Research on Set Theory Based on Paraconsistent Logic

Shi Jing

doi:doi:10.11648/j.ijp.20200802.13

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Research on Set Theory Based on Paraconsistent Logic

Shi Jing

Published in International Journal of Philosophy (Volume 8, Issue 2)

Received: 1 May 2020 Accepted: 25 May 2020 Published: 3 June 2020

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Abstract

Different from ZF axiomatic set theory, the paraconsistent set theory has changed the basic logic of set theory and selected paraconsistent logic which can accommodate or deal with contradictions, it effectively avoids the whole theory falling into a non-trivial dilemma when there are contradictions in set theory. In this paper, we first review the history and current situation of the praconsistent set theory; then, we give three kinds of paraconsistent logic which can be used to construct the praconsistent set theory among many kinds of paraconsistent logics. And then, we analyze the differences of methods of the paraconsistent set theory with strong or weak structure of paraconsistent logic and get different paraconsistent set theory. Finally, we verify that paraconsistent set theory is a new method to solve the paradox of set theor. The development of paraconsistent set theory can solve the difficulties in the development of set theory in a unique way, which is not only the extension of the application of paraconsistent logic, but also the new form and new trend of the development of set theory.

Published in	International Journal of Philosophy (Volume 8, Issue 2)
DOI	10.11648/j.ijp.20200802.13
Page(s)	43-48
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Set Theory, Paraconsistent Logic, Paradox

References

[1]	Li Na, “Research on Set Theory Based on Philosophical Logic”. Journal of Zhejiang University (Humanities and Social Sciences), 2017 (1): 215-216.
[2]	A. I. Arruda, da Costa. “A hierarchy of formal systems”, C. R. Acad. Sc. Paris, 259, 1964, 2943-2945.
[3]	A. I. Arruda, D. Batens, “Russell’s set versus the universal set in paraconsistent set theory”. Logique et Analyse, 1982 (98): 121-133.
[4]	A. Arruda, “Remarks on da Costa’s paraconsistent set theories”, Rev Colombiana Mat, 1985, 19 (1-2): 9-24.
[5]	G. Restall, “A Note on Naive Set Theory in LP”, Notre Dame Journal of Formal Logic, 1992, 33 (3): 422-432.
[6]	Da Costa, “On paraconsistent set theory”. Logique et Analyse, 1986, 29 (115): 361-371.
[7]	R. da C. Caiero, E. G. de Souza, “A New Paraconsistent Set Theory: ML1”, Logique & Analyse, 1997, 157: 115-141.
[8]	T. Libert, “ZF and the Axiom of Choice in Some Paraconsistent Set Theories”, Logic and Logical Philosophy, 2003, 11: 91-114.
[9]	T. Libert, “Models for Paraconsistent Set Theory”, Journal of Applied Logic, 3, 15-41.
[10]	G. Priest, “Paraconsistent Set Theory. Logic”, Mathematics, Philosophy, Vintage Enthusiasms. Springer Netherlands, 2011, 153-169.
[11]	Zach Weber. “Transfinite numbers in paraconsistent set theory”. The Review of Symbolic Logic, 2010, 3 (1): 71-73.
[12]	Zach Weber. “Transfinite cardinals in paraconsistent set theory”. The Review of Symbolic Logic, 2012, 5 (2): 269-293.
[13]	W. Carnielli, M. E. Coniglioin, “Paraconsistent Set Theory by Predicating on Consistency”, Journal of Logic and Computation, 2013.
[14]	Nick Thomas. “Expressive limitations of naive set theory in LP and minimally inconsistent LP”. The Review of Symbolic Logic, 2014, 7 (2): 341-350.
[15]	Greg Restall. “A note on naive set theory in LP”. Notre Dame Journal of Formal Logic, 1992, 33 (3).
[16]	Walter Carnielli and Marcelo E. Coniglio. “Paraconsistent set theory by predicating on consistency”. Journal of Logic and Computation, 2016, 26 (1): 97-116.
[17]	Zhang Jianjun, “Editor’s Guide: A few questions about paraconsistent logic”, Logic Research, 2018 (2): 1-8.
[18]	Li Na, He Jianfeng, “A New Method to Deal with the Paradox of Set Theory”, Philosophical Trends, 2017 (11): 95-97.
[19]	Zhang Qingyu, “paraconsistent logic”, China Society Press, Beijing, 2003.: 23.
[20]	W. Carnielli, M. E. Coniglio, J. Marcos, “Logics of Formal Inconsistency”, Hankbook of Philosophical Logic, Vol. 14, Springer, 2007: pp 1-94.
[21]	W. Carnielli, M. E. Coniglio, “Paraconsistent Set Theory By Predicating on Consistency”, Journal of Logic and Computation, 26 (1), 2013: 1-20.
[22]	G. Priest, “The Logic of Paradox”, Journal of Philosophical Logic, 8 (1), 1979, pp. 219-241.
[23]	Zhang Qingyu, “paraconsistent logic, An Exploration of Logic”, 1996 (11): 18.

