# Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

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## Abstract

**:**

## 1. Introduction

The standard modal square […] is valid with respect to any modal system at least as strong as the deontic system $\mathsf{KD}$, but invalid in any normal system strictly weaker than $\mathsf{KD}$.([44], p. 313, emphasis added)

## 2. Technical Background

#### 2.1. Modal Logic

**Prop**of atomic propositions, the modal language $\mathcal{L}$ is defined by means of the following grammar:

**Definition**

**1.**

**Definition**

**2.**

$\mathbb{M},w\models p$ | iff | $w\in V\left(p\right)$ |

$\mathbb{M},w\models \neg \phi $ | iff | $\mathbb{M},w$ $\phi $ |

$\mathbb{M},w\models \phi \wedge \psi $ | iff | $\mathbb{M},w\models \phi $and$\mathbb{M},w\models \psi $ |

$\mathbb{M},w\models \square \phi $ | iff | for all$v\in W$: if$wRv$then$\mathbb{M},v\models \phi $. |

**Definition**

**3.**

${\u22a2}_{\mathsf{K}}\phi $ | iff | ${\models}_{\mathsf{K}}\phi $ | ${\u22a2}_{\mathsf{KF}}\phi $ | iff | ${\models}_{\mathsf{KF}}\phi $ | |

${\u22a2}_{\mathsf{KD}}\phi $ | iff | ${\models}_{\mathsf{KD}}\phi $ | ${\u22a2}_{\mathsf{KDF}}\phi $ | iff | ${\models}_{\mathsf{KDF}}\phi $ |

**Definition**

**4.**

**Definition**

**5.**

$\mathbb{M},w\models \square \phi $ | iff | $\{v\in W\mid \mathbb{M},v\models \phi \}\in N\left(w\right)$. |

- For every augmented neighborhood frame $\mathbb{A}=\langle W,N\rangle $, there exists a modally equivalent Kripke frame ${\mathbb{A}}^{k}$, that is, for all valuations $V:\mathbf{Prop}\to \wp \left(W\right)$, states $w\in W$ and formulas $\phi \in \mathcal{L}$ we have $\langle \mathbb{A},V\rangle ,w\models \phi $ if $\langle {\mathbb{A}}^{k},V\rangle ,w\models \phi $;
- For every Kripke frame $\mathbb{K}=\langle W,R\rangle $, there exists a modally equivalent augmented neighborhood frame ${\mathbb{K}}^{n}$, that is, for all valuations $V:\mathbf{Prop}\to \wp \left(W\right)$, states $w\in W$ and formulas $\phi \in \mathcal{L}$ we have $\langle {\mathbb{K}}^{n},V\rangle ,w\models \phi $ if $\langle \mathbb{K},V\rangle ,w\models \phi $.

**Definition**

**6.**

${\u22a2}_{\mathsf{E}}\phi $ | iff | ${\models}_{\mathsf{E}}\phi $ | ${\u22a2}_{\mathsf{EMC}}\phi $ | iff | ${\models}_{\mathsf{EMC}}\phi $ | |

${\u22a2}_{\mathsf{EM}}\phi $ | iff | ${\models}_{\mathsf{EM}}\phi $ | ${\u22a2}_{\mathsf{EMN}}\phi $ | iff | ${\models}_{\mathsf{EMN}}\phi $ | |

${\u22a2}_{\mathsf{EC}}\phi $ | iff | ${\models}_{\mathsf{EC}}\phi $ | ${\u22a2}_{\mathsf{EMNC}}\phi $ | iff | ${\models}_{\mathsf{EMNC}}\phi $ |

#### 2.2. Logical Geometry

**Definition**

**7.**

$\mathsf{S}$-contradictory | iff | ${\models}_{\mathsf{S}}\neg (\phi \wedge \psi )$ | and | ${\models}_{\mathsf{S}}\phi \vee \psi $ |

$\mathsf{S}$-contrary | iff | ${\models}_{\mathsf{S}}\neg (\phi \wedge \psi )$ | and | $\phi \vee \psi $ |

$\mathsf{S}$-subcontrary | iff | $\neg (\phi \wedge \psi )$ | and | ${\models}_{\mathsf{S}}\phi \vee \psi $ |

in $\mathsf{S}$-subalternation | iff | ${\models}_{\mathsf{S}}\phi \to \psi $ | and | $\psi \to \phi $ |

**Definition**

**8.**

**Definition**

**9.**

- ${R}_{{\mathsf{S}}_{1}}(\phi ,\psi )$ iff ${R}_{{\mathsf{S}}_{2}}(f\left(\phi \right),f\left(\psi \right))$, for all Aristotelian relations R,
- $\phi {\equiv}_{{\mathsf{S}}_{1}}\psi $ iff $f\left(\phi \right){\equiv}_{{\mathsf{S}}_{2}}f\left(\psi \right)$.

