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Relevant First-Order Logic LP# and Curry’s Paradox Resolution

Received: 19 November 2014    Accepted: 22 November 2014    Published: 19 January 2015
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Abstract

In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s paraconsistent Set Theories〖NF〗_n^C

DOI 10.11648/j.pamj.s.2015040101.12
Published in Pure and Applied Mathematics Journal (Volume 4, Issue 1-1, January 2015)

This article belongs to the Special Issue Modern Combinatorial Set Theory and Large Cardinal Properties

Page(s) 6-12
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), 2024. Published by Science Publishing Group

Keywords

Curry's Paradox, Shaw-Kwei's, Paradox, Relevance Logics, Ƚukasiewicz Logic, Abelian Logic

References
[1] Curry, H. (1942): “The Inconsistency of certain formal logics”, Journal of Symbolic Logic 7, 115–17.
[2] M.Jago, Recent Work in Relevant Logic, Analysis (2013) 73 (3): 526-541. doi: 10.1093/analys/ant043
[3] Anderson, A.R. and N.D. Belnap, Jr., 1975, Entailment: The Logic of Relevance and Necessity, Princeton, Princeton University Press, Volume I. Anderson, A.R. N.D. Belnap, Jr. and J.M. Dunn (1992)Entailment, Volume II. [These are both collections of slightly modified articles on relevance logic together with a lot of material unique to these volumes. Excellent work and still the standard books on the subject. But they are very technical and quite difficult.]C. Smorynski, Handbook of mathematical logic, Edited by J. Barwise.North-Holland Publishing Company,1977
[4] Brady, R.T., 2005, Universal Logic, Stanford: CSLI Publications, 2005. [A difficult, but extremely important book, which gives details of Brady's semantics and his proofs that naïve set theory and higher order logic based on his weak relevant logic are consistent.]D.Marker,Model theory: an introduction.(Graduate Texts in Mathematics,Vol. 217). Springer 2002.
[5] Dunn, J.M., 1986, “Relevance Logic and Entailment” in F. Guenthner and D. Gabbay (eds.), Handbook of Philosophical Logic, Volume 3, Dordrecht: Reidel, pp. 117–24. [Dunn has rewritten this piece together with Greg Restall and the new version has appeared in volume 6 of the new edition of the Handbook of Philosophical Logic, Dordrecht: Kluwer, 2002,
[6] Mares, E.D., Relevant Logic: A Philosophical Interpretation, Cambridge: Cambridge University Press. U. R. Schmerl, Iterated Reflection Principles and the ω-Rule.The Journal ofSymbolic Logic,2004,Vol. 47, No. 4, pp.721-733.
[7] Mares, E.D. and R.K. Meyer, 2001, “Relevant Logics” in L. Goble (ed.), The Blackwell Guide to Philosophical Logic, Oxford: Blackwell.S. Feferman,Systems of predicative analysis. Journal of Symbolic Logic29:1-30.
[8] Paoli, F., 2002, Substructural Logics: A Primer, Dordrecht: Kluwer. [Excellent and clear introduction to a field of logic that includes relevance logic.]S. Feferman, C. Spector, Incompleteness along paths in progressionsof theories. Journal of Symbolic Logic 27:383--390.
[9] Priest, G., 2008, An Introduction to Non-Classical Logic: From If to Is, Cambridge: University of Cambridge Press. P.Lindstrom, "First order predicate logic and generalized quantifiers",Theoria, Vol. 32, No.3. pp. 186-195, December 1966.
[10] Read, S., 1988, Relevant Logic, Oxford: Blackwell.
[11] Restall, G., 2000, An Introduction to Substructural Logics, London: Routledge.
[12] [10] Routley, R., R.K. Meyer, V. Plumwood and R. Brady, 1983,Relevant Logics and its Rivals. Volume I.
[13] R.Brady (ed.), Relevant Logics and their Rivals. Volume II.
[14] Anderson, A.R., 1967, “Some Nasty Problems in the Formal Logic of Ethics,” Noûs, 1: 354–360.
[15] Belnap, N.D., 1982, “Display Logic,” Journal of Philosophical Logic, 11: 375–417.
[16] Robles, G., Méndez, J. M., Curry’s Paradox, Generalized Modus Ponens Axiom and Depth Relevance, StudiaLogicaFebruary 2014, Volume 102, Issue 1, pp. 185-217.
[17] G. Robles, J. M. Méndez, Blocking the Routes to Triviality with Depth Relevance, Journal of Logic, Language and InformationDecember 2014, Volume 23, Issue 4, pp. 493-526.
[18] Rogerson, S., Restall, G. Routes to Triviality, Journal of Philosophical Logic August 2004, Volume 33, Issue 4, pp. 421-436.
[19] Foukzon,J.,Relevant First-Order Logic LP# and Curry's Paradox. April 2008http://arxiv.org/abs/0804.4818
[20] Foukzon,J.,Paraconsistent First-Order Logic LP# with infinite hierarchy levels of contradiction.May 2008http://arxiv.org/abs/0805.1481
[21] Shaw-Kwei, M.,Logical paradoxes for many-valued systems. Journal of Symbolic Logic, pages 37-40, 1954.
[22] Restall. G., How to be really contraction free. StudiaLogica, 52(3):381{391,1993.
[23] Bacon,A,Curry’s Paradox and ω-Inconsistency,StudiaLogica February 2013, Volume 101, Issue 1, pp. 1-9.
[24] Hajek, P., Paris, J., and Shepherdson, J., The liar paradox and fuzzy logic. Journal of Symbolic Logic, 65(1):339{346, 2000.
[25] Restall, G., Arithmetic and truth in Lukasiewicz'sinnitely valued logic.LogiqueetAnalyse, 140:303{12, 1992.
[26] Restall,G., An introduction to substructural logics. Routledge, 2000.
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  • APA Style

    Jaykov Foukzon. (2015). Relevant First-Order Logic LP# and Curry’s Paradox Resolution. Pure and Applied Mathematics Journal, 4(1-1), 6-12. https://doi.org/10.11648/j.pamj.s.2015040101.12

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    ACS Style

    Jaykov Foukzon. Relevant First-Order Logic LP# and Curry’s Paradox Resolution. Pure Appl. Math. J. 2015, 4(1-1), 6-12. doi: 10.11648/j.pamj.s.2015040101.12

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    AMA Style

    Jaykov Foukzon. Relevant First-Order Logic LP# and Curry’s Paradox Resolution. Pure Appl Math J. 2015;4(1-1):6-12. doi: 10.11648/j.pamj.s.2015040101.12

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  • @article{10.11648/j.pamj.s.2015040101.12,
      author = {Jaykov Foukzon},
      title = {Relevant First-Order Logic LP# and Curry’s Paradox Resolution},
      journal = {Pure and Applied Mathematics Journal},
      volume = {4},
      number = {1-1},
      pages = {6-12},
      doi = {10.11648/j.pamj.s.2015040101.12},
      url = {https://doi.org/10.11648/j.pamj.s.2015040101.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040101.12},
      abstract = {In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s paraconsistent Set Theories〖NF〗_n^C},
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Author Information
  • Israel Institute of Technology, Department of Mathematics, Haifa, Israel

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