Valuations in G\"{o}del Logic, and the Euler Characteristic

TL;DR

This paper introduces a lattice-theoretic Euler characteristic for G"{o}del logic formulas, showing it encodes classical information and helps distinguish tautologies from non-tautologies, with implications for understanding many-valued logic valuations.

Contribution

It defines a novel Euler characteristic for G"{o}del logic formulas and demonstrates its effectiveness in classifying tautologies and analyzing the structure of many-valued logic valuations.

Findings

Euler characteristic coincides with classical Boolean logic.

Many-valued Euler characteristics distinguish tautologies from non-tautologies.

Set of many-valued characteristics is linearly independent in valuation space.

Abstract

Using the lattice-theoretic version of the Euler characteristic introduced by V. Klee and G.-C. Rota in the Sixties, we define the Euler characteristic of a formula in G\"{o}del logic (over finitely or infinitely many truth-values). We then prove that the information encoded by the Euler characteristic is classical, i.e. coincides with the analogous notion defined over Boolean logic. Building on this, we define many-valued versions of the Euler characteristic of a formula $\varphi$, and prove that they indeed provide information about the logical status of $\varphi$ in G\"{o}del logic. Specifically, our first main result shows that the many-valued Euler characteristics are invariants that separate many-valued tautologies from non-tautologies. Further, we offer an initial investigation of the linear structure of these generalised characteristics. Our second main result is that the collection of many-valued characteristics forms a linearly independent set in the real vector space of all valuations of G\"{o}del logic over finitely many propositional variables.