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

^{1}

^{2}

^{*}

## 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

## References

- Parsons, T. The Traditional Square of Opposition. In Stanford Encyclopedia of Philosophy; Zalta, E.N., Ed.; CSLI: Stanford, CA, USA, 2012. [Google Scholar]
- Read, S. John Buridan’s Theory of Consequence and His Octagons of Opposition. In Around and Beyond the Square of Opposition; Béziau, J.Y., Jacquette, D., Eds.; Springer: Basel, Switzerland, 2012; pp. 93–110. [Google Scholar]
- Lenzen, W. Leibniz’s Logic and the “Cube of Opposition”. Log. Univ.
**2016**, 10, 171–189. [Google Scholar] [CrossRef] - Kienzler, W. The Logical Square and the Table of Oppositions. Five Puzzles about the Traditional Square of Opposition Solved by Taking up a Hint from Frege. Log. Anal. Hist. Philos.
**2013**, 15, 398–413. [Google Scholar] - Beller, S. Deontic reasoning reviewed: Psychological questions, empirical findings, and current theories. Cognit. Process.
**2010**, 11, 123–132. [Google Scholar] [CrossRef] [PubMed] - Mikhail, J. Universal moral grammar: Theory, evidence and the future. Trends Cognit. Sci.
**2007**, 11, 143–152. [Google Scholar] [CrossRef] [PubMed] - Abrusci, V.M.; Casadio, C.; Medaglia, M.T.; Porcaro, C. Universal vs. Particular Reasoning: A Study with Neuroimaging Techniques. Log. J. IGPL
**2013**, 21, 1017–1027. [Google Scholar] [CrossRef] - Saurí, R.; Pustejovsky, J. FactBank: A Corpus Annotated with Event Factuality. Lang. Resour. Eval.
**2009**, 43, 227–268. [Google Scholar] [CrossRef] - Joerden, J. Logik im Recht; Springer: Berlin, Germany, 2010. [Google Scholar]
- O’Reilly, D. Using the Square of Opposition to Illustrate the Deontic and Alethic Relations Constituting Rights. Univ. Tor. Law J.
**1995**, 45, 279–310. [Google Scholar] [CrossRef] - Vranes, E. The Definition of ‘Norm Conflict’ in International Law and Legal Theory. Eur. J. Int. Law
**2006**, 17, 395–418. [Google Scholar] [CrossRef] - Dekker, P. Not Only Barbara. J. Log. Lang. Inf.
**2015**, 24, 95–129. [Google Scholar] [CrossRef] - Horn, L.R. A Natural History of Negation; University of Chicago Press: Chicago, IL, USA, 1989. [Google Scholar]
- Seuren, P.; Jaspers, D. Logico-Cognitive Structure in the Lexicon. Language
**2014**, 90, 607–643. [Google Scholar] [CrossRef] - Van der Auwera, J. Modality: The Three-layered Scalar Square. J. Semant.
**1996**, 13, 181–195. [Google Scholar] [CrossRef] - Glöckner, I. Fuzzy Quantifiers; Springer: Berlin, Germany, 2006. [Google Scholar]
- Murinová, P.; Novák, V. Analysis of Generalized Square of Opposition with Intermediate Quantifiers. Fuzzy Sets Syst.
**2014**, 242, 89–113. [Google Scholar] [CrossRef] - Murinová, P.; Novák, V. Graded Generalized Hexagon in Fuzzy Natural Logic. In Information Processing and Management of Uncertainty in Knowledge-Based Systems 2016, Part II; Carvalho, J.P., Lesot, M.J., Kaymak, U., Vieiram, S., Bouchon-Meunier, B., Yager, R.R., Eds.; CCIS 611; Springer: Berlin, Germany, 2016; pp. 36–47. [Google Scholar]
- Murinová, P.; Novák, V. Syllogisms and 5-Square of Opposition with Intermediate Quantifiers in Fuzzy Natural Logic. Log. Univ.
