \Rightarrow \neg \mathbf{S} \mathbf{x}) \Rightarrow ((\exists \mathbf{x}) ((M|z) \mathbf{x} \& (M|z') \mathbf{x}) \Rightarrow (\exists \mathbf{x}) (\neg \mathbf{S} \mathbf{x} \land P\mathbf{x})) \right) \end{array} \right) .\end{aligned}$$

By using the property of Łukasiewicz logic, we obtain:

$$\begin{aligned} \left( \begin{array}{c} T^{IQ} \vdash \left[ (\forall \mathbf{x}) ((M|z) \mathbf{x} \Rightarrow P\mathbf{x}) \land E\mathbf{v} \right] \Rightarrow \\ \Rightarrow \left( (\forall \mathbf{x}) ((M|z') \mathbf{x} \Rightarrow \neg \mathbf{S} \mathbf{x}) \Rightarrow ((\exists \mathbf{x}) ((M|z) \mathbf{x} \& (M|z') \mathbf{x}) \Rightarrow (\exists \mathbf{x}) (\neg \mathbf{S} \mathbf{x} \land P\mathbf{x})) \right) \end{array} \right) .\end{aligned}$$

By adjunction and the property of Łukasiewicz logic, we obtain:

$$\begin{split} &T^{l\mathbb{Q}} \vdash \left[ (\forall \mathbf{x}) (((M|z')\mathbf{x} \Rightarrow \neg \mathbf{S} \mathbf{x}) \land E\mathbf{v}' ) \Rightarrow \\ &\clubsuit \left( [(\forall \mathbf{x}) (((M|z)\mathbf{x} \Rightarrow P\mathbf{x}) \land E\mathbf{v}) \Rightarrow ((\exists \mathbf{x}) ((M|z)\mathbf{x} \& (M|z')\mathbf{x}) \Rightarrow (\exists \mathbf{x}) (\neg \mathbf{S} \mathbf{x} \land P\mathbf{x})) \right) \right]. \end{split}$$

By adjunction, we obtain:

$$\begin{split} \left| \begin{array}{c} T^{lQ} \vdash \left( [(\forall x)((M|z) \, \mathbf{x} \, \, \, \mathbf{Px}) \, \mathsf{A} \, \, \mathrm{Ev} \right) \& [(\forall x)((M|z') \, \mathbf{x} \, \, \neg \mathsf{Sx}) \, \mathsf{A} \, \mathsf{Ev}'] \right\| \& \\ \& (\exists x)((M|z) \, \mathbf{x} \, \mathsf{A} (M|z') \mathbf{x}) \end{array} \right| \& \left| \begin{array}{c} (\exists x)(\neg \mathsf{Sx} \, \mathsf{A} \, \mathsf{Px}) \, \mathsf{A} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{A} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{B} \, \mathsf{$$

By generalization (∀*z*), (∀*z*), and the properties of the quantifiers, we obtain:

$$\begin{split} \left( \left( \begin{array}{c} \left( \exists z \right) \left( \left( \exists z' \right) \left( \left( \left( \forall x \right) \left( \left( M|z \right) \texttt{x} \Rightarrow \text{Px} \right) \land Ev \right) \& \left[ \left( \forall x \right) \left( \left( M|z' \right) \texttt{x} \Rightarrow \neg \text{Sx} \right) \land Ev' \right] \& \right. \\ \left. \left( \exists x \right) \left( \left( \left( M|z \right) \texttt{x} \& \left( M|z' \right) \texttt{x} \right) \right) \Rightarrow \left( \exists x \right) \left( \left( \neg \texttt{Sx} \land P \right) \text{x} \right) . \end{array} \right) . \end{split}$$

If we put *Ev* := (*BiEx*)((*μ<sup>M</sup>*)(*M*|*z*)) and *Ev* := (¬*Sm*)((*μ<sup>M</sup>*)(*M*|*z*)), we obtain the strong validity of syllogism **\*(PG)o**-III. If we put *Ev* := (*BiVe*)((*μ<sup>M</sup>*)(*M*|*z*)) and

*Ev* := (*BiVe*)((*μ<sup>M</sup>*)(*M*|*z*)), we obtain the strong validity of syllogism **\*(TD)o**-III. If we put *Ev* := (¬*Sm*)((*μ<sup>M</sup>*)(*M*|*z*)) and *Ev* := (*BiEx*)((*μ<sup>M</sup>*)(*M*|*z*)), we obtain the strong validity of syllogism **\*(KB)o**-III.

