Krause, Décio
(2012)
Paraconsistent QuasiSet Theory.
[Preprint]
Abstract
Paraconsistent logics are logics that can be used to base inconsistent
but nontrivial 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. Quasiset theories are mathematical systems built for
dealing with collections of indiscernible elements. The basic motivation
for the development of quasiset 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 antisymmetric functions
(or vectors) in the relevant Hilbert spaces. But in standard mathematics,
such as that built in ZermeloFraenkel 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 quasiset 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 quasiset 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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