@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}, publisher = {Elsevier}, 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}, author = {Tommaso Flaminio and Llu\'{\i}s Godo and Sara Ugolini} }