# Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

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## Abstract

**:**

## 1. Introduction

## 2. The Boolean Algebra ${\mathbb{B}}_{4}$ and Its Polyhedral Hasse Diagram

contradictory ($\mathit{CD}$) | iff | ${b}_{1}\wedge {b}_{2}=0000$ | and | ${b}_{1}\vee {b}_{2}=1111$, |

contrary (C) | iff | ${b}_{1}\wedge {b}_{2}=0000$ | and | ${b}_{1}\vee {b}_{2}\ne 1111$, |

subcontrary ($\mathit{SC}$) | iff | ${b}_{1}\wedge {b}_{2}\ne 0000$ | and | ${b}_{1}\vee {b}_{2}=1111$, |

in subalternation ($\mathit{SA}$) | iff | ${b}_{1}\wedge {b}_{2}={b}_{1}$ | and | ${b}_{1}\vee {b}_{2}\ne {b}_{1}$. |

## 3. Polyhedral Aristotelian Diagrams for ${\mathbb{B}}_{4}$

#### 3.1. The Aristotelian Rhombic Dodecahedron for ${\mathbb{B}}_{4}$

#### 3.2. The Aristotelian Tetrakis Hexahedron for ${\mathbb{B}}_{4}$

#### 3.3. The Aristotelian Tetraicosahedron for ${\mathbb{B}}_{4}$

#### 3.4. The Aristotelian Nested Tetrahedron for ${\mathbb{B}}_{4}$

#### 3.5. Summary

## 4. A Comparative Analysis of Logical and Geometrical Distance

#### 4.1. Logical and Geometrical Distance in the Aristotelian Rhombic Dodecahedron for ${\mathbb{B}}_{4}$

#### 4.2. Logical and Geometrical Distance in the Aristotelian Tetrakis Hexahedron for ${\mathbb{B}}_{4}$

#### 4.3. Logical and Geometrical Distance in the Aristotelian Tetraicosahedron for ${\mathbb{B}}_{4}$

- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{2},\mathit{SA}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{1},{b}_{4}\in {L}_{3}$ and $\mathit{SA}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=1<2={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{TIH}}({b}_{1},{b}_{2})=2={d}_{\mathit{TIH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{2},{b}_{2}\in {L}_{3},\mathit{SA}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{1},{b}_{4}\in {L}_{3}$ and $\mathit{SA}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=1<2={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{TIH}}({b}_{1},{b}_{2})=2={d}_{\mathit{TIH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{2},\mathit{C}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{1},{b}_{4}\in {L}_{3}$ and $\mathit{CD}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=3<4={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{TIH}}({b}_{1},{b}_{2})=3.70>3.46={d}_{\mathit{TIH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{2},{b}_{3}\in {L}_{3},\mathit{SC}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{1},{b}_{4}\in {L}_{3}$ and $\mathit{CD}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=3<4={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{TIH}}({b}_{1},{b}_{2})=3.70>3.46={d}_{\mathit{TIH}}({b}_{3},{b}_{4})$.

- ${d}_{\mathit{TIH}}(1000,0110)=3.70>3.46={d}_{\mathit{TIH}}(1000,0111)={d}_{\mathit{TIH}}(1000,\neg 1000)$, so $\neg 1000\ne arg{max}_{x\in {\mathbb{B}}_{4}}{d}_{\mathit{TIH}}(1000,x)$,
- ${d}_{\mathit{TIH}}(0111,1100)=3.70>3.46={d}_{\mathit{TIH}}(0111,1000)={d}_{\mathit{TIH}}(0111,\neg 0111)$, so $\neg 0111\ne arg{max}_{x\in {\mathbb{B}}_{4}}{d}_{\mathit{TIH}}(0111,x)$.

