Analogical proportions link four items a, b, c and d by a relation stating that “a is to b as c is to d,” a, b, c, d being the formal representation of 4 real world entities such as profiles. As such, a, b, c, d could be atomic values (like Boolean, real or nominal values), or more generally vectors thereof. In this context, we may attach to a vector representing the profile of a group of individuals in a given population, the probability (frequency) of this group in the population. This raises the question of whether four profiles, whose descriptions form an analogical proportion, also have their attached probabilities forming an analogical proportion when viewed as real numbers. This issue is explored and illustrated in this paper, which also provides some new results on numerical analogical proportions.

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Analogical Proportions and Probabilities: Are They Compatible?

  • Henri Prade,
  • Gilles Richard

摘要

Analogical proportions link four items a, b, c and d by a relation stating that “a is to b as c is to d,” a, b, c, d being the formal representation of 4 real world entities such as profiles. As such, a, b, c, d could be atomic values (like Boolean, real or nominal values), or more generally vectors thereof. In this context, we may attach to a vector representing the profile of a group of individuals in a given population, the probability (frequency) of this group in the population. This raises the question of whether four profiles, whose descriptions form an analogical proportion, also have their attached probabilities forming an analogical proportion when viewed as real numbers. This issue is explored and illustrated in this paper, which also provides some new results on numerical analogical proportions.