Frank's triangular norms in Piaget's logical proportions

TL;DR

This paper explores defining analogical proportions between numerical values using Frank's triangular norms, linking Piaget's logical proportions with fuzzy logic concepts, and compares this approach to generalized means.

Contribution

It introduces a novel definition of analogical proportion based on triangular norms, connecting Piaget's logical proportions with fuzzy logic and providing a comparative analysis.

Findings

Frank's triangular norms effectively define analogical proportions between numbers.

The proposed approach offers a new perspective on logical proportions in Piaget's theory.

Comparison shows advantages over generalized means in certain contexts.

Abstract

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.