# A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

${P}_{1}:$ Almost all people do not have a plane. |

${P}_{2}:$ Most people have a phone. |

$C:$ Some people who have a phone do not have a plane. |

${Q}_{1}$Y is M |

${Q}_{2}$M is X |

$(\ge {Q}_{1}\otimes {Q}_{2})$Y are X. |

${P}_{1}$: $[5\%,10\%]$ students have a job. |

${P}_{2}$: $[5\%,10\%]$ students have a child. |

C: $[0\%,10\%]$ students have a child and have a job. |

Most children like computer games. |

Most cats like to sleep. |

#### 1.1. Application of Generalized Quantifiers

Most (many, few) analyzed time series stagnated recently, but their future trend is slightly increasing [20]. |

Most irises with both small-length sepals and petals have small-width petals. |

#### 1.2. Main Goals

Almost all students who do not like mathematics do not study technical fields. |

Most people who do not drink alcohol have healthy livers. |

${P}_{1}:$ Almost all people who do sports have healthy lungs. |

${P}_{2}:$ Almost all people who do sports do not have asthma. |

$C:$ Some people who do not have asthma have healthy lungs. |

#### 1.3. Application of New Forms of Fuzzy Intermediate Quantifiers

## 2. Main Methods

#### 2.1. Fuzzy Type Theory

**∧**, implication

**⇒**, negation

**¬**, strong conjunction

**&**, strong disjunction

**∇**, disjunction

**∨**, and delta $\mathbf{\Delta}$. The fuzzy type theory is complete, i.e., a theory T is consistent iff it has a (Henkin) model ($\mathcal{M}\vDash T$). We sometimes use the equivalent notion: $T\u22a2{A}_{o}$ iff $T\vDash {A}_{o}$.

#### 2.2. Evaluative Linguistic Expressions

- The constants $\top ,\perp \in {Form}_{o}$ for truth and falsity, and $\u2020\in {Form}_{o}$ for the middle truth value;
- A special constant $\sim \phantom{\rule{0.166667em}{0ex}}\in {Form}_{\left(oo\right)o}$ for an additional fuzzy equality on the set of truth values L;
- A set of special constants $\mathbf{\nu},\dots \in {Form}_{oo}$ for linguistic hedges. The ${J}^{\mathrm{Ev}}$ contains the following special constants: $\{Ex,Si,Ve$, $ML,Ro,QR,VR\}$’ these denote the linguistic hedges: (extremely, significantly, very, roughly, more or less, rather, quite roughly, and very roughly, respectively);
- A set of triples of next constants $({\mathbf{a}}_{\mathbf{\nu}},{\mathbf{b}}_{\mathbf{\nu}},{\mathbf{c}}_{\mathbf{\nu}}),\dots \in {Form}_{o}$, where each hedge $\mathbf{\nu}$ is uniquely connected with one triple of these constants.

**Lemma**

**1**

**.**The following ordering of the specific hedges can be proved.

**Theorem**

**1.**

**Proof.**

#### 2.3. Fuzzy Measure

**Definition**

**1.**

- A formula, $\mu \in {Form}_{o\left(o\alpha \right)\left(o\alpha \right)}$, defined by:$${\mu}_{o\left(o\alpha \right)\left(o\alpha \right)}:=\lambda {z}_{o\alpha}\phantom{\rule{0.166667em}{0ex}}\lambda {x}_{o\alpha}\phantom{\rule{0.166667em}{0ex}}\left(R{z}_{o\alpha}\right){x}_{o\alpha},$$represents a measure on fuzzy sets in the universe of type $\alpha \in Types$ if it has the following properties:
- 1.
- $\mathbf{\Delta}({x}_{o\alpha}\subseteq {z}_{o\alpha})\mathbf{\&}\mathbf{\Delta}({y}_{o\alpha}\subseteq {z}_{o\alpha})\mathbf{\&}\mathbf{\Delta}({x}_{o\alpha}\subseteq {y}_{o\alpha})\mathbf{\Rightarrow}(\left(\mu {z}_{o\alpha}\right){x}_{o\alpha}\mathbf{\Rightarrow}\left(\mu {z}_{o\alpha}\right){y}_{o\alpha})$;
- 2.
- $\mathbf{\Delta}({x}_{o\alpha}\subseteq {z}_{o\alpha})\mathbf{\Rightarrow}(\left(\mu {z}_{o\alpha}\right)({z}_{o\alpha}\setminus {x}_{o\alpha})\equiv \mathbf{\neg}\left(\mu {z}_{o\alpha}\right){x}_{o\alpha})$;
- 3.
- $\mathbf{\Delta}({x}_{o\alpha}\subseteq {y}_{o\alpha})\mathbf{\&}\mathbf{\Delta}({x}_{o\alpha}\subseteq {z}_{o\alpha})\mathbf{\&}\mathbf{\Delta}({y}_{o\alpha}\subseteq {z}_{o\alpha})\mathbf{\Rightarrow}(\left(\mu {z}_{o\alpha}\right){x}_{o\alpha}\mathbf{\Rightarrow}\left(\mu {y}_{o\alpha}\right){x}_{o\alpha})$.

