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RESEARCH PRODUCT

Compound conditionals, Fr\'echet-Hoeffding bounds, and Frank t-norms

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Abstract In this paper we consider compound conditionals, Frechet-Hoeffding bounds and the probabilistic interpretation of Frank t-norms. By studying the solvability of suitable linear systems, we show under logical independence the sharpness of the Frechet-Hoeffding bounds for the prevision of conjunctions and disjunctions of n conditional events. In addition, we illustrate some details in the case of three conditional events. We study the set of all coherent prevision assessments on a family containing n conditional events and their conjunction, by verifying that it is convex. We discuss the case where the prevision of conjunctions is assessed by Lukasiewicz t-norms and we give explicit solutions for the linear systems; then, we analyze a selected example. We obtain a probabilistic interpretation of Frank t-norms and t-conorms as prevision of conjunctions and disjunctions of conditional events, respectively. Then, we characterize the sets of coherent prevision assessments on a family containing n conditional events and their conjunction, or their disjunction, by using Frank t-norms, or Frank t-conorms. By assuming logical independence, we show that any Frank t-norm (resp., t-conorm) of two conditional events A | H and B | K , T λ ( A | H , B | K ) (resp., S λ ( A | H , B | K ) ), is a conjunction ( A | H ) ∧ ( B | K ) (resp., a disjunction ( A | H ) ∨ ( B | K ) ). Then, we analyze the case of logical dependence where A = B and we obtain the set of coherent assessments on A | H , A | K , ( A | H ) ∧ ( A | K ) ; moreover we represent it in terms of the class of Frank t-norms T λ , with λ ∈ [ 0 , 1 ] . By considering a family F containing three conditional events, their conjunction, and all pairwise conjunctions, we give some results on Frank t-norms and coherence of the prevision assessments on F . By assuming logical independence, we show that it is coherent to assess the previsions of all the conjunctions by means of Minimum and Product t-norms. In this case all the conjunctions coincide with the t-norms of the corresponding conditional events. We verify by a counterexample that, when the previsions of conjunctions are assessed by the Lukasiewicz t-norm, coherence is not assured. Then, the Lukasiewicz t-norm of conditional events may not be interpreted as their conjunction. Finally, we give two sufficient conditions for coherence and incoherence when using the Lukasiewicz t-norm.