Conceptual foundations of quantum mechanics: The role of evidence theory, quantum sets, and modal logic

Recognizing that syntactic and semantic structures of classical logic are n ot sufficient to understand the meaning of quantum phenomena, we propose in this paper a new interpretation of quantum mechanics based on evidence the ory. The connection between these two theories is obtained through a new la nguage, quantum set theory, built on a suggestion by J. Bell. Further, we g ive a modal logic interpretation of quantum mechanics and quantum set theor y by using Kripke's semantics of modal logic based on the concept of possib le worlds. This is grounded on previous work of a number of researchers (Re sconi, Klir, Harmanec) who shelved how to represent evidence theory and oth er uncertainty theories in terms of modal logic. Moreover, we also propose a reformulation of the many-worlds interpretation of quantum mechanics in t erms of Kripke's semantics. We thus show how three different theories - qua ntum mechanics, evidence theory, and modal logic - are interrelated. This o pens, on one hand, the way to new applications of quantum mechanics within domains different from the traditional ones, and, on the other hand, the po ssibility of building new generalizations of quantum mechanics itself.