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 |
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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 |

URI: | http://philsci-archive.pitt.edu/id/eprint/9053 |

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