DSm models and Non-Archimedean Reasoning

The Dezert-Smarandache theory of plausible and paradoxical reasoning is based on the premise that some elements θi of a frame Θ have a non-empty intersection. These elements are called exhaustive. In number theory, this property is observed only in non-Archimedean number systems, for example, in the ring Zp of p-adic integers, in the field Q of hyperrational numbers, in the field R of hyperreal numbers, etc. In this chapter, I show that non-Archimedean structures are infinite DSm models in that each positive exhaustive element is greater (or less) than each positive exclusive element. Then I consider three principal versions of the non-Archimedean logic: p-adic valued logic MZp , hyperrational valued logic M∗Q, hyperreal valued logic M∗R, and their applications to plausible reasoning. These logics are constructed for the first time.