×

Logical and geometrical distance in polyhedral Aristotelian diagrams in knowledge representation. (English) Zbl 1423.68489

Summary: Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study (connections between) various knowledge representation formalisms. In this paper, we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra \(\mathbb{B}_4\), viz. the rhombic dodecahedron, the tetrakis hexahedron, the tetraicosahedron and the nested tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation (or lack thereof) between logical (Hamming) and geometrical (Euclidean) distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance; the tetraicosahedron performs worse; and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design.

MSC:

68T30 Knowledge representation
03G05 Logical aspects of Boolean algebras
52B10 Three-dimensional polytopes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Parsons, T.; The Traditional Square of Opposition; Stanford Encyclopedia of Philosophy: Stanford, CA, USA 2012; . · Zbl 1140.03301
[2] Read, S.; John Buridan’s Theory of Consequence and His Octagons of Opposition; Around and Beyond the Square of Opposition: Basel, Switzerland 2012; ,93-110. · Zbl 1270.03015
[3] Lenzen, W.; Leibniz’s Logic and the “Cube of Opposition”; Log. Univ.: 2016; Volume 10 ,171-189. · Zbl 1360.03037
[4] 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; Volume 15 ,398-413.
[5] Beller, S.; Deontic reasoning reviewed: Psychological questions, empirical findings, and current theories; Cognit. Process.: 2010; Volume 11 ,123-132.
[6] Mikhail, J.; Universal moral grammar: Theory, evidence and the future; Trends Cognit. Sci.: 2007; Volume 11 ,143-152.
[7] Abrusci, V.M.; Casadio, C.; Medaglia, M.T.; Porcaro, C.; Universal vs. Particular Reasoning: A Study with Neuroimaging Techniques; Log. J. IGPL: 2013; Volume 21 ,1017-1027. · Zbl 1306.03003
[8] Saurí, R.; Pustejovsky, J.; FactBank: A Corpus Annotated with Event Factuality; Lang. Resour. Eval.: 2009; Volume 43 ,227-268.
[9] Joerden, J.; ; Logik im Recht: Berlin, Germany 2010; .
[10] O’Reilly, D.; Using the Square of Opposition to Illustrate the Deontic and Alethic Relations Constituting Rights; Univ. Tor. Law J.: 1995; Volume 45 ,279-310.
[11] Vranes, E.; The Definition of ‘Norm Conflict’ in International Law and Legal Theory; Eur. J. Int. Law: 2006; Volume 17 ,395-418.
[12] Dekker, P.; Not Only Barbara; J. Log. Lang. Inf.: 2015; Volume 24 ,95-129. · Zbl 1350.03027
[13] Horn, L.R.; ; A Natural History of Negation: Chicago, IL, USA 1989; .
[14] Seuren, P.; Jaspers, D.; Logico-Cognitive Structure in the Lexicon; Language: 2014; Volume 90 ,607-643.
[15] Van der Auwera, J.; Modality: The Three-layered Scalar Square; J. Semant.: 1996; Volume 13 ,181-195.
[16] Glöckner, I.; ; Fuzzy Quantifiers: Berlin, Germany 2006; . · Zbl 1089.03002
[17] Murinová, P.; Novák, V.; Analysis of Generalized Square of Opposition with Intermediate Quantifiers; Fuzzy Sets Syst.: 2014; Volume 242 ,89-113. · Zbl 1315.03038
[18] Murinová, P.; Novák, V.; Graded Generalized Hexagon in Fuzzy Natural Logic; Information Processing and Management of Uncertainty in Knowledge-Based Systems 2016, Part II: Berlin, Germany 2016; ,36-47. · Zbl 1455.03029
[19] Murinová, P.; Novák, V.; Syllogisms and 5-Square of Opposition with Intermediate Quantifiers in Fuzzy Natural Logic; Log. Univ.: 2016; Volume 10 ,339-357. · Zbl 1396.03061
[20] Trillas, E.; Seising, R.; Turning Around the Ideas of ‘Meaning’ and ‘Complement’; Fuzzy Technology: Berlin, Germany 2016; ,3-31.
