Multi-relational networks are used extensively to structure knowledge. Perhaps the most popular instance, due to the widespread adoption of the Semantic Web, is the Resource Description Framework (RDF). One of the primary purposes of a knowledge network is to reason; that is, to alter the topology of the network according to an algorithm that uses the existing topological structure as its input. There exist many such reasoning algorithms. With respect to the Semantic Web, the bivalent, monotonic reasoners of the RDF Schema (RDFS) and the Web Ontology Language (OWL) are the most prevalent. However, nothing prevents other forms of reasoning from existing in the Semantic Web. This article presents a non-bivalent, non-monotonic, evidential logic and reasoner that is an algebraic ring over a multi-relational network equipped with two binary operations that can be composed to execute various forms of inference. Given its multi-relational grounding, it is possible to use the presented evidential framework as another method for structuring knowledge and reasoning in the Semantic Web. The benefits of this framework are that it works with arbitrary, partial, and contradictory knowledge while, at the same time, it supports a tractable approximate reasoning process.
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