**Definition**

**10.**

#### 2.3. Bitstring Semantics

## 3. Logic-Sensitivity and Aristotelian Families

#### 3.1. Introduction

#### 3.2. Examples from Normal Modal Logic

#### 3.3. Examples from Non-Normal Modal Logic

## 4. Logic-Sensitivity and Logical Equivalence of Formulas

#### 4.1. Introduction

#### 4.2. Examples from Normal Modal Logic

#### 4.3. Examples from Non-Normal Modal Logic

#### 4.4. Theory and Further Examples

**Theorem**

**1.**

- 1.
- If the Aristotelian diagram for $(\mathcal{F},\mathsf{S})$ is a degenerate square, then the Aristotelian diagram for $({f}_{1}\left(\mathcal{F}\right),\mathsf{S})$ is a classical square (with an $\mathsf{S}$-subalternation from $\alpha \wedge \beta $ to α);
- 2.
- If the Aristotelian diagram for $(\mathcal{F},\mathsf{S})$ is a classical square (with an $\mathsf{S}$-subalternation from α to β), then the Aristotelian diagram for $({f}_{1}\left(\mathcal{F}\right),\mathsf{S})$ is a PCD (with $\alpha \wedge \beta {\equiv}_{\mathsf{S}}\alpha $).

**Proof.**

**1.**

## 5. Logic-Sensitivity and Contingency of Formulas

#### 5.1. Introduction

#### 5.2. Examples from Normal Modal Logic

#### 5.3. Examples from Non-Normal Modal Logic

#### 5.4. Theory and Further Examples

**Theorem**

**2.**

- 1.
- If the Aristotelian diagram for $(\mathcal{F},\mathsf{S})$ is a degenerate square, then the Aristotelian diagram for $({f}_{2}\left(\mathcal{F}\right),\mathsf{S})$ is a classical square (with an $\mathsf{S}$-subalternation from $\alpha \wedge \neg \beta $ to α);
- 2.
- If the Aristotelian diagram for $(\mathcal{F},\mathsf{S})$ is a classical square (with an $\mathsf{S}$-subalternation from α to β), then $\alpha \wedge \neg \beta $ and $\neg \alpha \vee \beta $ are not $\mathsf{S}$-contingent and the Aristotelian diagram for $({f}_{2}\left(\mathcal{F}\right),\mathsf{S})$ is a PCD.

**Proof.**

## 6. Logic-Sensitivity and Boolean Subfamilies

#### 6.1. Introduction

#### 6.2. Theory and Examples

**Theorem**

**3.**

- 1.
- If the Aristotelian diagram for $(\mathcal{F},\mathsf{S})$ is a degenerate square, then the Aristotelian diagram for $({f}_{3}\left(\mathcal{F}\right),\mathsf{S})$ is a weak JSB hexagon (with pairwise $\mathsf{S}$-contrarieties between $\alpha \wedge \beta $,$\neg \alpha \wedge \beta $ and $\neg \alpha \wedge \neg \beta $);
- 2.
- If the Aristotelian diagram for $(\mathcal{F},\mathsf{S})$ is a classical square (with an $\mathsf{S}$-subalternation from α to β), then the Aristotelian diagram for $({f}_{3}\left(\mathcal{F}\right),\mathsf{S})$ is a strong JSB hexagon (with the same pairwise $\mathsf{S}$-contrarieties).

**Proof.**

**1.**

## 7. Conclusions

- If $({\mathcal{F}}_{1},\mathsf{S})$ and $({\mathcal{F}}_{2},\mathsf{S})$ are both degenerate squares, then $({f}_{4}({\mathcal{F}}_{1},{\mathcal{F}}_{2}),\mathsf{S})$ is a weak Buridan octagon;
- If exactly one of $({\mathcal{F}}_{1},\mathsf{S})$ and $({\mathcal{F}}_{2},\mathsf{S})$ is a degenerate square and the other is a classical square, then $({f}_{4}({\mathcal{F}}_{1},{\mathcal{F}}_{2}),\mathsf{S})$ is an intermediate Buridan octagon;
- If $({\mathcal{F}}_{1},\mathsf{S})$ and $({\mathcal{F}}_{2},\mathsf{S})$ are both classical squares, then $({f}_{4}({\mathcal{F}}_{1},{\mathcal{F}}_{2}),\mathsf{S})$ is a strong Buridan octagon.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Aristotelian diagrams for $\{\square p,\u25cap,\square \neg p,\u25ca\neg p\}$ with respect to two systems of modal logic. Full, dashed, and dotted lines visualize contradiction, contrariety, and subcontrariety, respectively; arrows visualize subalternations.

