In this paper we investigate canonical extensions of conditional probabilities to Boolean algebras of conditionals. Before entering into the probabilistic setting, we first prove that the lattice order relation of every Boolean algebra of conditionals can be characterized in terms of the well-known order relation given by Goodman and Nguyen. Then, as an interesting methodological tool, we show that canonical extensions behave well with respect to conditional subalgebras. As a consequence, we prove that a canonical extension and its original conditional probability agree on basic conditionals. Moreover, we verify that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as defined in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. Finally we discuss relations of our approach with nonmonotonic reasoning based on an entailment relation among conditionals.

Flaminio, T., Gilio, A., Godo, L., Sanfilippo, G. (2023). On conditional probabilities and their canonical extensions to Boolean algebras of compound conditionals. INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, 159 [10.1016/j.ijar.2023.108943].

On conditional probabilities and their canonical extensions to Boolean algebras of compound conditionals

Sanfilippo, Giuseppe
2023-05-16

Abstract

In this paper we investigate canonical extensions of conditional probabilities to Boolean algebras of conditionals. Before entering into the probabilistic setting, we first prove that the lattice order relation of every Boolean algebra of conditionals can be characterized in terms of the well-known order relation given by Goodman and Nguyen. Then, as an interesting methodological tool, we show that canonical extensions behave well with respect to conditional subalgebras. As a consequence, we prove that a canonical extension and its original conditional probability agree on basic conditionals. Moreover, we verify that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as defined in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. Finally we discuss relations of our approach with nonmonotonic reasoning based on an entailment relation among conditionals.
16-mag-2023
Settore MAT/06 - Probabilita' E Statistica Matematica
Flaminio, T., Gilio, A., Godo, L., Sanfilippo, G. (2023). On conditional probabilities and their canonical extensions to Boolean algebras of compound conditionals. INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, 159 [10.1016/j.ijar.2023.108943].
File in questo prodotto:
File Dimensione Formato  
1-s2.0-S0888613X23000749-main.pdf

accesso aperto

Tipologia: Versione Editoriale
Dimensione 1.18 MB
Formato Adobe PDF
1.18 MB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/590564
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 5
  • ???jsp.display-item.citation.isi??? 4
social impact