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APA Style

Shi Jing. (2020). Research on Set Theory Based on Paraconsistent Logic. International Journal of Philosophy, 8(2), 43-48. https://doi.org/10.11648/j.ijp.20200802.13

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ACS Style

Shi Jing. Research on Set Theory Based on Paraconsistent Logic. Int. J. Philos. 2020, 8(2), 43-48. doi: 10.11648/j.ijp.20200802.13

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AMA Style

Shi Jing. Research on Set Theory Based on Paraconsistent Logic. Int J Philos. 2020;8(2):43-48. doi: 10.11648/j.ijp.20200802.13

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@article{10.11648/j.ijp.20200802.13,
  author = {Shi Jing},
  title = {Research on Set Theory Based on Paraconsistent Logic},
  journal = {International Journal of Philosophy},
  volume = {8},
  number = {2},
  pages = {43-48},
  doi = {10.11648/j.ijp.20200802.13},
  url = {https://doi.org/10.11648/j.ijp.20200802.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijp.20200802.13},
  abstract = {Different from ZF axiomatic set theory, the paraconsistent set theory has changed the basic logic of set theory and selected paraconsistent logic which can accommodate or deal with contradictions, it effectively avoids the whole theory falling into a non-trivial dilemma when there are contradictions in set theory. In this paper, we first review the history and current situation of the praconsistent set theory; then, we give three kinds of paraconsistent logic which can be used to construct the praconsistent set theory among many kinds of paraconsistent logics. And then, we analyze the differences of methods of the paraconsistent set theory with strong or weak structure of paraconsistent logic and get different paraconsistent set theory. Finally, we verify that paraconsistent set theory is a new method to solve the paradox of set theor. The development of paraconsistent set theory can solve the difficulties in the development of set theory in a unique way, which is not only the extension of the application of paraconsistent logic, but also the new form and new trend of the development of set theory.},
 year = {2020}
}

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TY  - JOUR
T1  - Research on Set Theory Based on Paraconsistent Logic
AU  - Shi Jing
Y1  - 2020/06/03
PY  - 2020
N1  - https://doi.org/10.11648/j.ijp.20200802.13
DO  - 10.11648/j.ijp.20200802.13
T2  - International Journal of Philosophy
JF  - International Journal of Philosophy
JO  - International Journal of Philosophy
SP  - 43
EP  - 48
PB  - Science Publishing Group
SN  - 2330-7455
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AB  - Different from ZF axiomatic set theory, the paraconsistent set theory has changed the basic logic of set theory and selected paraconsistent logic which can accommodate or deal with contradictions, it effectively avoids the whole theory falling into a non-trivial dilemma when there are contradictions in set theory. In this paper, we first review the history and current situation of the praconsistent set theory; then, we give three kinds of paraconsistent logic which can be used to construct the praconsistent set theory among many kinds of paraconsistent logics. And then, we analyze the differences of methods of the paraconsistent set theory with strong or weak structure of paraconsistent logic and get different paraconsistent set theory. Finally, we verify that paraconsistent set theory is a new method to solve the paradox of set theor. The development of paraconsistent set theory can solve the difficulties in the development of set theory in a unique way, which is not only the extension of the application of paraconsistent logic, but also the new form and new trend of the development of set theory.
VL  - 8
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Author Information

