Uncertain Linear Logic via Fibring of Probabilistic and Fuzzy Logic

Linear logic [Gir87] comprises a rich and fascinating formal system that summarizes, in a nuanced way, the way logical inference works if one treats the pool of potential premises of inferences as a resource to be meted out and accounted for. The linear logic abstractions can be applied to practical reasoning systems in a variety of different ways, and can be grounded in concrete domain-specific inference formalisms via multiple routes as well. Here we connect linear logic to uncertain reasoning based on observational semantics. Beginning with a simple semantics for propositions, based on counting observations, we argue that probabilistic and fuzzy logic correspond to two different heuristic assumptions regarding the combination of propositions whose evidence bases are not currently available. These two different heuristic assumptions lead to two different sets of formulas for propagating quantitative truth values through lattice operations. Given this setup, it becomes immediately apparents that these two sets of formulas instantiate the same algebraic and conceptual relationships as the multiplicative and additive operator-sets in linear logic. The standard rules of linear logic then emerge as consequences of the underlying semantics of fuzzy and probabilistic evidence management.