Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic

Chubaryan Anahit; Khamisyan Artur

doi:doi:10.11648/j.pamj.20170602.12

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Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic

Chubaryan Anahit, Khamisyan Artur

Published in Pure and Applied Mathematics Journal (Volume 6, Issue 2)

Received: 13 February 2017 Accepted: 15 March 2017 Published: 22 March 2017

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Abstract

This paper focuses on the problem of constructing of some standard Hilbert style proof systems for any version of many valued propositional logic. The generalization of Kalmar’s proof of deducibility for two valued tautologies inside classical propositional logic gives us a possibility to suggest some method for defining of two types axiomatic systems for any version of 3-valued logic, completeness of which is easy proved direct, without of loading into two valued logic. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-valued (k≥3) logics and may be for fuzzy logic also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics.

Published in	Pure and Applied Mathematics Journal (Volume 6, Issue 2)
DOI	10.11648/j.pamj.20170602.12
Page(s)	71-75
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), 2017. Published by Science Publishing Group

Keywords

Many-Valued Logics, Hilbert-Style Axiomatic Systems, Completeness of Formal System

References

[1]	E.Mendelson, Introduction to Mathematical Logic, Van Nostrand, Princeton, 1975.
[2]	An.Chubaryan, Relative efficiency of some proof systems for classical propositional logic, Proceedings of NASA RA, Vol.37, N5, 2002, and Journal of CMA (AAS),Vol.37, N5, 2002, 71-84.
[3]	A.A.Chubaryan, A.S.Tshitoyan, A.A.Khamisyan, On some proof systems for many-valued logics and on proof complexities in it, (in Russian) Reports of NASA RA, Vol. 116, N2, 2016, 18-24.
[4]	J.Lukasiewicz, O Logice Trojwartosciowej, Ruch filoseficzny (Lwow), Vol.5, 1920, 169-171.
[5]	E.Post, Introduction to a general theory of elementary propositions, Amer. Journ. Math., Vol.43, 1921, 163-185.
[6]	K.Godel, Zum intuitionistishen Aussagen kalkul, Akademie der Wissenshaften in Wien, Mathematische-naturwissenschaftliche Klasse, Auzeiger, Vol.69, 1932, 65-66.
[7]	S.Jaskowski, On the rules of suppositions in formal logic, Studia logica, No.1, Warsaw, 1934, 5-32.
[8]	S.C.Kleene, On the interpretation of intuitionistic number theory, Journ. Symb. Log. Vol.10, No.4, 1945, 109-123.
[9]	D.A. Bochvar, M. Bergmann, On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus, Journal History and Philosophy of Logic, Volume2, Issue1-2, 1981, 87-112.
[10]	N.D. Belnap, Auseful four-valued logic, in: Modern Uses of Multiple-valued Logic, Oriel Press., 1977, 35-56.
[11]	I.D. Zaslawskij, Symmetrical Konstructive Logic, Press. ASAR, (in Russian), 1978.

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APA Style

Chubaryan Anahit, Khamisyan Artur. (2017). Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic. Pure and Applied Mathematics Journal, 6(2), 71-75. https://doi.org/10.11648/j.pamj.20170602.12

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ACS Style

Chubaryan Anahit; Khamisyan Artur. Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic. Pure Appl. Math. J. 2017, 6(2), 71-75. doi: 10.11648/j.pamj.20170602.12

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AMA Style

Chubaryan Anahit, Khamisyan Artur. Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic. Pure Appl Math J. 2017;6(2):71-75. doi: 10.11648/j.pamj.20170602.12

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@article{10.11648/j.pamj.20170602.12,
  author = {Chubaryan Anahit and Khamisyan Artur},
  title = {Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic},
  journal = {Pure and Applied Mathematics Journal},
  volume = {6},
  number = {2},
  pages = {71-75},
  doi = {10.11648/j.pamj.20170602.12},
  url = {https://doi.org/10.11648/j.pamj.20170602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170602.12},
  abstract = {This paper focuses on the problem of constructing of some standard Hilbert style proof systems for any version of many valued propositional logic. The generalization of Kalmar’s proof of deducibility for two valued tautologies inside classical propositional logic gives us a possibility to suggest some method for defining of two types axiomatic systems for any version of 3-valued logic, completeness of which is easy proved direct, without of loading into two valued logic. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-valued (k≥3) logics and may be for fuzzy logic also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics.},
 year = {2017}
}

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TY  - JOUR
T1  - Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic
AU  - Chubaryan Anahit
AU  - Khamisyan Artur
Y1  - 2017/03/22
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170602.12
DO  - 10.11648/j.pamj.20170602.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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PB  - Science Publishing Group
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AB  - This paper focuses on the problem of constructing of some standard Hilbert style proof systems for any version of many valued propositional logic. The generalization of Kalmar’s proof of deducibility for two valued tautologies inside classical propositional logic gives us a possibility to suggest some method for defining of two types axiomatic systems for any version of 3-valued logic, completeness of which is easy proved direct, without of loading into two valued logic. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-valued (k≥3) logics and may be for fuzzy logic also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics.
VL  - 6
IS  - 2
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Author Information

Chubaryan Anahit

Department of Informatics and Applied Mathematics, Yerevan State University and Russian-Armenian University, Yerevan, Armenia
Khamisyan Artur

Department of Informatics and Applied Mathematics, Yerevan State University, Yerevan, Armenia

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Plain Text BibTeX RIS

APA Style

Chubaryan Anahit, Khamisyan Artur. (2017). Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic. Pure and Applied Mathematics Journal, 6(2), 71-75. https://doi.org/10.11648/j.pamj.20170602.12

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ACS Style

Chubaryan Anahit; Khamisyan Artur. Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic. Pure Appl. Math. J. 2017, 6(2), 71-75. doi: 10.11648/j.pamj.20170602.12

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AMA Style

Chubaryan Anahit, Khamisyan Artur. Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic. Pure Appl Math J. 2017;6(2):71-75. doi: 10.11648/j.pamj.20170602.12

Copy | Download

@article{10.11648/j.pamj.20170602.12,
  author = {Chubaryan Anahit and Khamisyan Artur},
  title = {Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic},
  journal = {Pure and Applied Mathematics Journal},
  volume = {6},
  number = {2},
  pages = {71-75},
  doi = {10.11648/j.pamj.20170602.12},
  url = {https://doi.org/10.11648/j.pamj.20170602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170602.12},
  abstract = {This paper focuses on the problem of constructing of some standard Hilbert style proof systems for any version of many valued propositional logic. The generalization of Kalmar’s proof of deducibility for two valued tautologies inside classical propositional logic gives us a possibility to suggest some method for defining of two types axiomatic systems for any version of 3-valued logic, completeness of which is easy proved direct, without of loading into two valued logic. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-valued (k≥3) logics and may be for fuzzy logic also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic
AU  - Chubaryan Anahit
AU  - Khamisyan Artur
Y1  - 2017/03/22
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170602.12
DO  - 10.11648/j.pamj.20170602.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 71
EP  - 75
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20170602.12
AB  - This paper focuses on the problem of constructing of some standard Hilbert style proof systems for any version of many valued propositional logic. The generalization of Kalmar’s proof of deducibility for two valued tautologies inside classical propositional logic gives us a possibility to suggest some method for defining of two types axiomatic systems for any version of 3-valued logic, completeness of which is easy proved direct, without of loading into two valued logic. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-valued (k≥3) logics and may be for fuzzy logic also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics.
VL  - 6
IS  - 2
ER  -

Copy | Download