Shi Jing

School of Culture and Media, Central University of Finance and Economics, Beijing, China

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APA Style

Shi Jing. (2020). Research on Set Theory Based on Paraconsistent Logic. International Journal of Philosophy, 8(2), 43-48. https://doi.org/10.11648/j.ijp.20200802.13

Copy | Download

ACS Style

Shi Jing. Research on Set Theory Based on Paraconsistent Logic. Int. J. Philos. 2020, 8(2), 43-48. doi: 10.11648/j.ijp.20200802.13

Copy | Download

AMA Style

Shi Jing. Research on Set Theory Based on Paraconsistent Logic. Int J Philos. 2020;8(2):43-48. doi: 10.11648/j.ijp.20200802.13

Copy | Download

@article{10.11648/j.ijp.20200802.13,
  author = {Shi Jing},
  title = {Research on Set Theory Based on Paraconsistent Logic},
  journal = {International Journal of Philosophy},
  volume = {8},
  number = {2},
  pages = {43-48},
  doi = {10.11648/j.ijp.20200802.13},
  url = {https://doi.org/10.11648/j.ijp.20200802.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijp.20200802.13},
  abstract = {Different from ZF axiomatic set theory, the paraconsistent set theory has changed the basic logic of set theory and selected paraconsistent logic which can accommodate or deal with contradictions, it effectively avoids the whole theory falling into a non-trivial dilemma when there are contradictions in set theory. In this paper, we first review the history and current situation of the praconsistent set theory; then, we give three kinds of paraconsistent logic which can be used to construct the praconsistent set theory among many kinds of paraconsistent logics. And then, we analyze the differences of methods of the paraconsistent set theory with strong or weak structure of paraconsistent logic and get different paraconsistent set theory. Finally, we verify that paraconsistent set theory is a new method to solve the paradox of set theor. The development of paraconsistent set theory can solve the difficulties in the development of set theory in a unique way, which is not only the extension of the application of paraconsistent logic, but also the new form and new trend of the development of set theory.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Research on Set Theory Based on Paraconsistent Logic
AU  - Shi Jing
Y1  - 2020/06/03
PY  - 2020
N1  - https://doi.org/10.11648/j.ijp.20200802.13
DO  - 10.11648/j.ijp.20200802.13
T2  - International Journal of Philosophy
JF  - International Journal of Philosophy
JO  - International Journal of Philosophy
SP  - 43
EP  - 48
PB  - Science Publishing Group
SN  - 2330-7455
UR  - https://doi.org/10.11648/j.ijp.20200802.13
AB  - Different from ZF axiomatic set theory, the paraconsistent set theory has changed the basic logic of set theory and selected paraconsistent logic which can accommodate or deal with contradictions, it effectively avoids the whole theory falling into a non-trivial dilemma when there are contradictions in set theory. In this paper, we first review the history and current situation of the praconsistent set theory; then, we give three kinds of paraconsistent logic which can be used to construct the praconsistent set theory among many kinds of paraconsistent logics. And then, we analyze the differences of methods of the paraconsistent set theory with strong or weak structure of paraconsistent logic and get different paraconsistent set theory. Finally, we verify that paraconsistent set theory is a new method to solve the paradox of set theor. The development of paraconsistent set theory can solve the difficulties in the development of set theory in a unique way, which is not only the extension of the application of paraconsistent logic, but also the new form and new trend of the development of set theory.
VL  - 8
IS  - 2
ER  -

Copy | Download