Valuations in Gödel Logic, and the Euler Characteristic

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 Gödel 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 $φ$, and prove that they indeed provide information about the logical status of $φ$ in Gödel 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 Gödel logic over finitely many propositional variables.