**Figure 3.**Aristotelian diagrams for ${\mathcal{F}}_{1a}:=\{\square p,\u25cap,\square \neg p,\u25ca\neg p\}$ in three normal modal logics.

**Figure 4.**Aristotelian diagrams for ${\mathcal{F}}_{1b}:=\{\square (p\wedge q),\square p\wedge \square q,\u25ca\neg p\vee \u25ca\neg q,\u25ca(\neg p\vee \neg q)\}$ in any normal modal logic and in three non-normal modal logics.

**Figure 5.**Aristotelian diagrams for ${\mathcal{F}}_{1a}=\{\square p,\u25cap,\square \neg p,\u25ca\neg p\}$ in two normal modal logics.

**Figure 6.**Aristotelian diagrams for ${\mathcal{F}}_{1b}=\{\square (p\wedge q),\square p\wedge \square q,\u25ca\neg p\vee \u25ca\neg q,\u25ca(\neg p\vee \neg q)\}$ in two non-normal modal logics.

**Figure 7.**Aristotelian diagrams for ${f}_{1}\left({\mathcal{F}}_{1a}\right)=\{\square p\wedge \u25cap,\square p,\u25ca\neg p,\square \neg p\vee \u25ca\neg p\}$ in two normal modal logics.

**Figure 8.**Aristotelian diagrams for ${f}_{1}\left({\mathcal{F}}_{1b}\right)=\{\square (p\wedge q)\wedge \square p\wedge \square q,\square (p\wedge q),\u25ca(\neg p\vee \neg q),\u25ca(\neg p\vee \neg q)\vee \u25ca\neg p\vee \u25ca\neg q\}$ in two non-normal modal logics.

**Figure 9.**Aristotelian diagrams for ${\mathcal{F}}_{2a}=\{\u25cap,\u25ca\top ,\square \perp ,\square \neg p\}$ in two normal modal logics.

**Figure 10.**Invalid (!) diagrams that approximate the PCDs for $({\mathcal{F}}_{1a},\mathsf{KDF})$ and $({\mathcal{F}}_{2a},\mathsf{KD})$.

**Figure 11.**Aristotelian diagrams for ${\mathcal{F}}_{2b}=\{\square p,\square \top ,\u25ca\perp ,\u25ca\neg p\}$ in two non-normal modal logics.

**Figure 12.**Aristotelian diagrams for ${f}_{2}\left({\mathcal{F}}_{1a}\right)=\{\square \perp ,\square p,\u25ca\neg p,\u25ca\top \}$ in two normal modal logics.

**Figure 13.**Aristotelian diagrams for ${f}_{2}\left({\mathcal{F}}_{1b}\right)=\{\square (p\wedge q)\wedge (\u25ca\neg p\vee \u25ca\neg q),\square (p\wedge q),\u25ca(\neg p\vee \neg q),\u25ca(\neg p\vee \neg q)\vee (\square \neg p\wedge \square \neg q)\}$ in two non-normal modal logics.

**Figure 14.**Aristotelian diagrams for ${f}_{3}\left({\mathcal{F}}_{1a}\right)=\{\square p\wedge \u25cap,\u25cap\wedge \u25ca\neg p,\square \neg p\wedge \u25ca\neg p,\square p\vee \u25cap,\square p\vee \square \neg p,\square \neg p\vee \u25ca\neg p\}$ in two normal modal logics.

**Figure 15.**Aristotelian diagrams for $\begin{array}{cc}{f}_{3}\left({\mathcal{F}}_{1b}\right)=& \{\square (p\wedge q)\wedge \square p\wedge \square q,\phantom{\rule{4pt}{0ex}}\square p\wedge \square q\wedge \u25ca(\neg p\vee \neg q),\phantom{\rule{4pt}{0ex}}(\u25ca\neg p\vee \phantom{\rule{0ex}{0ex}}\u25ca\neg q)\wedge \u25ca(\neg p\vee \neg q),\square (p\wedge q)\vee (\square p\wedge \square q),\phantom{\rule{4pt}{0ex}}\square (p\wedge q)\vee \u25ca\neg p\vee \u25ca\neg q,\phantom{\rule{4pt}{0ex}}\u25ca\neg p\vee \u25ca\neg q\vee \u25ca(\neg p\vee \neg q\left)\right\}\end{array}$ in two non-normal modal logics.

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Demey, L.
Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics. *Axioms* **2021**, *10*, 128.
https://doi.org/10.3390/axioms10030128

**AMA Style**

Demey L.
Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics. *Axioms*. 2021; 10(3):128.
https://doi.org/10.3390/axioms10030128

**Chicago/Turabian Style**

Demey, Lorenz.
2021. "Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics" *Axioms* 10, no. 3: 128.
https://doi.org/10.3390/axioms10030128