**Theorem 24.** *Let syllogisms* **AOo***-III,* **\*(PG)o***-III,* **\*(TD)o***-III,* **\*(KB)o***-III, and* **IEo***-III be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** From the strongly valid logical syllogism **AOo**-III, and using the monotonicity (Theorem A2), we prove the strong validity of the syllogisms in the first column by transitivity. From the strongly valid syllogism **IEo**-III, and by monotonicity (Theorem A2), we can verify the strong validity of the syllogisms in the first row by transitivity. Analogously, using the strongly valid syllogism **\*(PG)o**-III and by monotonicity (Theorem A2), we can verify the strong validity of the syllogisms in the second column by transitivity. The syllogisms in the third and the fourth column can be proven analogously.

**Theorem 25.** *Syllogisms* **aoO***-III,* **\*(pg)O***-III,* **\*(td)O***-III,* **\*(kb)O***-III, and* **ieO***-III are strongly valid in TIQ.*

**Proof.** This can be proven analogously to the proof of Theorem 23, by replacing each formula with its negation.

**Theorem 26.** *Let syllogisms* **aoO***-III,* **\*(pg)O***-III,* **\*(td)O***-III,* **\*(kb)O***-III, and* **ieO***-III be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** This can be proven by monotonicity (Theorem 4), similarly to Theorem 24.

Next, we will consider the strong validity of some syllogisms without concrete proofs. These proofs are similar to the proofs in Theorem 23.

**Theorem 27.** *Let syllogisms* **EOi***-III,* **\*(BG)i***-III,* **\*(DD)i***-III,* **\*(GB)i***-III, and* **OEi***-III be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*

> *E(\*E)i (\*B)Ei (\*D)Ei (\*G)Ei (\*V)Ei (\*Z)Ei OEi E(\*B)i \*(BB)i \*(DB)i \*(GB)i E(\*D)i \*(BD)i \*(DD)i E(\*G)i \*(BG)i E(\*V)i E(\*Z)i EOi*

**Proof.** This can be proven by monotonicity (Theorem A2), similarly to Theorem 24. **Theorem 28.** *Let syllogisms* **oeI***-III,* **\*(gb)I***-III,* **\*(dd)I***-III,* **\*(bg)I***-III, and* **eoI***-III be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*

```
e(*e)I (*b)eI (*d)eI (*g)eI (*v)eI (*z)eI oeI
e(*b)I *(bb)I *(db)I *(gb)I
e(*d)I *(bd)I *(dd)I
e(*g)I *(bg)I
e(*v)I
e(*z)I
 eoI
```
**Proof.** This can be proven by monotonicity (Theorem 4), similarly to Theorem 24.

The following syllogisms are also strongly valid in Figure III.

**Theorem 29.** *Let syllogisms* **eAe***-III,* **EaE***-III,* **aAa***-III,* **AaA***-III,* **eEA***-III,* **Eea***-III,* **aEE***-III, and* **Aee***-III be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** From the strongly valid syllogism **eAe**-III, and from monotonicity (Theorem 4), we can prove, by transitivity, the strong validity of the syllogisms in the first column. Analogously, from monotonicity (Theorems A2 and 4), we can prove the strong validity of the syllogisms in the other columns.

**Theorem 30.** *Let syllogisms* **eAe***-III,* **EaE***-III,* **aAa***-III,* **AaA***-III,* **eEA***-III,* **Eea***-III,* **aEE***-III, and* **Aee***-III be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:* **(\*e)AO***-III,* **(\*E)ao***-III,* **(\*a)AI***-III,* **(\*A)ai***-III,* **(\*e)Ei***-III,* **(\*E)eI***-III,* **(\*a)Eo***-III, and* **(\*A)eO***-III.*

**Proof.** This can be proven by transitivity and by Proposition 2.

*3.7. New Forms of Figure IV*

> Firstly, we show a proof of the strongly valid syllogisms of this figure.

