Analogical proportions are statements of the form a is to b as c is to d. They are supposed to satisfy some postulates such as reflexivity, symmetry, stability under central permutation, or unicity of d when a, b, c are given. A simple Boolean model that satisfy all these postulates has been proposed fifteen years ago. In this model, analogical proportions are also transitive, which may be found more debatable. This short paper discusses how these postulates or properties (except reflexivity and symmetry) could be challenged, in agreement with a focused version of the Boolean model. By focused version we mean that the evaluation of an analogical proportion heavily depends on the set of attributes that should be considered. By varying this set, some properties such as the unicity of d given a, b, and c, or the transitivity may be lost. This type of observation also calls for a weighting of the importance of attributes or groups of attributes. It is argued that Sugeno integrals are of interest for this task.

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Multiple Viewpoint Analogical Proportions and Their Weighting

  • Henri Prade,
  • Gilles Richard

摘要

Analogical proportions are statements of the form a is to b as c is to d. They are supposed to satisfy some postulates such as reflexivity, symmetry, stability under central permutation, or unicity of d when a, b, c are given. A simple Boolean model that satisfy all these postulates has been proposed fifteen years ago. In this model, analogical proportions are also transitive, which may be found more debatable. This short paper discusses how these postulates or properties (except reflexivity and symmetry) could be challenged, in agreement with a focused version of the Boolean model. By focused version we mean that the evaluation of an analogical proportion heavily depends on the set of attributes that should be considered. By varying this set, some properties such as the unicity of d given a, b, and c, or the transitivity may be lost. This type of observation also calls for a weighting of the importance of attributes or groups of attributes. It is argued that Sugeno integrals are of interest for this task.