# Paraconsistent Quasi-Set Theory

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

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## Abstract

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
Creators:
CreatorsEmailORCID
Krause, Déciodeciokrause@gmail.com
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 14:56
Item ID: 9053
Subjects: Specific Sciences > Physics > Quantum Mechanics
General Issues > Structure of Theories
Date: 12 March 2012
URI: https://philsci-archive.pitt.edu/id/eprint/9053