by Lydia Castronovo, Tommaso Flaminio, Lluis Godo, Giuseppe Sanfilippo
Abstract:
The present paper is about iterated conditionals, i.e., expressions of the form (a|b)|(c|d) that read as ``if c holds conditionally to d, then a holds conditionally to b''. Firstly, we introduce algebraic structures for iterated conditionals by repeating twice the construction of Boolean algebras of conditionals, where one can represent basic conditionals (a|b) and their Boolean combinations. Then, from the probabilistic perspective, we show that relevant properties of a probability Q on these Boolean algebras of conditionals can be characterized in terms of satisfiability of known principles of its ``canonical extensions'' \$\$\backslashmu \_Q\$\$$\mu$Qto the algebra of iterated conditionals. Precisely, we show that Q satisfies a property called ``separability'' if and only if \$\$\backslashmu \_Q\$\$$\mu$Qsatisfies a weak version of the Import-Export principle. Likewise, Q satisfies the McGee formula for the conjunction of basic conditionals if and only if a ``conjunction rationality principle'' holds for its canonical extension \$\$\backslashmu \_Q\$\$$\mu$Qon the algebra of iterated conditionals.
Reference:
Lydia Castronovo, Tommaso Flaminio, Lluis Godo, Giuseppe Sanfilippo, "Towards an Algebraic and Probabilistic Setting for Iterated Boolean Conditionals", Chapter in Symbolic and Quantitative Approaches to Reasoning with Uncertainty, , Springer Nature Switzerland, Cham, pp. 331-346, 2026.
Bibtex Entry:
@INCOLLECTION{ECSQARU2025,
author="Castronovo, Lydia
and Flaminio, Tommaso
and Godo, Lluis
and Sanfilippo, Giuseppe",
editor="Sauerwald, Kai
and Thimm, Matthias",
title="Towards an Algebraic and Probabilistic Setting for Iterated Boolean Conditionals",
booktitle="Symbolic and Quantitative Approaches to Reasoning with Uncertainty",
year="2026",
publisher="Springer Nature Switzerland",
address="Cham",
pages="331--346",
abstract="The present paper is about iterated conditionals, i.e., expressions of the form (a|b)|(c|d) that read as ``if c holds conditionally to d, then a holds conditionally to b''. Firstly, we introduce algebraic structures for iterated conditionals by repeating twice the construction of Boolean algebras of conditionals, where one can represent basic conditionals (a|b) and their Boolean combinations. Then, from the probabilistic perspective, we show that relevant properties of a probability Q on these Boolean algebras of conditionals can be characterized in terms of satisfiability of known principles of its ``canonical extensions'' {\$}{\$}{\backslash}mu {\_}Q{\$}{\$}$\mu$Qto the algebra of iterated conditionals. Precisely, we show that Q satisfies a property called ``separability'' if and only if {\$}{\$}{\backslash}mu {\_}Q{\$}{\$}$\mu$Qsatisfies a weak version of the Import-Export principle. Likewise, Q satisfies the McGee formula for the conjunction of basic conditionals if and only if a ``conjunction rationality principle'' holds for its canonical extension {\$}{\$}{\backslash}mu {\_}Q{\$}{\$}$\mu$Qon the algebra of iterated conditionals.",
isbn="978-3-032-05134-9",
doi="https://doi.org/10.1007/978-3-032-05134-9_23",
}