**Example**

**1.**

#### 2.4. Formal Definition of Intermediate Quantifiers

**Proposition**

**1.**

**Definition**

**2.**

“$\langle Quantifier\rangle \phantom{\rule{4pt}{0ex}}B$s are A”.

**Definition**

**3**

**.**Let $\mathit{Ev}$ be a formula representing an evaluative expression, x be a variable, and $A,B,z$ be formulas. Then, either of the formulas:

“$\langle *Quantifier\rangle \phantom{\rule{4pt}{0ex}}B$s are A”.

***A, *E, *P, *B, *T, *D, *K, *G, *F, *V, *S, *Z, *I,**and

***O**.

## 3. Results

#### 3.1. Formal Structure of Peterson’s Syllogisms Related to Peterson’s Square

**Definition**

**4**

**.**A syllogism is a triple $\langle {P}_{1},{P}_{2},C\rangle $ 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.

**Definition**

**5.**

**Definition**

**6.**

**Theorem**

**2**

**.**The following syllogisms are strongly valid in ${T}^{IQ}$:

**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. |

**Theorem**

**3**

**.**The following syllogisms are strongly valid in ${T}^{IQ}$:

#### 3.2. New Forms of Fuzzy Intermediate Quantifiers Related to Graded Peterson’s Cube

**Definition**

**7**

**.**Let $\mathit{Ev}$ be a formula representing an evaluative expression, x be a variable, and $A,B,z$ be formulas. Then, for either of the formulas:

“$\langle quantifier\rangle $ not Bs are not A”.

***a, *e, *p, *b, *t, *d, *k, *g, *f, *v, *s, *z, *i,**and

***o**

**Theorem**

**4.**

- 1.
- $\begin{array}{c}{T}^{IQ}\u22a2\mathit{a}\mathbf{\Rightarrow}\mathit{p},{T}^{IQ}\u22a2\mathit{p}\mathbf{\Rightarrow}\mathit{t},{T}^{IQ}\u22a2\mathit{t}\mathbf{\Rightarrow}\mathit{k},\\ {T}^{IQ}\u22a2\mathit{k}\mathbf{\Rightarrow}\mathit{f},{T}^{IQ}\u22a2\mathit{f}\mathbf{\Rightarrow}\mathit{s},{T}^{IQ}\phantom{\rule{0.277778em}{0ex}}\u22a2\mathit{s}\mathbf{\Rightarrow}\mathit{i};\end{array}$
- 2.
- $\begin{array}{c}{T}^{IQ}\u22a2\mathit{e}\mathbf{\Rightarrow}\mathit{b},{T}^{IQ}\u22a2\mathit{b}\mathbf{\Rightarrow}\mathit{d},{T}^{IQ}\u22a2\mathit{d}\mathbf{\Rightarrow}\mathit{g},\\ {T}^{IQ}\u22a2\mathit{g}\mathbf{\Rightarrow}\mathit{v},{T}^{IQ}\u22a2\mathit{v}\mathbf{\Rightarrow}\mathit{z},{T}^{IQ}\u22a2\mathit{z}\mathbf{\Rightarrow}\mathit{o}.\end{array}$

