Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams

2017

Symmetry

Self Cite

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 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.

Section: Introductionmentioning

confidence: 99%

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

2017

Symmetry

Self Cite

“…In Subsection 6.1, however, we showed that these relations can also be used to define an Aristotelian square, but whether this square is classical or degenerated, depends on whether the relations' first argument is assumed to be satisfiable; compare Figure 24(a) and (b). Similarly, the six relations in the duality diagram shown in Figure 33(a) can also be used to define an Aristotelian hexagon, but whether this is a JSB or a U4 hexagon again depends on whether the relations' first argument is assumed to be satisfiable; compare Figure 25 This sharp contrast between Aristotelian and duality diagrams can be seen as the metalogical manifestation of a more general phenomenon that is well-understood at the object-logical level: the Aristotelian relations are sensitive to the deductive power of the underlying logical system, but the duality relations are entirely insensitive to this [23]. 51 Consider, for example, the formulas p and ¬p from modal logic.…”

Section: Resultsmentioning

confidence: 99%

“…the Aristotelian diagram that is obtained by adding the complements of all relations that occur in the original diagram. 23 The resulting diagram is a dodecagon, which is shown in Figure 18. 20 The systematic study of such non-standard Aristotelian diagrams is still in its infancy; some preliminary results can be found in [60,61].…”

mentioning

confidence: 99%

“…However, this reformulation has to be taken with a grain of salt: Béziau's hexagon is incorrect in its contradictions and subcontrarieties, so it is not a proper JSB hexagon to begin with, and thus the question whether it is a strong or a weak one does not even arise, strictly speaking. 23 For most Aristotelian diagrams, this operation does not make much sense, since they are already closed under negation, and are thus their own complement-closure. Comparing this dodecagon to the Aristotelian RDH that was described in Subsection 4.1, we see that it lacks only two relations, namely precisely those that were originally forgotten by Béziau: NCD and its contradictory, CD ∪ C ∪ SC .…”

mentioning

confidence: 99%

“…37 The resulting metalogical octagon is shown in Figure 26(b). 38 This type of Aristotelian octagon has hitherto not been studied in any systematic way, but it can be found in [23,37,45,47], where it receives an object-logical decoration consisting of categorical statements from syllogistics. In [31] is is shown that the octagon's object-logical (syllogistic) decoration is intimately related to the metalogical decoration that has been discussed here.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Metalogical Decorations of Logical Diagrams

2016

Log. Univers.

Self Cite

Abstract. In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of (sub)contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object-and metalogical diagrams. Overall, the paper studies two types of logical diagrams (viz. Aristotelian and duality diagrams) and four kinds of metalogical decorations (viz. those based on the opposition, implication, Aristotelian and duality relations).Mathematics Subject Classification (2010). Primary 03A10, 03B45; Secondary 03B65, 03B80.

The Interaction Between Logic and Geometry in Aristotelian Diagrams

Lecture Notes in Computer Science

2016

Self Cite

We develop a systematic approach for dealing with informationally equivalent Aristotelian diagrams, based on the interaction between the logical properties of the visualized information and the geometrical properties of the concrete polygon/polyhedron. To illustrate the account's fruitfulness, we apply it to all Aristotelian families of 4-formula fragments that are closed under negation (comparing square and rectangle) and to all Aristotelian families of 6-formula fragments that are closed under negation (comparing hexagon and octahedron).