**2016**, 10, 339–357. [Google Scholar] [CrossRef] - Trillas, E.; Seising, R. Turning Around the Ideas of ‘Meaning’ and ‘Complement’. In Fuzzy Technology; Collan, M., Fedrizzi, M., Kacprzyk, J., Eds.; SFSC 335; Springer: Berlin, Germany, 2016; pp. 3–31. [Google Scholar]
- Carnielli, W.; Pizzi, C. Modalities and Multimodalities; Springer: Dordrecht, The Netherlands, 2008. [Google Scholar]
- Demey, L. Structures of Oppositions for Public Announcement Logic. In Around and Beyond the Square of Opposition; Béziau, J.Y., Jacquette, D., Eds.; Springer: Basel, Switzerland, 2012; pp. 313–339. [Google Scholar]
- Fitting, M.; Mendelsohn, R.L. First-Order Modal Logic; Kluwer: Dordrecht, The Netherlands, 1998. [Google Scholar]
- Lenzen, W. How to Square Knowledge and Belief. In Around and Beyond the Square of Opposition; Béziau, J.Y., Jacquette, D., Eds.; Springer: Basel, Switzerland, 2012; pp. 305–311. [Google Scholar]
- Luzeaux, D.; Sallantin, J.; Dartnell, C. Logical Extensions of Aristotle’s Square. Log. Univ.
**2008**, 2, 167–187. [Google Scholar] [CrossRef] - Gilio, A.; Pfeifer, N.; Sanfilippo, G. Transitivity in Coherence-Based Probability Logic. J. Appl. Log.
**2016**, 14, 46–64. [Google Scholar] [CrossRef] - Pfeifer, N.; Sanfilippo, G. Square of Opposition under Coherence. In Soft Methods for Data Science; AISC 456; Springer: Berlin, Germany, 2017; pp. 407–414. [Google Scholar]
- Pfeifer, N.; Sanfilippo, G. Probabilistic Squares and Hexagons of Opposition under Coherence. Int. J. Approx. Reason.
**2017**, 88, 282–294. [Google Scholar] [CrossRef] - Amgoud, L.; Besnard, P.; Hunter, A. Foundations for a Logic of Arguments. In Logical Reasoning and Computation: Essays Dedicated to Luis Fariñas del Cerro; Cabalar, P., Herzig, M.D.A., Pearce, D., Eds.; IRIT: Toulouse, France, 2016; pp. 95–107. [Google Scholar]
- Amgoud, L.; Prade, H. Can AI Models Capture Natural Language Argumentation? Int. J. Cognit. Inf. Nat. Intell.
**2012**, 6, 19–32. [Google Scholar] [CrossRef] - Amgoud, L.; Prade, H. Towards a Logic of Argumentation. In Scalable Uncertainty Management 2012; LNCS 7520; Springer: Berlin, Germany, 2012; pp. 558–565. [Google Scholar]
- Amgoud, L.; Prade, H. A Formal Concept View of Formal Argumentation. In Symbolic and Quantiative Approaches to Resoning with Uncertainty (ECSQARU 2013); van der Gaag, L.C., Ed.; LNCS 7958; Springer: Berlin, Germany, 2013; pp. 1–12. [Google Scholar]
- Ciucci, D.; Dubois, D.; Prade, H. Structures of Opposition in Fuzzy Rough Sets. Fundam. Inform.
**2015**, 142, 1–19. [Google Scholar] [CrossRef] - Ciucci, D.; Dubois, D.; Prade, H. Structures of opposition induced by relations. The Boolean and the gradual cases. Ann. Math. Artif. Intell.
**2016**, 76, 351–373. [Google Scholar] [CrossRef] - Dubois, D.; Prade, H. Gradual Structures of Oppositions. In Enric Trillas: A Passion for Fuzzy Sets; Magdalena, L., Verdegay, J.L., Esteva, F., Eds.; SFSC 322; Springer: Berlin, Germany, 2015; pp. 79–91. [Google Scholar]
- Dubois, D.; Prade, H.; Rico, A. Graded Cubes of Opposition and Possibility Theory with Fuzzy Events. Int. J. Approx. Reason.
**2017**, in press. [Google Scholar] [CrossRef] - Ciucci, D.; Dubois, D.; Prade, H. The Structure of Oppositions in Rough Set Theory and Formal Concept Analysis—Toward a New Bridge between the Two Settings. In Foundations of Information and Knowledge Systems (FoIKS 2014); Beierle, C., Meghini, C., Eds.; LNCS 8367; Springer: Berlin, Germany, 2014; pp. 154–173. [Google Scholar]
- Dubois, D.; Prade, H. From Blanché’s Hexagonal Organization of Concepts to Formal Concept Analysis and Possibility Theory. Log. Univ.
