Graded cubes of opposition and possibility theory with fuzzy events

Dubois, Didier; Prade, Henri; Rico, Agnès

doi:10.1016/j.ijar.2017.02.006

Cited by 32 publications

(24 citation statements)

References 22 publications

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“…Within the latter field, Aristotelian diagrams have been used to study various logic-based approaches to knowledge representation, including fuzzy logic [16][17][18][19][20], modal-epistemic logic [21][22][23][24][25] and probabilistic logic [26][27][28]. Furthermore, Aristotelian diagrams are also used extensively to study (the connections between) other types of knowledge representation formalisms, such as formal argumentation theory [29][30][31][32], fuzzy set theory [33][34][35][36], formal concept analysis and possibility theory [37][38][39], rough set theory [37,40,41], multiple-criteria decision-making [42][43][44] and the theory of logical and analogical proportions [45][46][47][48][49]. In sum, then, Aristotelian diagrams have come to serve as visual tools that greatly facilitate communication, research and teaching in a wide variety of disciplines that deal with logical reasoning in all its facets.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the rhombic dodecahedron has been used in research on modal logic and dynamic epistemic logic [22,61], the tetrakis hexahedron and tetraicosahedron have been used to investigate modal logic [25,60,64,65], and the nested tetrahedron has been used to study (connections between) the knowledge representation formalisms of rough set theory, formal concept analysis and possibility theory [38,40]. (However, it should also be noted that our focus on B 4 precludes us from studying Aristotelian diagrams for knowledge representation formalisms that go beyond a Boolean setup, such as fuzzy logic and set theory [16][17][18][19][20][33][34][35][36]. The systematic (logical and diagrammatic) investigation of such non-classical Aristotelian diagrams is a matter of ongoing research.…”

Section: Introductionmentioning

confidence: 99%

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Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Demey

Smessaert

2017

Symmetry

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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.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Demey

Smessaert

2017

Symmetry

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“…A gradual square of opposition can be defined by attaching variables α, ǫ, o, ι valued on a totally ordered set V to vertices A, E, O, I respectively, so as to respect the following constraints [12]:…”

Section: Gradual Square and Hexagonmentioning

confidence: 99%

“…Proof. Based on results from Proposition 17 in [12], the two conditions π i = 1 for some i, and (1−a) → 0 ≤ 1−(a → 0) are sufficient to get…”

Section: Approximate Equality and Weighted Attributesmentioning

confidence: 99%

“…We have shown that the geometrical structure called hexagon of opposition, which organizes the relationship between various comparison modalities can survive under various gradual extensions of such modalities. A study of conditions needed for the gradual hexagon of opposition (as we did for the cube of opposition in [12]) in a more general algebraic setting has been carried out, introducing weak and full-fledged graded versions of the hexagon. Further investigation is still needed to characterize proper algebraic settings that support the gradual hexagon.…”

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confidence: 99%

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Fuzzy Extensions of Conceptual Structures of Comparison

Dubois¹,

Prade²,

Rico³

2018

Communications in Computer and Information Science

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Comparing two items (objects, images) involves a set of relevant attributes whose values are compared. Such a comparison may be expressed in terms of different modalities such as identity, similarity, difference, opposition, analogy. Recently J.-Y. Béziau has proposed an "analogical hexagon" that organizes the relations linking these modalities. The hexagon structure extends the logical square of opposition invented in Aristotle time (in relation with the theory of syllogisms). The interest of these structures has been recently advocated in logic and in artificial intelligence. When non-Boolean attributes are involved, elementary comparisons may be a matter of degree. Moreover, attributes may not have the same importance. One might only consider most attributes rather than all of them, using operators such as ordered weighted min and max. The paper studies in which ways the logical hexagon structure may be preserved in such gradual extensions. As an illustration, we start with the hexagon of equality and inequality due to Blanché and extend it with fuzzy equality and fuzzy inequality.

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Representations of Uncertainty in Artificial Intelligence: Probability and Possibility

Denœux

Dubois

Prade

2020

A Guided Tour of Artificial Intelligence Research

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Due to its major focus on knowledge representation and reasoning, artificial intelligence was bound to deal with various frameworks for the handling of uncertainty: probability theory, but more recent approaches as well: possibility theory, evidence theory, and imprecise probabilities. The aim of this chapter is to provide an introductive survey that lays bare specific features of two basic frameworks for representing uncertainty: probability theory and possibility theory, while highlighting the main issues that the task of representing uncertainty is faced with. This purpose also provides the opportunity to position related topics, such as rough sets and fuzzy sets, respectively motivated by the need to account for the granularity of representations as induced by the choice of a language, and the gradual nature of natural language predicates. Moreover, this overview includes concise presentations of yet other theoretical representation frameworks such as formal concept analysis, conditional events and ranking functions, and also possibilistic logic, in connection with the uncertainty frameworks addressed here. The next chapter in this volume is devoted to more complex frameworks: belief functions and imprecise probabilities.

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Graded cubes of opposition and possibility theory with fuzzy events

Cited by 32 publications

References 22 publications

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Fuzzy Extensions of Conceptual Structures of Comparison

Representations of Uncertainty in Artificial Intelligence: Probability and Possibility

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