**Proof.**

g: Many animals which are not mammals are fish. |

A: All dolphins are mammals. |

o: Some animals which are not dolphins are fish. |

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

**Theorem**

**5.**

aaa | ||||||

aap | app | |||||

aat | apt | att | ||||

aak | apk | atk | akk | |||

aaf | apf | atf | akf | aff | ||

aas | aps | ats | aks | afs | ass | |

a(*a)i | a(*p)i | a(*t)i | a(*k)i | a(*f)i | a(*s)i | aii |

**Proof.**

**Theorem**

**6.**

eae | ||||||

eab | epb | |||||

ead | epd | etd | ||||

eag | epg | etg | ekg | |||

eav | epv | etv | ekv | efv | ||

eaz | epz | etz | ekz | efz | esz | |

e(*a)o | e(*p)o | e(*t)o | e(*k)o | e(*f)o | e(*s)o | eio |

**Proof.**

#### 3.4. New Forms of Figure I

**Theorem**

**7.**

**aEE-I, aBB-I, aDD-I, aGG-I, aVV-I, aZZ-I,**and

**aOO-I**are strongly valid in ${T}^{IQ}$.

**Proof.**

aEE-I: | $(\forall x)(\mathbf{\neg}Mx\mathbf{\Rightarrow}\mathbf{\neg}Px)$ |

$(\forall x)(Sx\mathbf{\Rightarrow}\mathbf{\neg}Mx)$ | |

$(\forall x)(Sx\mathbf{\Rightarrow}\mathbf{\neg}Px)$. |

**Proof.**

aOO-I: | $(\forall x)(\mathbf{\neg}Mx\mathbf{\Rightarrow}\mathbf{\neg}Px)$ |

$(\exists x)(Sx\mathbf{\wedge}\mathbf{\neg}Mx)$ | |

$(\exists x)(Sx\mathbf{\wedge}\mathbf{\neg}Px)$. |

**Proof.**

aBB-I: | $(\forall x)(\mathbf{\neg}Mx\mathbf{\Rightarrow}\mathbf{\neg}Px)$ |

$(\exists z)\left[\right(\forall x\left)\right(\left(S\right|z)x\mathbf{\Rightarrow}\mathbf{\neg}Mx)\mathbf{\wedge}\left(BiEx\right)\left(\right(\mu S\left)\right(S\left|z\right)\left)\right]$ | |

$(\exists z)\left[\right(\forall x\left)\right(\left(S\right|z)x\mathbf{\Rightarrow}\mathbf{\neg}Px)\mathbf{\wedge}\left(BiEx\right)\left(\right(\mu S\left)\right(S\left|z\right)\left)\right]$. |

**aBB**-I. If we denote $Ev:=\left(BiVe\right)\left(\right(\mu S\left)\right(S\left|z\right))$, we obtain the strong validity of

**aDD**-I. If we put $Ev:=(\mathbf{\neg}Sm)\left(\right(\mu S\left)\right(S\left|z\right))$, we have the strong validity of syllogism

**aGG**-I. By denoting $Ev:=\left(SmSi\right)\left(\right(\mu S\left)\right(S\left|z\right))$, we obtain the strong validity of syllogism

**aVV**-I. Finally, if we denote $Ev:=\left(SmVe\right)\left(\right(\mu S\left)\right(S\left|z\right))$, we conclude that the syllogism

**aZZ**-I is strongly valid. □

**Theorem**

**8.**

**aEE**-I,

**aBB**-I,

**aDD**-I,

**aGG**-I,

**aVV**-I,

**aZZ**-I,

**aOO**-I be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

aEE | ||||||

aEB | aBB | |||||

aED | aBD | aDD | ||||

aEG | aBG | aDG | aGG | |||

aEV | aBV | aDV | aGV | aVV | ||

aEZ | aBZ | aDZ | aGZ | aVZ | aZZ | |

a(*E)O | a(*B)O | a(*D)O | a(*G)O | a(*V)O | a(*Z)O | aOO |

**Proof.**

**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.**

**Aee**-I,

**Abb**-I,

**Add**-I,

**Agg**-I

**Avv**-I,

**Azz**-I,

**Aoo**-I are strongly valid in ${T}^{IQ}$.

**Proof.**

**Theorem**

**10.**

**Aee**-I,

**Abb**-I,

**Add**-I,

**Agg**-I,

**Avv**-I,

**Azz**-I, and

**Aoo**-I be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

Aee | ||||||

Aeb | Abb | |||||

Aed | Abd | Add | ||||

Aeg | Abg | Adg | Agg | |||

Aev | Abv | Adv | Agv | Avv | ||

Aez | Abz | Adz | Agz | Avz | Azz | |

A(*e)o | A(*b)o | A(*d)o | A(*g)o | A(*v)o | A(*z)o | Aoo |

**Proof.**

**Theorem**

**11.**

**Eea**-I,

**Ebp**-I,

**Edt**-I,

**Egk**-I,

**Evf**-I,

**Ezs**-I, and

**Eoi**-I be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

Eea | ||||||

Eep | Ebp | |||||

Eet | Ebt | Edt | ||||

Eek | Ebk | Edk | Egk | |||

Eef | Ebf | Edf | Egf | Evf | ||

Ees | Ebs | Eds | Egs | Evs | Ezs | |

E(*e)i | E(*b)i | E(*d)i | E(*g)i | E(*v)i | E(*z)i | Eoi |

**Proof.**

**Theorem**

**12.**

**eEA**-I,

**eBP**-I,

**eDT**-I,

**eGK**-I,

**eVF**-I,

**eZS**-I, and

**eOI**-I be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

eEA | ||||||

eEP | eBP | |||||

eET | eBT | eDT | ||||

eEK | eBK | eDK | eGK | |||

eEF | eBF | eDF | eGF | eVF | ||

eES | eBS | eDS | eGS | eVS | eZS | |

e(*E)I | e(*B)I | e(*D)I | e(*G)I | e(*V)I | e(*Z)I | eOI |

**Proof.**

**Proposition**

**2**

- (a)
- ${T}^{IQ}\u22a2\mathbf{A}\mathbf{\Rightarrow}\mathbf{i}$;
- (b)
- ${T}^{IQ}\u22a2\mathbf{E}\mathbf{\Rightarrow}\mathbf{o}$;
- (c)
- ${T}^{IQ}\u22a2\mathbf{a}\mathbf{\Rightarrow}\mathbf{I}$;
- (d)
- ${T}^{IQ}\u22a2\mathbf{e}\mathbf{\Rightarrow}\mathbf{O}$.

