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Compounds of conditionals and iterated conditioning under coherence


We discuss the problem of defining logical operations among conditional events. Differently from many authors, we define the conjunction and disjunction in the setting of conditional random quantities. In probability theory and in probability logic a relevant problem, largely discussed by many authors, is that of defining logical operations among conditional events. In the many works concerning these operations, the conjunction and disjunction have been usually defined as suitable conditional events. In Kaufmann 2009 it has been proposed a theory for the compounds of conditionals which has been framed in the setting of coherence in (Gilio and Sanfilippo , 2013, 2014) In this framework, which is quantitative rather than a logical one, conjunction and disjunction of conditional events are interpreted as conditional random quantities (c.r.q.'s), which can sometimes reduce to conditional events given logical dependencies. In our approach, given a random quantity $X$ and a non impossible event $H$ (we use the same symbols to refer to the event $H$ and its indicator), we look at a c.r.q. $X|H$ as an extended quantity $XH + mu o{H}$, where $mu$ is the conditional prevision $pr(X|H)$ which is subjectively evaluated. In particular (the indicator of) a conditional event $A|H$ is looked at as $AH + P(A|H) o{H}$. In this way, based on the betting scheme of de Finetti and its generalizations, the c.r.q. $X|H$ may be interpreted as the amount that you receive (resp., pay) in a bet on $X$ conditional on $H$, if you agree to pay (resp., to receive) $pr(X|H)$. This extended notion allows algebraic developments among c.r.q.'s also when the conditioning events are different. Then, among other things, we can give a meaning to the conjunction and disjunction of conditional events and we can define the notion of iterated conditionals; in particular we obtain that the usual probabilistic properties are preserved. Therefore, we consider the coherent prevision of these suitable random quantities as the generalization of the corresponding coherent probabilistic versions of unconditional events. These notions are relevant in natural language for a probabilistic interpretation of the conjunction, disjunction and iteration of conditionals. In particular they are useful for the probabilistic approaches in the psychology of reasoning and in the conditional syllogisms.