**Theorem 31.** *Syllogisms* **aII***-IV,* **Aii***-IV,* **EOi***-IV,* **eoI***-IV,* **aOo***-IV, and* **AoO***-IV are strongly valid in TIQ.*

**Proof.** Let us assume the syllogism as follows:

$$\text{аII-IV:} \begin{array}{c} (\forall x)(\neg Px \Rightarrow \neg Mx) \\ (\exists x)(Mx \land Sx) \end{array}$$

$$\frac{(\exists x)(Sx \land Sx)}{(\exists x)(Sx \land Px).}$$

We know that:

> *TIQ* (¬*Px* ⇒ ¬*Mx*) ⇒ (*Mx* ⇒ *Px*). (18)

We also know that:

$$T^{IQ} \vdash (M\mathbf{x} \Rightarrow P\mathbf{x}) \Rightarrow ((M\mathbf{x} \land S\mathbf{x}) \Rightarrow (S\mathbf{x} \land P\mathbf{x})).\tag{19}$$

By the application of transitivity on (18) and (19), we obtain:

$$T^{IQ} \vdash (\neg P\mathbf{x} \Rightarrow \neg M\mathbf{x}) \Rightarrow ((M\mathbf{x} \land S\mathbf{x}) \Rightarrow (S\mathbf{x} \land P\mathbf{x})) .$$

By generalization (∀*x*) and quatifier properties, we obtain:

> *TIQ* (∀*x*)( ¬*Px* ⇒ ¬*Mx*) ⇒ ((<sup>∃</sup>*x*)(*Mx* ∧ *Sx*) ⇒ (<sup>∃</sup>*x*)(*Sx* ∧ *Px*)).

The strong validity of other syllogisms can be proven similarly.

**Theorem 32.** *Let syllogisms* **aII***-IV,* **Aii***-IV,* **EOi***-IV,* **eoI***-IV,* **aOo***-IV, and* **AoO***-IV be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** From the strongly valid syllogism **aII**-IV, and from monotonicity (Theorem A2), we prove the strongly valid syllogisms in the first column by transitivity. We can prove the syllogisms in the other columns analogously by using monotonicity (Theorem A2, Theorem 4).

We will consider the strong validity of some syllogisms without concrete proofs. These proofs are similar to the previous proofs in this article.

**Theorem 33.** *Let syllogisms* **AAa***-IV,* **aaA***-IV,* **eAe***-IV,* **EaE***-IV,* **eEA***-IV, and* **Eea***-IV be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** From the strongly valid syllogism **AAa**-IV and monotonicity (Theorem 4), we prove, by transitivity, the strongly valid syllogism in the first column. Similarly, we can prove the strong validity of the syllogisms in the other columns by monotonicity (Theorems A2 and 4).

**Theorem 34.** *Let syllogisms* **eAe***-IV,* **EaE***-IV,* **eEA***-IV, and* **Eea***-IV be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:* **(\*e)AO***-IV,* **(\*E)ao***-IV,* **(\*e)Ei***-IV, and* **(\*E)eI***-IV.*

**Proof.** This can be proven by transitivity and by Proposition 2.

**Theorem 35.** *Let syllogisms* **oAO***-IV,* **Oao***-IV,* **oEi***-IV,* **OeI***-IV,* **IEo***-IV, and* **ieO***-IV be strongly valid in TIQ. Then, the following syllogisms are strongly valid in TIQ:*


**Proof.** From the strongly valid syllogism **oAO**-IV and monotonicity (Theorem 4), we prove the strongly valid syllogisms in the first column by transitivity. The syllogisms in the other columns can be proven similarly, by using monotonicity (Theorems A2 and 4).

#### *3.8. Examples of Logical Syllogisms in Finite Model*

The model theory deals with the relation between syntax and semantics. In finite model theory, an interpretation is restricted to finite structures which have finite universes. In our examples, the models are restricted to finite universes—the universe in the first example consists of six elements and the universe in the second example consists of eight elements.

