Frank's triangular norms in Piaget's logical proportions
This work addresses a theoretical problem in mathematical logic and AI for researchers, but it appears incremental as it builds on existing concepts of logical proportions and triangular norms.
The paper tackles the problem of defining analogical proportions between numerical values by proposing a definition based on triangular norms, specifically using Frank's family, and compares it with an existing approach using generalized means.
Starting from the Boolean notion of logical proportion in Piaget's sense, which turns out to be equivalent to analogical proportion, this note proposes a definition of analogical proportion between numerical values based on triangular norms (and dual co-norms). Frank's family of triangular norms is particularly interesting from this perspective. The article concludes with a comparative discussion with another very recent proposal for defining analogical proportions between numerical values based on the family of generalized means.