On valuations in Gödel and nilpotent minimum logics

Some decades ago, V. Klee and G.-C. Rota introduced a lattice-theoretic analogue of the Euler characteristic, the celebrated topological invariant of polyhedra. In [1], using the Klee-Rota defi nition, we introduce the Euler characteristic of a formula in Goedel logic, the extension of intuitionistic logic via the prelinearity axiom (A->B)V(B->A). We then prove that the Euler characteristic of a formula A over n propositional variables coincides with the number of Boolean assignments to these n variables that satisfy A. Building on this, we generalise this notion to other invariants of A that provide additional information about the satis ability of A in Goedel logic. Specifically, the Euler characteristic does not determine non-classical tautologies: the maximum value of the characteristic of A(X1,...,Xn) is 2^n, and this can be attained even when A is not a tautology in Goedel logic. By contrast, we prove that these new invariants do. In this talk, we present the aforementioned results and compare what has been obtained for Goedel logic with analogous results for a diff erent many-valued logic, namely, the logic of Nilpotent Minimum. This logic can also be described as the extension of Nelson logic by the prelinearity axiom. The latter results are joint work with D. Valota. [1] P. Codara, O. M. D'Antona, V. Marra, Valuations in Goedel Logic, and the Euler Characteristic, Journal of Multiple-Valued Logic and Soft Computing 19(1-3) (2012), 71-84.