Open Partitions and Probability Assignments in Gödel Logic

In the elementary case of finitely many events, we generalise to Gödel (propositional infinite-valued) logic — one of the fundamental fuzzy logics in the sense of Hájek — the classical correspondence between partitions, quotient measure spaces, and push-forward measures. To achieve this end, appropriate Gödelian analogues of the Boolean notions of probability assignment and partition are needed. Concerning the former, we use a notion of probability assignment introduced in the literature by the third-named author et al. Concerning the latter, we introduce and use open partitions, whose definition is justified by independent considerations on the relational semantics of Gödel logic (or, more generally, of the finite slice of intuitionistic logic). Our main result yields a construction of finite quotient measure spaces in the Gödelian setting that closely parallels its classical counterpart.