[21] Carnielli, W.; Pizzi, C.; ; Modalities and Multimodalities: Dordrecht, The Netherlands 2008; . · Zbl 1210.03017
[22] Demey, L.; Structures of Oppositions for Public Announcement Logic; Around and Beyond the Square of Opposition: Basel, Switzerland 2012; ,313-339. · Zbl 1263.03010
[23] Fitting, M.; Mendelsohn, R.L.; ; First-Order Modal Logic: Dordrecht, The Netherlands 1998; . · Zbl 1025.03001
[24] Lenzen, W.; How to Square Knowledge and Belief; Around and Beyond the Square of Opposition: Basel, Switzerland 2012; ,305-311. · Zbl 1280.03019
[25] Luzeaux, D.; Sallantin, J.; Dartnell, C.; Logical Extensions of Aristotle’s Square; Log. Univ.: 2008; Volume 2 ,167-187. · Zbl 1138.03314
[26] Gilio, A.; Pfeifer, N.; Sanfilippo, G.; Transitivity in Coherence-Based Probability Logic; J. Appl. Log.: 2016; Volume 14 ,46-64. · Zbl 1436.03146
[27] Pfeifer, N.; Sanfilippo, G.; Square of Opposition under Coherence; Soft Methods for Data Science: Berlin, Germany 2017; ,407-414.
[28] Pfeifer, N.; Sanfilippo, G.; Probabilistic Squares and Hexagons of Opposition under Coherence; Int. J. Approx. Reason.: 2017; Volume 88 ,282-294. · Zbl 1422.03041
[29] Amgoud, L.; Besnard, P.; Hunter, A.; Foundations for a Logic of Arguments; Logical Reasoning and Computation: Essays Dedicated to Luis Fariñas del Cerro: Toulouse, France 2016; ,95-107.
[30] Amgoud, L.; Prade, H.; Can AI Models Capture Natural Language Argumentation?; Int. J. Cognit. Inf. Nat. Intell.: 2012; Volume 6 ,19-32.
[31] Amgoud, L.; Prade, H.; Towards a Logic of Argumentation; Scalable Uncertainty Management 2012: Berlin, Germany 2012; ,558-565.
[32] Amgoud, L.; Prade, H.; A Formal Concept View of Formal Argumentation; Symbolic and Quantiative Approaches to Resoning with Uncertainty (ECSQARU 2013): Berlin, Germany 2013; ,1-12. · Zbl 1390.68606
[33] Ciucci, D.; Dubois, D.; Prade, H.; Structures of Opposition in Fuzzy Rough Sets; Fundam. Inform.: 2015; Volume 142 ,1-19. · Zbl 1350.03039
[34] Ciucci, D.; Dubois, D.; Prade, H.; Structures of opposition induced by relations. The Boolean and the gradual cases; Ann. Math. Artif. Intell.: 2016; Volume 76 ,351-373. · Zbl 1336.03034
[35] Dubois, D.; Prade, H.; Gradual Structures of Oppositions; Enric Trillas: A Passion for Fuzzy Sets: Berlin, Germany 2015; ,79-91. · Zbl 1362.03024
[36] Dubois, D.; Prade, H.; Rico, A.; Graded Cubes of Opposition and Possibility Theory with Fuzzy Events; Int. J. Approx. Reason.: 2017; . · Zbl 1419.68161
[37] 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; Foundations of Information and Knowledge Systems (FoIKS 2014): Berlin, Germany 2014; ,154-173. · Zbl 1407.68478
[38] Dubois, D.; Prade, H.; From Blanché’s Hexagonal Organization of Concepts to Formal Concept Analysis and Possibility Theory; Log. Univ.: 2012; Volume 6 ,149-169. · Zbl 1272.03015
[39] Dubois, D.; Prade, H.; Formal Concept Analysis from the Standpoint of Possibility Theory; Formal Concept Analysis (ICFCA 2015): Berlin, Germany 2015; ,21-38. · Zbl 1312.68186
[40] Ciucci, D.; Dubois, D.; Prade, H.; Oppositions in Rough Set Theory; Rough Sets and Knowledge Technology: Berlin, Germany 2012; ,504-513.
[41] Yao, Y.; Duality in Rough Set Theory Based on the Square of Opposition; Fundam. Inform.: 2013; Volume 127 ,49-64. · Zbl 1315.03105
[42] Dubois, D.; Prade, H.; Rico, A.; The Cube of Opposition—A Structure underlying many Knowledge Representation Formalisms; Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015): Palo Alto, CA, USA 2015; ,2933-2939.
[43] Dubois, D.; Prade, H.; Rico, A.; The Cube of Opposition and the Complete Appraisal of Situations by Means of Sugeno Integrals; Foundations of Intelligent Systems (ISMIS 2015): Berlin, Germany 2015; ,197-207.