Below, we introduce several examples of new forms of logical syllogisms. We will show examples of valid syllogisms in the simple model with the finite set *M-* of elements. Details of the constructed model can be found in [16]. The built model is M = (*<sup>M</sup><sup>α</sup>*, <sup>=</sup>*α*)*α*<sup>∈</sup>*Types*, LΔ, where *Mo* = [0, 1] is based on the standard Łukasiewicz MVΔ-algebra. The Łukasiewicz biresiduation ↔ represents the fuzzy equality <sup>=</sup>*o*. The logical implication is represented by the Łukasiewicz residuation →.

We will assume the example of the fuzzy measure which was proposed in Section 2.3 by Equation (6). For model M, it holds true that M | = *TIQ*. The syllogism in model M is valid if

$$
\mathcal{M}(P\_1) \otimes \mathcal{M}(P\_2) \le \mathcal{M}(\mathbb{C}).
$$

3.8.1. Example of Valid Syllogism of Figure III

Let us assume the following syllogism:

> *P*1 : All flu are viral diseases.

**AOo**: *P*2 : Some flu are not diseases transmittable to humans *C* Somediseaseswhicharenot transmittabletohumansareviraldiseases.

We assume the frame which is described in the previous subsection. Let *M-* be a set of diseases. We consider six diseases *U* = {*<sup>u</sup>*1, *u*2, *u*3, *u*4, *u*5, *<sup>u</sup>*6}. We interpret the formulas in the considered model as follows: Let Flu*o-* be the formula "flu", with the interpretation M(Flu*o-*) = *F* ⊂ ∼*M-* defined by

$$F = \{ 0.3/\mathfrak{u}\_1, 0.8/\mathfrak{u}\_2, 0.4/\mathfrak{u}\_3, 0.7/\mathfrak{u}\_4, 0.4/\mathfrak{u}\_5, 0.5/\mathfrak{u}\_6 \}.$$

Let Vir*o-* be the formula "viral diseases", with the interpretation M(Vir*o-*) = *V* ⊂ ∼ *M-* defined by

$$V = \{0.8/\mathfrak{u}\_1, 0.2/\mathfrak{u}\_2, 0.3/\mathfrak{u}\_3, 0.9/\mathfrak{u}\_4, 0.7/\mathfrak{u}\_5, 0.6/\mathfrak{u}\_6\}.$$

Let Tran*o-* be the formula "diseases transmittable to humans", with the interpretation M(Tran*o-*) = *T* ⊂ ∼*M-* defined by

$$T = \{0.2/\mathfrak{u}\_1, 0.1/\mathfrak{u}\_2, 0.7/\mathfrak{u}\_3, 0.3/\mathfrak{u}\_4, 0.1/\mathfrak{u}\_5, 0.6/\mathfrak{u}\_6\}.$$

Let Ntran*o-* be the formula "diseases not transmittable to humans", with the interpretation M(Ntran*o-*) = ¬ *T* ⊂ ∼*M-* defined by

$$\neg T = \{ 0.8/\mathfrak{u}\_1, 0.9/\mathfrak{u}\_2, 0.3/\mathfrak{u}\_3, 0.7/\mathfrak{u}\_4, 0.9/\mathfrak{u}\_5, 0.4/\mathfrak{u}\_6 \}. $$

**Major premise**: "All flu are viral diseases" is the formula:

$$\mathcal{Q}\_{\mathrm{Bid}}^{\vee}(\mathrm{Flu}\_{\mathrm{oc}}, \mathrm{Vir}\_{\mathrm{oc}}) := (\forall \mathfrak{x}\_{\mathfrak{c}}) (\mathrm{Flu}\_{\mathrm{oc}}(\mathfrak{x}\_{\mathfrak{c}}) \Rightarrow \mathrm{Vir}\_{\mathrm{oc}}(\mathfrak{x}\_{\mathfrak{c}}))\_{\mathfrak{c}}$$

which is interpreted by:

$$\mathcal{M}(Q\_{\text{Bid}}^\vee(\text{Flu}\_{\text{oc}}, \text{Vir}\_{\text{oc}})) = \bigwedge\_{m \in M\_{\text{c}}} \left( \mathcal{M}(\text{Flu}\_{\text{oc}}(m)) \to \mathcal{M}(\text{Vir}\_{\text{oc}}(m)) \right) = 0.4. \tag{20}$$

