Towards an Algebraic and Probabilistic Setting for Iterated Boolean Conditionals

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” μQ to the algebra of iterated conditionals. Precisely, we show that Q satisfies a property called “separability” if and only if μQ satisfies 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 μQ on the algebra of iterated conditionals.