Frank's triangular norms in Piaget's logical proportions
Henri Prade, Gilles Richard
TL;DR
This work extends Piaget's Boolean logical proportion to numerical values by replacing logical connectives with pairs of triangular norms and dual co-norms, focusing on Frank's t-norm family for its advantageous arithmetic property $T_F^p(a,b)+S^p_{T_F}(a,b)=a+b$. It defines an all-or-nothing analogical proportion $A_T(a,b,c,d)$ via $T(a,d)=T(b,c)$ and $S_T(a,d)=S_T(b,c)$, showing it preserves core analogical postulates and aligns with the six Boolean patterns when inputs are in $\{0,1\}$. The paper compares this framework to a recent generalized-means approach and discusses how Frank's family allows interpolation between conservative and liberal perspectives on analogical relations while offering a potential path toward unification of numerical analogies. Overall, the approach provides a principled, logic-based method for capturing analogical relations in numerical data with clear avenues for further theoretical and empirical development.
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.