Toward a probability theory for product logic: states, integral representation and reasoning

Tommaso Flaminio; Lluis Godo; Sara Ugolini

doi:10.1016/j.ijar.2017.11.003

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arXiv · 1803.03208

Toward a probability theory for product logic: states, integral representation and reasoning

Tommaso Flaminio·Lluis Godo·Sara Ugolini

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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.

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BibTeX RIS

Tommaso Flaminio, Lluis Godo, Sara Ugolini. 2018-03-08. Toward a probability theory for product logic: states, integral representation and reasoning. https://doi.org/10.1016/j.ijar.2017.11.003

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