First-order Nilpotent Minimum Logics: first steps

TL;DR

This paper investigates the properties of first-order Nilpotent Minimum logic, focusing on decidability, tautologies, and the relationship between formulas and algebraic structures, extending prior work on G"odel logics.

Contribution

It provides the first systematic analysis of first-order NM logic, including decidability, tautology sets, and algebraic connections, filling a gap in the logical landscape.

Findings

Decidability and tautology inclusion results for NM subalgebras

Connection established between formula validity and order type of NM-chains

Analysis of axiomatizability, undecidability, and monadic fragments

Abstract

Following the lines of the analysis done in [BPZ07, BCF07] for first-order G\"odel logics, we present an analogous investigation for Nilpotent Minimum logic NM. We study decidability and reciprocal inclusion of various sets of first-order tautologies of some subalgebras of the standard Nilpotent Minimum algebra. We establish a connection between the validity in an NM-chain of certain first-order formulas and its order type. Furthermore, we analyze axiomatizability, undecidability and the monadic fragments.