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From Blanché’s hexagonal organization of concepts to formal concept analysis and possibility theory. (English) Zbl 1272.03015
Summary: The paper first introduces a cube of opposition that associates the traditional square of opposition with the dual square obtained by Piaget’s reciprocation. It is then pointed out that Blanché’s extension of the square-of-opposition structure into an conceptual hexagonal structure always relies on an abstract tripartition. Considering quadripartitions leads to organize the 16 binary connectives into a regular tetrahedron. Lastly, the cube of opposition, once interpreted in modal terms, is shown to account for a recent generalization of formal concept analysis, where noticeable hexagons are also laid bare. This generalization of formal concept analysis is motivated by a parallel with bipolar possibility theory. The latter, albeit graded, is indeed based on four graded set functions that can be organized in a similar structure.

03A05 Philosophical and critical aspects of logic and foundations
03B05 Classical propositional logic
68T30 Knowledge representation
68T37 Reasoning under uncertainty in the context of artificial intelligence
Full Text: DOI
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