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.

Open Partitions and Probability Assignments in Gödel Logic / P. Codara, O. M. D'Antona, V. Marra - In: Symbolic and quantitative approaches to reasoning with uncertainty : 10th European Conference, ECSQARU 2009, Verona, Italy, July 1-3, 2009 : proceedings / [a cura di] C. Sossai, G. Chemello. - Berlin : Springer, 2009. - ISBN 9783642029059. - pp. 911-922 (( Intervento presentato al 10th. convegno European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2009) tenutosi a Verona, Italy nel 2009 [10.1007/978-3-642-02906-6_78].

### Open Partitions and Probability Assignments in Gödel Logic

#### Abstract

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.
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Gödel logic ; partition ; probability
2009
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2434/72808`
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