**Minor premise**: "Some flu are not diseases transmittable to humans" is the formula:

$$Q\_{Bi\Delta}^{\preceq}(\text{Flu}\_{\text{oc}}, \text{Ntran}\_{\text{oc}}) := (\exists x\_{\mathfrak{c}})(\text{Flu}\_{\text{oc}}(x\_{\mathfrak{c}}) \, \mathsf{N} \, \mathsf{N} \text{tran}\_{\text{oc}}(x\_{\mathfrak{c}})),$$

which is interpreted by:

$$\mathcal{M}(Q\_{\text{Bi}\Delta}^{\exists}(\text{Flu}\_{\text{05}}, \text{Ntran}\_{\text{05}})) = \bigvee\_{m \in M\_{\text{\mathcal{Q}}}} (\mathcal{M}(\text{Flu}\_{\text{05}}(m)) \wedge \mathcal{M}(\text{Ntran}\_{\text{05}}(m))) = 0.8. \tag{21}$$

**Conclusion**: "Some diseases which are not transmittable to humans are viral diseases" is the formula:

$$Q\_{\mathrm{Bi}}^{\exists}(\mathrm{Ntran}\_{\mathrm{oc}}, \mathrm{Vir}\_{\mathrm{oc}}) := (\exists \mathbf{x}\_{\mathrm{c}})(\mathrm{Ntran}\_{\mathrm{oc}}(\mathbf{x}\_{\mathrm{c}}) \, \mathsf{A} \, \mathrm{Vir}\_{\mathrm{oc}}(\mathbf{x}\_{\mathrm{c}})),$$

which is interpreted by:

$$\mathcal{M}(Q^{\exists}\_{\text{Bid}}(\text{Ntran}\_{\text{05}}, \text{Vir}\_{\text{05}})) = \bigvee\_{m \in M\_{\text{0}}} (\mathcal{M}(\text{Ntran}\_{\text{05}}(m)) \wedge \mathcal{M}(\text{Vir}\_{\text{05}}(m))) = 0.8. \tag{22}$$

From (20)–(22), we can see that the condition of validity in our model is satisfied because M(*<sup>P</sup>*1) ⊗ M(*<sup>P</sup>*2) = 0.4 ⊗ 0.8 = 0.2 ≤ M(*C*) = 0.8.

3.8.2. Example of Valid Syllogism of Figure IV

> *P*1: Many diseases which are not lethal are virus diseases.

**(\*g)Ei**: *P*2: All virus diseases can not be cured by antibiotics.

*P*3: Some diseases which can not be cured by antibiotics are not lethal diseases.

We suppose the same frame and the fuzzy measure as in the previous example. Let *M-* be a set of diseases. We consider eight diseases *V* = {*<sup>v</sup>*1, *v*2, *v*3, *v*4, *v*5, *v*6, *v*7, *<sup>v</sup>*8}. We interpret the formulas in the considered model as follows: Let Nlethal*o-* be the formula "diseases which are not lethal", with the interpretation M(Nlethal*o-*) = ¬*L* ⊂∼*M-* defined by

$$\neg L = \{1/\upsilon\_1, 0.8/\upsilon\_2, 0.3/\upsilon\_3, 0.5/\upsilon\_4, 0.7/\upsilon\_5, 0.4/\upsilon\_6, 0.4/\upsilon\_7, 0.4/\upsilon\_8\}.$$

Let Vir*o-* be the formula "virus diseases", with the interpretation M(Vir*o-*) = *V* ⊂∼ *M-* defined by

$$V = \{0.6/v\_1, 0.4/v\_2, 0.2/v\_3, 0.6/v\_4, 0.7/v\_5, 0.5/v\_6, 0.3/v\_7, 0.2/v\_8\}.$$

