The aim of this short note is to report on a counter-example by Stefano Aguzzoli (private communication) showing that a claim made in a recent paper of ours [2, Proposition 5.2], stating that the class of states of a free product algebra is closed, is in fact not true. That claim was used in turn in the proof of one of the main results of the same paper [2, Theorem 5.4]. However, we also provide in this note an alternative proof for that result, so that it keeps holding true.

}, doi = {10.1016/j.ijar.2018.09.010}, url = {https://doi.org/10.1016/j.ijar.2018.09.010}, author = {Tommaso Flaminio and Llu{\'\i}s Godo and Sara Ugolini} } @article {5618, title = {Towards a probability theory for product logic: states, integral representation and reasoning}, journal = {Internationa Journal of Approximate Reasoning}, volume = {93}, year = {2018}, pages = {199-218}, abstract = {The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one-one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the measure is a state in the above sense.}, url = {https://www.sciencedirect.com/science/article/pii/S0888613X17302360} } @conference {5606, title = {States of free product algebras and their integral representation}, booktitle = {37th Linz Seminar on Fuzzy Set Theory}, year = {2017}, pages = {35-38} }