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Paraconsistent Quasi-Set Theory

Krause, Décio (2012) Paraconsistent Quasi-Set Theory. [Preprint]

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    Paraconsistent logics are logics that can be used to base inconsistent
    but non-trivial systems. In paraconsistent set theories, we can quan-
    tify over sets that in standard set theories (that are based on classical
    logic), if consistent, would lead to contradictions, such as the Russell set,
    R = fx : x =2 xg. Quasi-set theories are mathematical systems built for
    dealing with collections of indiscernible elements. The basic motivation
    for the development of quasi-set theories came from quantum physics,
    where indiscernible entities need to be considered (in most interpreta-
    tions). Usually, the way of dealing with indiscernible objects within clas-
    sical logic and mathematics is by restricting them to certain structures,
    in a way so that the relations and functions of the structure are not sufficient to individuate the objects; in other words, such structures are not
    rigid. In quantum physics, this idea appears when symmetry conditions
    are introduced, say by choosing symmetric and anti-symmetric functions
    (or vectors) in the relevant Hilbert spaces. But in standard mathematics,
    such as that built in Zermelo-Fraenkel set theory (ZF), any structure can
    be extended to a rigid structure. That means that, although we can deal
    with certain objects as they were indiscernible, we realize that from out-
    side of these structures these objects are no more indiscernible, for they
    can be individualized in the extended rigid structures: ZF is a theory
    of individuals, distinguishable objects. In quasi-set theory, it seems that
    there are structures that cannot be extended to rigid ones, so it seems that
    they provide a natural mathematical framework for expressing quantum
    facts without certain symmetry suppositions. There may be situations,
    however, in which we may need to deal with inconsistent bits of infor-
    mation in a quantum context, even if these informations are concerned
    with ways of speech. Furthermore, some authors think that superposi-
    tions may be understood in terms of paraconsistent logics, and even the
    notion of complementarity was already treated by such a means. This is,
    apparently, a nice motivation to try to merge these two frameworks. In
    this work, we develop the technical details, by basing our quasi-set theory
    in the paraconsistent system C1. We also elaborate a new hierarchy of
    paraconsistent calculi, the paraconsistent calculi with indiscernibility. For
    the finalities of this work, some philosophical questions are outlined, but
    this topic is left to a future work.

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    Item Type: Preprint
    Keywords: paraconsistent logic, quasi-set theory, indistinguishable quanta, quantum physics.
    Subjects: Specific Sciences > Physics > Quantum Mechanics
    General Issues > Structure of Theories
    Depositing User: Décio Krause
    Date Deposited: 14 Mar 2012 10:56
    Last Modified: 13 Sep 2015 11:54
    Item ID: 9053

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