**Theorem**

**13.**

**AAA**-I,

**aaa**-I,

**EAE**-I,

**eae**-I,

**Aee**-I,

**aEE**-I,

**Eea**-I, and

**eEA**-I be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

**(*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.**

**Theorem**

**14.**

**oAo**-I,

**iAi**-I,

**oeI**-I,

**ieO**-I,

**IEo**-I,

**OEi**-I,

**IaI**-I, and

**OaO**-I are strongly valid in ${T}^{IQ}$.

**Proof.**

oAo-I: | $(\exists x)(\mathbf{\neg}Mx\mathbf{\wedge}Px)$ |

$(\forall x)(Sx\mathbf{\Rightarrow}Mx)$ | |

$(\exists x)(\mathbf{\neg}Sx\mathbf{\wedge}Px)$. |

**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.**

**oAo**-I,

**iAi**-I,

**oeI**-I,

**ieO**-I,

**IEo**-I,

**OEi**-I,

**IaI**-I, and

**OaO**-I be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

(*e)Ao | (*E)aO | (*a)Ai | (*A)aI | (*e)eI | (*E)Ei | (*a)eO | (*A)Eo |

(*b)Ao | (*B)aO | (*p)Ai | (*P)aI | (*b)eI | (*B)Ei | (*p)eO | (*P)Eo |

(*d)Ao | (*D)aO | (*t)Ai | (*T)aI | (*d)eI | (*D)Ei | (*t)eO | (*T)Eo |

(*g)Ao | (*G)aO | (*k)Ai | (*K)aI | (*g)eI | (*G)Ei | (*k)eO | (*K)Eo |

(*v)Ao | (*V)aO | (*f)Ai | (*F)aI | (*v)eI | (*V)Ei | (*f)eO | (*F)Eo |

(*z)Ao | (*Z)aO | (*s)Ai | (*S)aI | (*z)eI | (*Z)Ei | (*s)eO | (*S)Eo |

oAo | OaO | iAi | IaI | oeI | OEi | ieO | IEo |

**Proof.**

**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

**Theorem**

**16.**

**aAA**-II,

**aPP**-II,

**aTT**-II,

**aKK**-II,

**aFF**-II,

**aSS**-II, and

**aII**-II be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

aAA | ||||||

aAP | aPP | |||||

aAT | aPT | aTT | ||||

aAK | aPK | aTK | aKK | |||

aAF | aPF | aTF | aKF | aFF | ||

aAS | aPS | aTS | aKS | aFS | aSS | |

a(*A)I | a(*P)I | a(*T)I | a(*K)I | a(*F)I | a(*S)I | aII |

**Proof.**

**Theorem**

**17.**

**Aaa**-II,

**App**-II,

**Att**-II,

**Akk**-II,

**Aff**-II,

**Ass**-II, and

**Aii**-II be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

Aaa | ||||||

Aap | App | |||||

Aat | Apt | Att | ||||

Aak | Apk | Atk | Akk | |||

Aaf | Apf | Atf | Akf | Aff | ||

Aas | Aps | Ats | Aks | Afs | Ass | |

A(*a)i | A(*p)i | A(*t)i | A(*k)i | A(*f)i | A(*s)i | Aii |

**Proof.**

**Theorem**

**18.**

**eEA**-II,

**eBP**-II,

**eDT**-II,

**eGK**-II,

**eVF**-II,

**eZS**-II, and

**eOI**-II be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

eEA | ||||||

eEP | eBP | |||||

eET | eBT | eDT | ||||

eEK | eBK | eDK | eGK | |||

eEF | eBF | eDF | eGF | eVF | ||

eES | eBS | eDS | eGS | eVS | eZS | |

e(*E)I | e(*B)I | e(*D)I | e(*G)I | e(*V)I | e(*Z)I | eOI |

**Proof.**

**Theorem**

**19.**

**Eea**-II,

**Ebp**-II,

**Edt**-II,

**Egk**-II,

**Evf**-II,

**Ezs**-II, and

**Eoi**-II be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

Eea | ||||||

Eep | Ebp | |||||

Eet | Ebt | Edt | ||||

Eek | Ebk | Edk | Egk | |||