**2012**, 6, 149–169. [Google Scholar] [CrossRef] - Dubois, D.; Prade, H. Formal Concept Analysis from the Standpoint of Possibility Theory. In Formal Concept Analysis (ICFCA 2015); Baixeries, J., Sacarea, C., Ojeda-Aciego, M., Eds.; LNCS 9113; Springer: Berlin, Germany, 2015; pp. 21–38. [Google Scholar]
- Ciucci, D.; Dubois, D.; Prade, H. Oppositions in Rough Set Theory. In Rough Sets and Knowledge Technology; Li, T., Nguyen, H.S., Wang, G., Grzymala-Busse, J., Janicki, R., Hassanien, A.E., Yu, H., Eds.; LNCS 7414; Springer: Berlin, Germany, 2012; pp. 504–513. [Google Scholar]
- Yao, Y. Duality in Rough Set Theory Based on the Square of Opposition. Fundam. Inform.
**2013**, 127, 49–64. [Google Scholar] - Dubois, D.; Prade, H.; Rico, A. The Cube of Opposition—A Structure underlying many Knowledge Representation Formalisms. In Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015); Yang, Q., Wooldridge, M., Eds.; AAAI Press: Palo Alto, CA, USA, 2015; pp. 2933–2939. [Google Scholar]
- Dubois, D.; Prade, H.; Rico, A. The Cube of Opposition and the Complete Appraisal of Situations by Means of Sugeno Integrals. In Foundations of Intelligent Systems (ISMIS 2015); LNCS 9384; Springer: Berlin, Germany, 2015; pp. 197–207. [Google Scholar]
- Dubois, D.; Prade, H.; Rico, A. Organizing Families of Aggregation Operators into a Cube of Opposition. In Granular, Soft and Fuzzy Approaches for Intelligent Systems; Kacprzyk, J., Filev, D., Beliakov, G., Eds.; Springer: Berlin, Germany, 2017; pp. 27–45. [Google Scholar]
- Miclet, L.; Prade, H. Analogical Proportions and Square of Oppositions. In Information Processing and Management of Uncertainty in Knowledge-Based Systems 2014, Part II; CCIS 442; Springer: Berlin, Germany, 2014; pp. 324–334. [Google Scholar]
- Prade, H.; Richard, G. From Analogical Proportion to Logical Proportions. Log. Univ.
**2013**, 7, 441–505. [Google Scholar] [CrossRef] [Green Version] - Prade, H.; Richard, G. Picking the one that does not fit—A matter of logical proportions. In Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-13); Pasi, G., Montero, J., Ciucci, D., Eds.; Atlantis Press: Amsterdam, The Netherlands, 2013; pp. 392–399. [Google Scholar]
- Prade, H.; Richard, G. On Different Ways to be (dis)similar to Elements in a Set. Boolean Analysis and Graded Extension. In Information Processing and Management of Uncertainty in Knowledge-Based Systems 2016, Part II; CCIS 611; Springer: Berlin, Germany, 2016; pp. 605–618. [Google Scholar]
- Prade, H.; Richard, G. From the Structures of Opposition Between Similarity and Dissimilarity Indicators to Logical Proportions. In Representation and Reality in Humans, Other Living Organisms and Intelligent Machines; Dodig-Crnkovic, G., Giovagnoli, R., Eds.; Springer: Berlin, Germany, 2017; pp. 279–299. [Google Scholar]
- Smessaert, H.; Demey, L. Logical Geometries and Information in the Square of Opposition. J. Log. Lang. Inf.
**2014**, 23, 527–565. [Google Scholar] [CrossRef] - Demey, L.; Smessaert, H. Combinatorial Bitstring Semantics for Arbitrary Logical Fragments. J. Philos. Log.