#### 4.4. Logical and Geometrical Distance in the Aristotelian Nested Tetrahedron for ${\mathbb{B}}_{4}$

- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{2},\mathit{SA}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{2},{b}_{4}\in {L}_{2}$ and $\mathit{Un}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=1<2={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=1.41={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{2},\mathit{SA}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{2},{b}_{4}\in {L}_{3}$ and $\mathit{SC}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=1<3={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=1.41={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{2},{b}_{2}\in {L}_{2},\mathit{Un}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{2},{b}_{4}\in {L}_{3}$ and $\mathit{SC}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=2<3={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=1.41={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{2},\mathit{SA}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{3},{b}_{4}\in {L}_{3}$ and $\mathit{SC}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=1<2={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=1.41>0.94={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{3},\mathit{SA}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{2},{b}_{4}\in {L}_{3}$ and $\mathit{SC}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=2<3={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=1.63>1.41={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{1},\mathit{C}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{1},{b}_{4}\in {L}_{2}$ and $\mathit{C}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=2<3={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=2.83>2.45={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{1},\mathit{C}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{2},{b}_{4}\in {L}_{3}$ and $\mathit{SC}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=2<3={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=2.83>1.41={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{1},\mathit{C}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{1},{b}_{4}\in {L}_{3}$ and $\mathit{CD}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=2<4={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=2.83>2.31={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{1},\mathit{C}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{2},{b}_{4}\in {L}_{2}$ and $\mathit{CD}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=2<4={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=2.83>2={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$,
- if ${b}_{1}\in {L}_{1},{b}_{2}\in {L}_{2},\mathit{C}({b}_{1},{b}_{2}),{b}_{3}\in {L}_{1},{b}_{4}\in {L}_{3}$ and $\mathit{CD}({b}_{3},{b}_{4})$, then ${d}_{H}({b}_{1},{b}_{2})=3<4={d}_{H}({b}_{3},{b}_{4})$ and yet ${d}_{\mathit{NTH}}({b}_{1},{b}_{2})=2.45>2.31={d}_{\mathit{NTH}}({b}_{3},{b}_{4})$.

- ${d}_{\mathit{NTH}}(1000,0110)=2.45>2.31={d}_{\mathit{NTH}}(1000,0111)={d}_{\mathit{NTH}}(1000,\neg 1000)$, so $\neg 1000\ne arg{max}_{x\in {\mathbb{B}}_{4}}{d}_{\mathit{NTH}}(1000,x)$,
- ${d}_{\mathit{NTH}}(1100,0001)=2.45>2={d}_{\mathit{NTH}}(1100,0011)={d}_{\mathit{NTH}}(1100,\neg 1100)$, so $\neg 1100\ne arg{max}_{x\in {\mathbb{B}}_{4}}{d}_{\mathit{NTH}}(1100,x)$.

#### 4.5. Summary

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**(

**a**) Two-dimensional Hasse diagram for ${\mathbb{B}}_{4}$; (

**b**) Three-dimensional Hasse diagram for ${\mathbb{B}}_{4}$, based on a rhombic dodecahedron (with the planes corresponding to the logical levels shown in gray).

**Figure 3.**(

**a**) Aristotelian rhombic dodecahedron for ${\mathbb{B}}_{4}$; (

**b**) The rhombic dodecahedron as a cube with pyramids put onto each of its six faces.

**Figure 5.**(

**a**) Tetrahedron for the ${L}_{1}$-bitstrings; (

**b**) octahedron for the ${L}_{2}$-bitstrings and (

**c**) tetrahedron for the ${L}_{3}$-bitstrings inside the Aristotelian rhombic dodecahedron for ${\mathbb{B}}_{4}$.

**Figure 6.**(

**a**) Aristotelian tetrakis hexahedron for ${\mathbb{B}}_{4}$; (

**b**) The tetrakis hexahedron as a cube with pyramids put onto each of its six faces.

**Figure 7.**(

**a**) Aristotelian tetraicosahedron for ${\mathbb{B}}_{4}$; (

**b**) The tetraicosahedron as a cube with pyramids put onto each of its six faces.

**Figure 8.**(

**a**) Aristotelian nested tetrahedron for ${\mathbb{B}}_{4}$, sitting on one of its faces; (

**b**) Aristotelian nested tetrahedron for ${\mathbb{B}}_{4}$, sitting on one of its edges.

$\mathit{b}\in {\mathbb{B}}_{4}$ | ${\mathit{c}}_{\mathit{RDH}}(\mathit{b})$ | ${\mathit{c}}_{\mathit{THH}}(\mathit{b})$ | ${\mathit{c}}_{\mathit{TIH}}(\mathit{b})$ | ${\mathit{c}}_{\mathit{NTH}}(\mathit{b})$ |
---|---|---|---|---|