Let Natb*o-* be the formula "diseases which cannot be cured by antibiotics", with the interpretation M(Natb*o-*) = ¬*A* ⊂∼*M-* defined by

$$\neg A = \{0.7/\upsilon\_1, 0.5/\upsilon\_2, 0.3/\upsilon\_3, 0.7/\upsilon\_4, 0.7/\upsilon\_5, 0.5/\upsilon\_6, 0.3/\upsilon\_7, 0.5/\upsilon\_8\}.\}$$

**Major premise**: "Many diseases which are not lethal are virus diseases." can be represented in our model as:

M((<sup>∃</sup>*zo-*)[(∀*<sup>x</sup>-*)((Nlethal*o-*|*zo-*)(*<sup>x</sup>-*) ⇒ Vir*o-*(*<sup>x</sup>-*))&

&(<sup>∃</sup>*<sup>x</sup>-*)(Nlethal*o-*|*zo-*)(*<sup>x</sup>-*) ∧ (¬*Sm*)((*μ* Nlethal*o-*)(Nlethal*o-*|*zo-*))]). (23)

This leads us to find the fuzzy set M(Nlethal*o-*|*zo-*) = *C* ⊂∼ *M-*, which gives us the greatest degree in (23). It can be confirmed that fuzzy set *C* = {0.5/*<sup>v</sup>*4, 0.7/*<sup>v</sup>*5, 0.4/*<sup>v</sup>*6} ⊂∼ ¬*L* leads to the greatest degree in (23).

$$\mathcal{M}(Q\_{-,\text{Sim}}^{\forall}(\text{Nlethal}\_{\text{oc}}, \text{Vir}\_{\text{oc}})) = 1 \otimes 0.7 \land 1 = 0.7. \tag{24}$$

**Minor premise**: "All virus diseases can not be cured by antibiotics" is the formula:

$$Q\_{\mathrm{Bi}\mathbf{A}}^{\vee}(\mathrm{Vir}\_{\mathrm{oc}\prime}\mathrm{Natb}\_{\mathrm{oc}}) := (\forall \mathbf{x}\_{\mathfrak{c}}) (\mathrm{Vir}\_{\mathrm{oc}}(\mathbf{x}\_{\mathfrak{c}}) \Rightarrow \mathrm{Natb}\_{\mathrm{oc}}(\mathbf{x}\_{\mathfrak{c}})),$$

which is interpreted by:

$$\mathcal{M}(Q\_{\text{Bid}}^{\circ}(\text{Vir}\_{\text{oc}}, \text{Natb}\_{\text{oc}})) = \bigwedge\_{m \in M\_{\text{c}}} \left( \mathcal{M}(\text{Vir}\_{\text{oc}}(m)) \to \mathcal{M}(\text{Natb}\_{\text{oc}}(m)) \right) = 1. \tag{25}$$

**Conclusion**: "Some diseases which can not be cured by antibiotics are not lethal diseases" is the formula:

$$Q\_{\mathrm{BiA}}^{\exists}(\mathrm{Natb}\_{\mathrm{oc}}, \mathrm{Nlethal}\_{\mathrm{oc}}) := (\exists x\_{\mathrm{c}})(\mathrm{Natb}\_{\mathrm{oc}}(x\_{\mathrm{c}}) \, \mathsf{Nlethal}\_{\mathrm{oc}}(x\_{\mathrm{c}})),$$

which is interpreted by:

$$\mathcal{M}(Q\_{\text{Bid}}^{\exists}(\text{Natb}\_{\text{ox}}.\text{Nlethal}\_{\text{ox}})) = \bigvee\_{m \in M\_{\text{\\_}}} (\mathcal{M}(\text{Natb}\_{\text{ox}}(m)) \wedge \mathcal{M}(\text{Nlethal}\_{\text{ox}}(m))) = 0.7. \tag{26}$$

From (24)–(26), we can see that the condition of validity is satisfied in our model because M(*<sup>P</sup>*1) ⊗ M(*<sup>P</sup>*2) = 0.7 ⊗ 1 = 0.7 ≤ M(*C*) = 0.7.