[44] Dubois, D.; Prade, H.; Rico, A.; Organizing Families of Aggregation Operators into a Cube of Opposition; Granular, Soft and Fuzzy Approaches for Intelligent Systems: Berlin, Germany 2017; ,27-45. · Zbl 1380.68356
[45] Miclet, L.; Prade, H.; Analogical Proportions and Square of Oppositions; Information Processing and Management of Uncertainty in Knowledge-Based Systems 2014, Part II: Berlin, Germany 2014; ,324-334. · Zbl 1418.68194
[46] Prade, H.; Richard, G.; From Analogical Proportion to Logical Proportions; Log. Univ.: 2013; Volume 7 ,441-505. · Zbl 1323.03011
[47] Prade, H.; Richard, G.; Picking the one that does not fit—A matter of logical proportions; Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-13): Amsterdam, The Netherlands 2013; ,392-399.
[48] Prade, H.; Richard, G.; On Different Ways to be (dis)similar to Elements in a Set. Boolean Analysis and Graded Extension; Information Processing and Management of Uncertainty in Knowledge-Based Systems 2016, Part II: Berlin, Germany 2016; ,605-618. · Zbl 1460.62016
[49] Prade, H.; Richard, G.; From the Structures of Opposition Between Similarity and Dissimilarity Indicators to Logical Proportions; Representation and Reality in Humans, Other Living Organisms and Intelligent Machines: Berlin, Germany 2017; ,279-299.
[50] Smessaert, H.; Demey, L.; Logical Geometries and Information in the Square of Opposition; J. Log. Lang. Inf.: 2014; Volume 23 ,527-565. · Zbl 1306.03005
[51] Demey, L.; Smessaert, H.; Combinatorial Bitstring Semantics for Arbitrary Logical Fragments; J. Philos. Log.: 2017; . · Zbl 1436.03110
[52] Demey, L.; Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams; Modeling and Using Context: Berlin, Germany 2015; ,331-345.
[53] Demey, L.; Smessaert, H.; Shape Heuristics in Aristotelian Diagrams; Shapes 3.0 Proceedings: Aachen, Germany 2016; ,35-45. · Zbl 1384.03023
[54] Demey, L.; Smessaert, H.; The Interaction between Logic and Geometry in Aristotelian Diagrams; Diagrammatic Representation and Inference: Berlin, Germany 2016; ,67-82. · Zbl 1394.68369
[55] Smessaert, H.; Demey, L.; Visualising the Boolean Algebra B 4 in 3D; Diagrammatic Representation and Inference: Berlin, Germany 2016; ,289-292.
[56] Demey, L.; Smessaert, H.; The Relationship between Aristotelian and Hasse Diagrams; Diagrammatic Representation and Inference: Berlin, Germany 2014; ,213-227.
[57] Demey, L.; Smessaert, H.; Geometric and Cognitive Differences between Aristotelian Diagrams for the Boolean Algebra B 4 ; 2017; . · Zbl 1384.03023
[58] Kruja, E.; Marks, J.; Blair, A.; Waters, R.; A Short Note on the History of Graph Drawing; Graph Drawing (GD 2001): Berlin, Germany 2002; ,272-286. · Zbl 1054.68500
[59] Ford, B.J.; ; Images of Science: A History of Scientific Illustration: New York, NY, USA 1993; .
[60] Moretti, A.; The Geometry of Logical Opposition; Ph.D. Thesis: Neuenburg, Switzerland 2009; .
[61] Smessaert, H.; On the 3D Visualisation of Logical Relations; Log. Univ.: 2009; Volume 3 ,303-332. · Zbl 1255.03032
[62] Béziau, J.Y.; New light on the square of oppositions and its nameless corner; Log. Investig.: 2003; Volume 10 ,218-232. · Zbl 1053.03005
[63] Smessaert, H.; Demey, L.; Béziau’s Contributions to the Logical Geometry of Modalities and Quantifiers; The Road to Universal Logic: Basel, Switzerland 2015; ,475-493. · Zbl 1376.03021
[64] Pellissier, R.; Setting n-Opposition; Log. Univ.: 2008; Volume 2 ,235-263. · Zbl 1156.03021
[65] Moretti, A.; The Geometry of Standard Deontic Logic; Log. Univ.: 2009; Volume 3 ,19-57. · Zbl 1255.03031
[66] Tversky, B.; Prolegomenon to Scientific Visualizations; Visualization in Science Education: Dordrecht, The Netherlands 2005; ,29-42.