**2017**. [Google Scholar] [CrossRef] - Demey, L. Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams. In Modeling and Using Context; Christiansen, H., Stojanovic, I., Papadopoulos, G., Eds.; LNCS 9405; Springer: Berlin, Germany, 2015; pp. 331–345. [Google Scholar]
- Demey, L.; Smessaert, H. Shape Heuristics in Aristotelian Diagrams. In Shapes 3.0 Proceedings; Kutz, O., Borgo, S., Bhatt, M., Eds.; Workshop Proceedings 1616; CEUR: Aachen, Germany, 2016; pp. 35–45. [Google Scholar]
- Demey, L.; Smessaert, H. The Interaction between Logic and Geometry in Aristotelian Diagrams. In Diagrammatic Representation and Inference; Jamnik, M., Uesaka, Y., Elzer Schwartz, S., Eds.; LNCS 9781; Springer: Berlin, Germany, 2016; pp. 67–82. [Google Scholar]
- Smessaert, H.; Demey, L. Visualising the Boolean Algebra ${\mathbb{B}}_{4}$ in 3D. In Diagrammatic Representation and Inference; Jamnik, M., Uesaka, Y., Elzer Schwartz, S., Eds.; LNCS 9781; Springer: Berlin, Germany, 2016; pp. 289–292. [Google Scholar]
- Demey, L.; Smessaert, H. The Relationship between Aristotelian and Hasse Diagrams. In Diagrammatic Representation and Inference; Dwyer, T., Purchase, H., Delaney, A., Eds.; LNCS 8578; Springer: Berlin, Germany, 2014; pp. 213–227. [Google Scholar]
- Demey, L.; Smessaert, H. Geometric and Cognitive Differences between Aristotelian Diagrams for the Boolean Algebra ${\mathbb{B}}_{4}$.
**2017**. submitted. [Google Scholar] - Kruja, E.; Marks, J.; Blair, A.; Waters, R. A Short Note on the History of Graph Drawing. In Graph Drawing (GD 2001); Mutzel, P., Jünger, M., Leipert, S., Eds.; LNCS 2265; Springer: Berlin, Germany, 2002; pp. 272–286. [Google Scholar]
- Ford, B.J. Images of Science: A History of Scientific Illustration; Oxford University Press: New York, NY, USA, 1993. [Google Scholar]
- Moretti, A. The Geometry of Logical Opposition. Ph.D. Thesis, University of Neuchâtel, Neuenburg, Switzerland, 2009. [Google Scholar]
- Smessaert, H. On the 3D Visualisation of Logical Relations. Log. Univ.
**2009**, 3, 303–332. [Google Scholar] [CrossRef] - Béziau, J.Y. New light on the square of oppositions and its nameless corner. Log. Investig.
**2003**, 10, 218–232. [Google Scholar] - Smessaert, H.; Demey, L. Béziau’s Contributions to the Logical Geometry of Modalities and Quantifiers. In The Road to Universal Logic; Koslow, A., Buchsbaum, A., Eds.; Springer: Basel, Switzerland, 2015; pp. 475–493. [Google Scholar]
- Pellissier, R. Setting n-Opposition. Log. Univ.
**2008**, 2, 235–263. [Google Scholar] [CrossRef] - Moretti, A. The Geometry of Standard Deontic Logic. Log. Univ.
**2009**, 3, 19–57. [Google Scholar] [CrossRef] - Tversky, B. Prolegomenon to Scientific Visualizations. In Visualization in Science Education; Gilbert, J.K., Ed.; Springer: Dordrecht, The Netherlands, 2005; pp. 29–42. [Google Scholar]
- Tversky, B. Visualizing Thought. Top. Cognit. Sci.
**2011**, 3, 499–535. [Google Scholar] [CrossRef] [PubMed] - Moretti, A. Was Lewis Carroll an Amazing Oppositional Geometer? Hist. Philos. Log.
**2014**, 35, 383–409. [Google Scholar] [CrossRef] - Smessaert, H.; Demey, L. Logical and Geometrical Complementarities between Aristotelian Diagrams. In Diagrammatic Representation and Inference; Dwyer, T., Purchase, H., Delaney, A., Eds.; LNCS 8578; Springer: Berlin, Germany, 2014; pp. 246–260. [Google Scholar]
- Givant, S.; Halmos, P. Introduction to Boolean Algebras; Springer: New York, NY, USA, 2009. [Google Scholar]
- Smessaert, H.; Demey, L. The Unreasonable Effectiveness of Bitstrings in Logical Geometry. In The Square of Opposition: A Cornerstone of Thought; Béziau, J.Y., Basti, G., Eds.; Springer: Basel, Switzerland, 2017; pp. 197–214. [Google Scholar]
- Demey, L.; Smessaert, H. Metalogical Decorations of Logical Diagrams. Log. Univ.