Eef | Ebf | Edf | Egf | Evf | ||

Ees | Ebs | Eds | Egs | Evs | Ezs | |

E(*e)i | E(*b)i | E(*d)i | E(*g)i | E(*v)i | E(*z)i | Eoi |

**Proof.**

**Theorem**

**20.**

**EAE**-II,

**eae**-II,

**AEE**-II,

**aee**-II,

**aAA**-II,

**Aaa**-II,

**Eea**-II, and

**eEA**-II be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

**(*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.**

**Theorem**

**21.**

**OAo**-II,

**oaO**-II,

**IEo**-II,

**ieO**-II,

**OeI**-II,

**oEi**-II,

**IaI**-II, and

**iAi**-II be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

(*E)Ao | (*e)aO | (*A)Eo | (*a)eO | (*E)eI | (*e)Ei | (*a)Ai | (*A)aI |

(*B)Ao | (*b)aO | (*P)Eo | (*p)eO | (*B)eI | (*b)Ei | (*p)Ai | (*P)aI |

(*D)Ao | (*d)aO | (*T)Eo | (*t)eO | (*D)eI | (*d)Ei | (*t)Ai | (*T)aI |

(*G)Ao | (*g)aO | (*K)Eo | (*k)eO | (*G)eI | (*g)Ei | (*k)Ai | (*K)aI |

(*V)Ao | (*v)aO | (*F)Eo | (*f)eO | (*V)eI | (*v)Ei | (*f)Ai | (*F)aI |

(*Z)Ao | (*z)aO | (*S)Eo | (*s)eO | (*Z)eI | (*z)Ei | (*s)Ai | (*S)aI |

OAo | oaO | IEo | ieO | OeI | oEi | iAi | IaI |

**Proof.**

**Theorem**

**22.**

**(*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 ${T}^{IQ}$.

**Proof.**

(*A)Ai-II: | $(\forall x)(Px\mathbf{\Rightarrow}Mx)\mathbf{\&}(\exists x)(\mathbf{\neg}Mx)$ |

$(\forall x)(Sx\mathbf{\Rightarrow}Mx)$ | |

$(\exists x)(\mathbf{\neg}Sx\mathbf{\wedge}\mathbf{\neg}Px)$. |

#### 3.6. New Forms of Figure III

**Theorem**

**23.**

**AOo**-III,

***(PG)o**-III,

***(TD)o**-III,

***(KB)o**-III, and

**IEo**-III are strongly valid in ${T}^{IQ}$.

**Proof.**

AOo-III: | $(\forall x)(Mx\mathbf{\Rightarrow}Px)$ |

$(\exists x)(Mx\mathbf{\wedge}\mathbf{\neg}Sx)$ | |

$(\exists x)(\mathbf{\neg}Sx\mathbf{\wedge}Px)$. |

**Proof.**

IEo-III: | $(\exists x)(Mx\mathbf{\wedge}Px)$ |

$(\forall x)(Mx\mathbf{\Rightarrow}\mathbf{\neg}Sx)$ | |

$(\exists x)(\mathbf{\neg}Sx\mathbf{\wedge}Px)$. |

**Proof.**

PGo-III: | $(\exists z)\left[\right(\forall x\left)\right(\left(M\right|z)x\mathbf{\Rightarrow}Px)\mathbf{\wedge}\left(BiEx\right)\left(\right(\mu M\left)\right(M\left|z\right)\left)\right]$ |

$(\exists {z}^{\prime})[(\forall x)(\left(M\right|{z}^{\prime})x\mathbf{\Rightarrow}\mathbf{\neg}Sx)\mathbf{\wedge}(\mathbf{\neg}Sm)\left(\left(\mu M\right)\left(M\right|{z}^{\prime})\right)]$ | |

$(\exists x)(\mathbf{\neg}Sx\mathbf{\wedge}Px)$. |

***(PG)o**-III. If we put $Ev:=\left(BiVe\right)\left(\right(\mu M\left)\right(M\left|z\right))$ and $E{v}^{\prime}:=\left(BiVe\right)\left(\left(\mu M\right)\left(M\right|{z}^{\prime})\right)$, we obtain the strong validity of syllogism

***(TD)o**-III. If we put $Ev:=(\mathbf{\neg}Sm)\left(\right(\mu M\left)\right(M\left|z\right))$ and $E{v}^{\prime}:=\left(BiEx\right)\left(\left(\mu M\right)\left(M\right|{z}^{\prime})\right)$, we obtain the strong validity of syllogism