0000 | $(0,0,0)$ | $(0,0,0)$ | $(0,0,0)$ | $(0,0,0)$ |

1000 | $(-1,1,-1)$ | $(-1,1,-1)$ | $(-1,1,-1)$ | $(-1,1,-1)$ |

0100 | $(-1,-1,1)$ | $(-1,-1,1)$ | $(-1,-1,1)$ | $(-1,-1,1)$ |

0010 | $(1,-1,-1)$ | $(1,-1,-1)$ | $(1,-1,-1)$ | $(1,-1,-1)$ |

0001 | $(1,1,1)$ | $(1,1,1)$ | $(1,1,1)$ | $(1,1,1)$ |

1100 | $(-2,0,0)$ | $(-\frac{3}{2},0,0)$ | $(-1-\sqrt{2},0,0)$ | $(-1,0,0)$ |

1010 | $(0,0,-2)$ | $(0,0,-\frac{3}{2})$ | $(0,0,-1-\sqrt{2})$ | $(0,0,-1)$ |

1001 | $(0,2,0)$ | $(0,\frac{3}{2},0)$ | $(0,1+\sqrt{2},0)$ | $(0,1,0)$ |

0110 | $(0,-2,0)$ | $(0,-\frac{3}{2},0)$ | $(0,-1-\sqrt{2},0)$ | $(0,-1,0)$ |

0101 | $(0,0,2)$ | $(0,0,\frac{3}{2})$ | $(0,0,1+\sqrt{2})$ | $(0,0,1)$ |

0011 | $(2,0,0)$ | $(\frac{3}{2},0,0)$ | $(1+\sqrt{2},0,0)$ | $(1,0,0)$ |

1110 | $(-1,-1,-1)$ | $(-1,-1,-1)$ | $(-1,-1,-1)$ | $(-\frac{1}{3},-\frac{1}{3},-\frac{1}{3})$ |

1101 | $(-1,1,1)$ | $(-1,1,1)$ | $(-1,1,1)$ | $(-\frac{1}{3},\frac{1}{3},\frac{1}{3})$ |

1011 | $(1,1,-1)$ | $(1,1,-1)$ | $(1,1,-1)$ | $(\frac{1}{3},\frac{1}{3},-\frac{1}{3})$ |

0111 | $(1,-1,1)$ | $(1,-1,1)$ | $(1,-1,1)$ | $(\frac{1}{3},-\frac{1}{3},\frac{1}{3})$ |

1111 | $(0,0,0)$ | $(0,0,0)$ | $(0,0,0)$ | $(0,0,0)$ |

**Table 2.**Vertices, edges and faces of the four polyhedral Aristotelian diagrams studied in this paper.

Elements | Rhombic Dodecahedron | Tetrakis Hexahedron | Tetraicosahedron | Nested Tetrahedron |
---|---|---|---|---|

(RDH) | (THH) | (TIH) | (NTH) | |

vertices | 14 | 14 | 14 | 4 |

edges | 24 | 36 | 36 | 6 |

faces | 12 | 24 | 24 | 4 |

**Table 3.**Logical and geometrical distance in the four polyhedral Aristotelian diagrams for ${\mathbb{B}}_{4}$.

${\mathit{d}}_{\mathit{H}}(\mathit{b},{\mathit{b}}^{\prime})$ | $\mathit{L}(\mathit{b})$ | $\mathit{L}({\mathit{b}}^{\prime})$ | $\mathit{R}(\mathit{b},{\mathit{b}}^{\prime})$ | Example | ${\mathit{d}}_{\mathit{RDH}}(\mathit{b},{\mathit{b}}^{\prime})$ | ${\mathit{d}}_{\mathit{THH}}(\mathit{b},{\mathit{b}}^{\prime})$ | ${\mathit{d}}_{\mathit{TIH}}(\mathit{b},{\mathit{b}}^{\prime})$ | ${\mathit{d}}_{\mathit{NTH}}(\mathit{b},{\mathit{b}}^{\prime})$ |
---|---|---|---|---|---|---|---|---|

1 | 1 | 2 | $\mathit{SA}$ | 1000-1100 | 1.73 | 1.5 | 2 | 1.41 |

1 | 2 | 3 | $\mathit{SA}$ | 1100-1110 | 1.73 | 1.5 | 2 | 0.82 |

2 | 1 | 3 | $\mathit{SA}$ | 1000-1110 | 2 | 2 | 2 | 1.63 |

2 | 1 | 1 | $\mathit{C}$ | 1000-0001 | 2.83 | 2.83 | 2.83 | 2.83 |

2 | 3 | 3 | $\mathit{SC}$ | 1110-0111 | 2.83 | 2.83 | 2.83 | 0.94 |

2 | 2 | 2 | $\mathit{Un}$ | 1100-0110 | 2.83 | 2.12 | 3.41 | 1.41 |

3 | 1 | 2 | $\mathit{C}$ | 1000-0110 | 3.32 | 2.87 | 3.70 | 2.45 |

3 | 2 | 3 | $\mathit{SC}$ | 1100-0111 | 3.32 | 2.87 | 3.70 | 1.41 |

4 | 1 | 3 | $\mathit{CD}$ | 1000-0111 | 3.46 | 3.46 | 3.46 | 2.31 |

4 | 2 | 2 | $\mathit{CD}$ | 1100-0011 | 4 | 3 | 4.83 | 2 |

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## Share and Cite

**MDPI and ACS Style**

Demey, L.; Smessaert, H.
Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation. *Symmetry* **2017**, *9*, 204.
https://doi.org/10.3390/sym9100204

**AMA Style**

Demey L, Smessaert H.
Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation. *Symmetry*. 2017; 9(10):204.
https://doi.org/10.3390/sym9100204

**Chicago/Turabian Style**

Demey, Lorenz, and Hans Smessaert.
2017. "Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation" *Symmetry* 9, no. 10: 204.
https://doi.org/10.3390/sym9100204