[67] Tversky, B.; Visualizing Thought; Top. Cognit. Sci.: 2011; Volume 3 ,499-535.
[68] Moretti, A.; Was Lewis Carroll an Amazing Oppositional Geometer?; Hist. Philos. Log.: 2014; Volume 35 ,383-409. · Zbl 1369.03015
[69] Smessaert, H.; Demey, L.; Logical and Geometrical Complementarities between Aristotelian Diagrams; Diagrammatic Representation and Inference: Berlin, Germany 2014; ,246-260.
[70] Givant, S.; Halmos, P.; ; Introduction to Boolean Algebras: New York, NY, USA 2009; . · Zbl 1168.06001
[71] Smessaert, H.; Demey, L.; The Unreasonable Effectiveness of Bitstrings in Logical Geometry; The Square of Opposition: A Cornerstone of Thought: Basel, Switzerland 2017; ,197-214.
[72] Demey, L.; Smessaert, H.; Metalogical Decorations of Logical Diagrams; Log. Univ.: 2016; Volume 10 ,233-292. · Zbl 1384.03023
[73] Demey, L.; Metalogic, Metalanguage and Logical Geometry; 2017; .
[74] Davey, B.; Priestley, H.; ; Introduction to Lattices and Order: Cambridge, UK 2002; . · Zbl 1002.06001
[75] Kauffman, L.H.; The Mathematics of Charles Sanders Peirce; Cybern. Hum. Knowing: 2001; Volume 8 ,79-110.
[76] Zellweger, S.; Untapped potential in Peirce’s iconic notation for the sixteen binary connectives; Studies in the Logic of Charles Peirce: Bloomington, IN, USA 1997; ,334-386.
[77] Harary, F.; Hayes, J.P.; Wu, H.J.; A Survey of the Theory of Hypercube Graphs; Comput. Math. Appl.: 1988; Volume 15 ,277-289. · Zbl 0645.05061
[78] Coxeter, H.S.M.; ; Regular Polytopes: New York, NY, USA 1973; .
[79] Larkin, J.; Simon, H.; Why a Diagram is (Sometimes) Worth Ten Thousand Words; Cognit. Sci.: 1987; Volume 11 ,65-99.
[80] Conway, J.H.; Burgiel, H.; Goodman-Strauss, C.; ; The Symmetries of Things: Boca Raton, FL, USA 2008; . · Zbl 1173.00001
[81] Wenninger, M.; ; Polyhedron Models: Cambridge, UK 1974; . · Zbl 0299.50005
[82] Wenninger, M.; ; Dual Models: Cambridge, UK 1983; . · Zbl 0521.52006
[83] Coxeter, H.S.M.; Regular and Semiregular Polyhedra; Shaping Space. Exploring Polyhedra in Nature, Art, and the Geometrical Imagination: New York, NY, USA 2013; ,41-52. · Zbl 1267.52002
[84] Walter, M.; Pedersen, J.; Wenninger, M.; Schattschneider, D.; Loeb, A.L.; Demaine, E.; Demaine, M.; Hart, V.; Six Recipes for Making Polyhedra; Shaping Space. Exploring Polyhedra in Nature, Art, and the Geometrical Imagination: New York, NY, USA 2013; ,13-40.
[85] Sauriol, P.; Remarques sur la Théorie de l’hexagone logique de Blanché; Dialogue: 1968; Volume 7 ,374-390.
[86] Johnson, N.W.; Convex Polyhedra with Regular Faces; Can. J. Math.: 1966; Volume 18 ,169-200. · Zbl 0132.14603
[87] Carroll, L.; ; Symbolic Logic. Edited, with Annotations and an Introduction by William Warren Bartley III: New York, NY, USA 1977; .
[88] Roth, R.M.; ; Introduction to Coding Theory: Cambridge, UK 2006; . · Zbl 1092.94001
[89] Deza, M.M.; Deza, E.; ; Encyclopedia of Distances: Dordrecht, The Netherlands 2009; . · Zbl 1167.51001
[90] Demey, L.; Smessaert, H.; Logische geometrie en pragmatiek; Patroon en Argument: Leuven, Belgium 2014; ,553-564.
[91] Peterson, P.; On the Logic of “Few”, “Many”, and “Most”; Notre Dame J. Form. Log.: 1979; Volume 20 ,155-179. · Zbl 0299.02012
[92] Demey, L.; Smessaert, H.; The Logical Geometry of the Boolean Algebra B 4 ; 2017; . · Zbl 1376.03021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.