***(KB)o**-III. □

**Theorem**

**24.**

**AOo**-III,

***(PG)o**-III,

***(TD)o**-III,

***(KB)o**-III, and

**IEo**-III be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

(*A)Eo | (*P)Eo | (*T)Eo | (*K)Eo | (*F)Eo | (*S)Eo | IEo |

A(*B)o | *(PB)o | *(TB)o | *(KB)o | |||

A(*D)o | *(PD)o | *(TD)o | ||||

A(*G)o | *(PG)o | |||||

A(*V)o | ||||||

A(*Z)o | ||||||

AOo |

**Proof.**

**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.**

**aoO**-III,

***(pg)O**-III,

***(td)O**-III,

***(kb)O**-III, and

**ieO**-III are strongly valid in ${T}^{IQ}$.

**Proof.**

**Theorem**

**26.**

**aoO**-III,

***(pg)O**-III,

***(td)O**-III,

***(kb)O**-III, and

**ieO**-III be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

(*a)eO | (*p)eO | (*t)eO | (*k)eO | (*f)eO | (*s)eO | ieO |

a(*b)O | *(pb)O | *(tb)O | *(kb)O | |||

a(*d)O | *(pd)O | *(td)O | ||||

a(*g)O | *(pg)O | |||||

a(*v)O | ||||||

a(*z)O | ||||||

aoO |

**Proof.**

**Theorem**

**27.**

**EOi**-III,

***(BG)i**-III,

***(DD)i**-III,

***(GB)i**-III, and

**OEi**-III be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

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.**

**Theorem**

**28.**

**oeI**-III,

***(gb)I**-III,

***(dd)I**-III,

***(bg)I**-III, and

**eoI**-III be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

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.**

**Theorem**

**29.**

**eAe**-III,

**EaE**-III,

**aAa**-III,

**AaA**-III,

**eEA**-III,

**Eea**-III,

**aEE**-III, and

**Aee**-III be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

eAe | EaE | aAa | AaA | eEA | Eea | aEE | Aee |

eAb | EaB | aAp | AaP | eEP | Eep | aEB | Aeb |

eAd | EaD | aAt | AaT | eET | Eet | aED | Aed |

eAg | EaG | aAk | AaK | eEK | Eek | aEG | Aeg |

eAv | EaV | aAf | AaF | eEF | Eef | aEV | Aev |

eAz | EaZ | aAs | AaS | eES | Ees | aEZ | Aez |

e(*A)o | E(*a)O | a(*A)i | A(*a)I | e(*E)I | E(*e)i | a(*E)O | A(*e)o |

**Proof.**

**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.**

**eAe**-III,

**EaE**-III,

**aAa**-III,

**AaA**-III,

**eEA**-III,

**Eea**-III,

**aEE**-III, and

**Aee**-III be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

**(*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.**

#### 3.7. New Forms of Figure IV

**Theorem**

**31.**

**aII**-IV,

**Aii**-IV,

**EOi**-IV,

**eoI**-IV,

**aOo**-IV, and

**AoO**-IV are strongly valid in ${T}^{IQ}$.

**Proof.**

aII-IV: | $(\forall x)(\mathbf{\neg}Px\mathbf{\Rightarrow}\mathbf{\neg}Mx)$ |

$(\exists x)(Mx\mathbf{\wedge}Sx)$ | |

$(\exists x)(Sx\mathbf{\wedge}Px)$. |

**Theorem**

**32.**

**aII**-IV,

**Aii**-IV,

**EOi**-IV,

**eoI**-IV,

**aOo**-IV, and

**AoO**-IV be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

a(*A)I | A(*a)i | E(*E)i | e(*e)I | a(*E)o | A(*e)O |

a(*P)I | A(*p)i | E(*B)i | e(*b)I | a(*B)o | A(*b)O |

a(*T)I | A(*t)i | E(*D)i | e(*d)I | a(*D)o | A(*d)O |

a(*K)I | A(*k)i | E(*G)i | e(*g)I | a(*G)o | A(*g)O |

a(*F)I | A(*f)i | E(*V)i | e(*v)I | a(*V)o | A(*v)O |

a(*S)I | A(*s)i | E(*Z)i | e(*z)I | a(*Z)o | A(*z)O |

aII | Aii | EOi | eoI | aOo | AoO |

**Proof.**

**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). □

**Theorem**

**33.**

**AAa**-IV,

**aaA**-IV,

**eAe**-IV,

**EaE**-IV,

**eEA**-IV, and

**Eea**-IV be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

AAa | aaA | eAe | EaE | eEA | Eea |

AAp | aaP | eAb | EaB | eEP | Eep |

AAt | aaT | eAd | EaD | eET | Eet |

AAk | aaK | eAg | EaG | eEK | Eek |

AAf | aaF | eAv | EaV | eEF | Eef |

AAs | aaS | eAz | EaZ | eES | Ees |

A(*A)i | a(*a)I | e(*A)o | E(*a)O | e(*E)I | E(*e)i |