**2016**, 10, 233–292. [Google Scholar] [CrossRef] [Green Version] - Demey, L. Metalogic, Metalanguage and Logical Geometry.
**2017**. submitted. [Google Scholar] - Davey, B.; Priestley, H. Introduction to Lattices and Order; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Kauffman, L.H. The Mathematics of Charles Sanders Peirce. Cybern. Hum. Knowing
**2001**, 8, 79–110. [Google Scholar] - Zellweger, S. Untapped potential in Peirce’s iconic notation for the sixteen binary connectives. In Studies in the Logic of Charles Peirce; Houser, N., Roberts, D.D., Van Evra, J., Eds.; Indiana University Press: Bloomington, IN, USA, 1997; pp. 334–386. [Google Scholar]
- Harary, F.; Hayes, J.P.; Wu, H.J. A Survey of the Theory of Hypercube Graphs. Comput. Math. Appl.
**1988**, 15, 277–289. [Google Scholar] [CrossRef] - Coxeter, H.S.M. Regular Polytopes; Dover Publications: New York, NY, USA, 1973. [Google Scholar]
- Larkin, J.; Simon, H. Why a Diagram is (Sometimes) Worth Ten Thousand Words. Cognit. Sci.
**1987**, 11, 65–99. [Google Scholar] [CrossRef] - Conway, J.H.; Burgiel, H.; Goodman-Strauss, C. The Symmetries of Things; CRC Press: Boca Raton, FL, USA, 2008. [Google Scholar]
- Wenninger, M. Polyhedron Models; Cambridge University Press: Cambridge, UK, 1974. [Google Scholar]
- Wenninger, M. Dual Models; Cambridge University Press: Cambridge, UK, 1983. [Google Scholar]
- Coxeter, H.S.M. Regular and Semiregular Polyhedra. In Shaping Space. Exploring Polyhedra in Nature, Art, and the Geometrical Imagination; Senechal, M., Ed.; Springer: New York, NY, USA, 2013; pp. 41–52. [Google Scholar]
- Walter, M.; Pedersen, J.; Wenninger, M.; Schattschneider, D.; Loeb, A.L.; Demaine, E.; Demaine, M.; Hart, V. Six Recipes for Making Polyhedra. In Shaping Space. Exploring Polyhedra in Nature, Art, and the Geometrical Imagination; Senechal, M., Ed.; Springer: New York, NY, USA, 2013; pp. 13–40. [Google Scholar]
- Sauriol, P. Remarques sur la Théorie de l’hexagone logique de Blanché. Dialogue
**1968**, 7, 374–390. [Google Scholar] [CrossRef] - Johnson, N.W. Convex Polyhedra with Regular Faces. Can. J. Math.
**1966**, 18, 169–200. [Google Scholar] [CrossRef] - Carroll, L. Symbolic Logic. Edited, with Annotations and an Introduction by William Warren Bartley III; Clarkson N. Potter: New York, NY, USA, 1977. [Google Scholar]
- Roth, R.M. Introduction to Coding Theory; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Deza, M.M.; Deza, E. Encyclopedia of Distances; Springer: Dordrecht, The Netherlands, 2009. [Google Scholar]
- Demey, L.; Smessaert, H. Logische geometrie en pragmatiek. In Patroon en Argument; Van De Velde, F., Smessaert, H., Van Eynde, F., Verbrugge, S., Eds.; Leuven University Press: Leuven, Belgium, 2014; pp. 553–564. [Google Scholar]
- Peterson, P. On the Logic of “Few”, “Many”, and “Most”. Notre Dame J. Form. Log.
**1979**, 20, 155–179. [Google Scholar] [CrossRef] - Demey, L.; Smessaert, H. The Logical Geometry of the Boolean Algebra ${\mathbb{B}}_{4}$.
**2017**. Unpublished work. [Google Scholar]

**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 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## 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