On rearrangement inequalities for T-norm logics

TL;DR

This paper extends the classical rearrangement inequality to multi-valued logic systems using T-norms and T-conorms, demonstrating that similar ordering principles hold under these generalized operations.

Contribution

It introduces a novel generalization of the rearrangement inequality within T-norm based logics, including cases derived from Archimedean copulas.

Findings

Rearrangement inequalities hold for T-norms and T-conorms in multi-valued logic.

The inequalities are valid when T-norms and T-conorms are derived from Archimedean copulas.

The results unify classical inequalities with fuzzy logic and probabilistic models.

Abstract

The rearrangement inequality states that the sum of products of permutations of 2 sequences of real numbers are maximized when the terms are similarly ordered and minimized when the terms are ordered in opposite order. We show that similar inequalities exist for multi-valued logic with the multiplication and addition operation replaced with various T-norms and T-conorms respectively. For instance, we show that the rearrangement inequality holds when the T-norms and T-conorms are derived from Archimedean copulas.