**Proof.**

**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.**

**eAe**-IV,

**EaE**-IV,

**eEA**-IV, and

**Eea**-IV be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

**(*e)AO**-IV,

**(*E)ao**-IV,

**(*e)Ei**-IV, and

**(*E)eI**-IV.

**Proof.**

**Theorem**

**35.**

**oAO**-IV,

**Oao**-IV,

**oEi**-IV,

**OeI**-IV,

**IEo**-IV, and

**ieO**-IV be strongly valid in ${T}^{IQ}$. Then, the following syllogisms are strongly valid in ${T}^{IQ}$:

(*e)AO | (*E)ao | (*e)Ei | (*E)eI | (*A)Eo | (*a)eO |

(*b)AO | (*B)ao | (*b)Ei | (*B)eI | (*P)Eo | (*p)eO |

(*d)AO | (*D)ao | (*d)Ei | (*D)eI | (*T)Eo | (*t)eO |

(*g)AO | (*G)ao | (*g)Ei | (*G)eI | (*K)Eo | (*k)eO |

(*v)AO | (*V)ao | (*v)Ei | (*V)eI | (*F)Eo | (*f)eO |

(*z)AO | (*Z)ao | (*z)Ei | (*Z)eI | (*S)Eo | (*s)eO |

oAO | Oao | oEi | OeI | IEo | ieO |

**Proof.**

**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

#### 3.8.1. Example of Valid Syllogism of Figure III

AOo: | ${P}_{1}:$ All flu are viral diseases. |

${P}_{2}:$ Some flu are not diseases transmittable to humans | |

C Some diseases which are not transmittable to humans are viral diseases. |

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

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

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

#### 3.8.2. Example of Valid Syllogism of Figure IV

(*g)Ei: | ${P}_{1}$: Many diseases which are not lethal are virus diseases. |

${P}_{2}$: All virus diseases can not be cured by antibiotics. | |

${P}_{3}$: Some diseases which can not be cured by antibiotics are not lethal diseases. |

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

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

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

## 4. Discussion

#### 4.1. Figure I

#### 4.2. Figure II

**(*A)Ai-II**, in which we can see that its presupposition is a formula $(\exists x)(\mathbf{\neg}Mx)$, but the middle formula in this syllogism is $\left(Mx\right)$. This is a consequence of the property of contraposition (Lemma A1(h)). The formula representing the presupposition is related to the assumption that all formulas are not empty.

#### 4.3. Figure III

#### 4.4. Figure IV

## 5. Conclusion and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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

**Lemma**

**A1.**

- (a)
- $\u22a2\left(\right(A\mathbf{\&}B)\mathbf{\Rightarrow}C)\equiv (A\mathbf{\Rightarrow}(B\mathbf{\Rightarrow}C\left)\right)$;
- (b)
- $\u22a2(A\mathbf{\Rightarrow}(B\mathbf{\Rightarrow}C\left)\right)\mathbf{\Rightarrow}(B\mathbf{\Rightarrow}(A\mathbf{\Rightarrow}C\left)\right)$;
- (c)
- $\u22a2(A\mathbf{\&}B)\mathbf{\Rightarrow}A;(A\mathbf{\wedge}B)\mathbf{\Rightarrow}A;(A\mathbf{\&}B)\mathbf{\Rightarrow}(A\mathbf{\wedge}B)$;
- (d)
- $\u22a2(A\mathbf{\&}B)\equiv (B\mathbf{\&}A)$;
- (e)
- $\u22a2(B\mathbf{\Rightarrow}C)\mathbf{\Rightarrow}\left(\right(A\mathbf{\Rightarrow}B)\mathbf{\Rightarrow}(A\mathbf{\Rightarrow}C\left)\right)$;
- (f)
- $\u22a2(C\mathbf{\Rightarrow}A)\mathbf{\Rightarrow}\left(\right(C\mathbf{\Rightarrow}B)\mathbf{\Rightarrow}(C\mathbf{\Rightarrow}(B\mathbf{\wedge}A)\left)\right)$;
- (g)
- $\u22a2(A\mathbf{\Rightarrow}B)\mathbf{\Rightarrow}\left(\right(A\mathbf{\wedge}C)\mathbf{\Rightarrow}(B\mathbf{\wedge}C\left)\right)$;
- (h)
- $\u22a2(A\mathbf{\Rightarrow}B)\mathbf{\Rightarrow}(\mathbf{\neg}B\mathbf{\Rightarrow}\mathbf{\neg}A)$;
- (i)
- $\u22a2\left(\right(A\mathbf{\Rightarrow}B)\mathbf{\&}(C\mathbf{\Rightarrow}D\left)\right)\mathbf{\Rightarrow}\left(\right(A\mathbf{\&}C)\mathbf{\Rightarrow}(B\mathbf{\&}D\left)\right)$.

**Lemma**

**A2.**

- (a)
- $\u22a2(\forall {x}_{\alpha})(A\mathbf{\Rightarrow}B)\mathbf{\Rightarrow}((\forall {x}_{\alpha})A\mathbf{\Rightarrow}(\forall {x}_{\alpha})B)$;
- (b)
- $\u22a2(\forall {x}_{\alpha})(A\mathbf{\Rightarrow}B)\mathbf{\Rightarrow}((\exists {x}_{\alpha})A\mathbf{\Rightarrow}(\exists {x}_{\alpha})B)$;
- (c)
- $\u22a2(\forall {x}_{\alpha})(A\mathbf{\Rightarrow}B)\mathbf{\Rightarrow}(A\mathbf{\Rightarrow}(\forall {x}_{\alpha})B)$, ${x}_{\alpha}$ is not free in A;
- (d)
- $\u22a2(\forall {x}_{\alpha})(A\mathbf{\Rightarrow}B)\mathbf{\Rightarrow}((\exists {x}_{\alpha})A\mathbf{\Rightarrow}B)$, ${x}_{\alpha}$ is not free in B.

**Lemma**

**A3.**

**Theorem**

**A1**

**.**Let T be a theory, and $A,B\in \phantom{\rule{4.pt}{0ex}}{Form}_{o}$ and $\alpha \in Types$.

- If $T\u22a2A$ and $T\u22a2A\mathbf{\Rightarrow}B$, then $T\u22a2B$;
- If $T\u22a2A$, then $T\u22a2(\forall {x}_{\alpha})A$.

#### Appendix A.2. Graded Peterson’s Square of Opposition

**Definition**

**A1.**

- ${P}_{1}$ and ${P}_{2}$ are contraries if $\mathcal{M}\left({P}_{1}\right)\otimes \mathcal{M}\left({P}_{2}\right)=0$;
- ${P}_{1}$ and ${P}_{2}$ are sub-contraries if $\mathcal{M}\left({P}_{1}\right)\oplus \mathcal{M}\left({P}_{2}\right)=1$;
- ${P}_{1}$ and ${P}_{2}$ are contradictories if both $\mathcal{M}(\mathbf{\Delta}{P}_{1})\otimes \mathcal{M}(\mathbf{\Delta}{P}_{2})=0$ and $\mathcal{M}(\mathbf{\Delta}{P}_{1})\oplus \mathcal{M}(\mathbf{\Delta}{P}_{2})=1$;
- ${P}_{2}$ is a sub-altern of ${P}_{1}$ if $\mathcal{M}\left({P}_{1}\right)\le \mathcal{M}\left({P}_{2}\right).$

**Theorem**

**A2.**

**A,…,O**be intermediate quantifiers. Then, the following set of implications is provable in ${T}^{IQ}$:

- 1.
- $\begin{array}{c}{T}^{IQ}\u22a2\mathit{A}\mathbf{\Rightarrow}\mathit{P},{T}^{IQ}\u22a2\mathit{P}\mathbf{\Rightarrow}\mathit{T},{T}^{IQ}\u22a2\mathit{T}\mathbf{\Rightarrow}\mathit{K},\\ {T}^{IQ}\u22a2\mathit{K}\mathbf{\Rightarrow}\mathit{F},{T}^{IQ}\u22a2\mathit{F}\mathbf{\Rightarrow}\mathit{S},{T}^{IQ}\phantom{\rule{0.277778em}{0ex}}\u22a2\mathit{S}\mathbf{\Rightarrow}\mathit{I};\end{array}$
- 2.
- $\begin{array}{c}{T}^{IQ}\u22a2\mathit{E}\mathbf{\Rightarrow}\mathit{B},{T}^{IQ}\u22a2\mathit{B}\mathbf{\Rightarrow}\mathit{D},{T}^{IQ}\u22a2\mathit{D}\mathbf{\Rightarrow}\mathit{G},\\ {T}^{IQ}\u22a2\mathit{G}\mathbf{\Rightarrow}\mathit{V},{T}^{IQ}\u22a2\mathit{V}\mathbf{\Rightarrow}\mathit{Z},{T}^{IQ}\u22a2\mathit{Z}\mathbf{\Rightarrow}\mathit{O}.\end{array}$

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Fiala, K.; Murinová, P.
A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube. *Mathematics* **2022**, *10*, 906.
https://doi.org/10.3390/math10060906

**AMA Style**

Fiala K, Murinová P.
A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube. *Mathematics*. 2022; 10(6):906.
https://doi.org/10.3390/math10060906

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Fiala, Karel, and Petra Murinová.
2022. "A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube" *Mathematics* 10, no. 6: 906.
https://doi.org/10.3390/math10060906