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Extensionality and Restriction in Naive Set Theory

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Abstract

The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads to triviality.

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References

  1. Asmus, Conrad, ‘Restricted Arrow’, Journal of Philosophical Logic, 38(4), 2009.

  2. Austin, David F., (ed.), Philosophical Analysis, Kluwer, 1988.

  3. Batens, D., C. Mortensen, G. Priest, and J.-P. van Bendegem, (eds.), Frontiers of Paraconsistent Logic, Kluwer Academic Publishers, 2000.

  4. Beall, J.C., R. T. Brady, A.P. Hazen, G. Priest, and G. Restall, ‘Relevant restricted quantification’, Journal of Philosophical Logic, 35:587–598, 2006.

    Article  Google Scholar 

  5. Bero, Francesco, ‘Adunaton and material exclusion’, Australasian Journal of Philosophy, 86(2):165–190, 2008.

    Article  Google Scholar 

  6. Brady, Ross, ‘The consistency of the axioms of the axioms of abstraction and extensionality in a three valued logic’, Notre Dame Journal of Formal Logic, 12:447–453, 1971.

    Article  Google Scholar 

  7. Brady, Ross, Universal Logic, CSLI, 2006.

  8. Brady, Ross T., ‘The non-triviality of dialectical set theory’, in Priest et al. [38], pp. 437–470.

  9. Brady, Ross T., and Richard Routley, ‘The non-triviality of extensional dialectical set theory’, in Priest et al. [38], pp. 415–436.

  10. Cantor, Georg, ‘Letter to Dedekind’, in van Heijenoort [44].

  11. Chang, C.C., ‘The axiom of comprehension in infinite valued logic’, Math. Scand., 13:9–30, 1963.

    Google Scholar 

  12. Curry, Haskell B., ‘The inconsistency of certain formal logics’, Journal of Symbolic Logic, 7:115–117, 1942.

    Article  Google Scholar 

  13. da Costa, Newton, ‘Paraconsistent mathematics’, in Batens et al. [3], pp. 165–180.

  14. Dunn, J. Michael, ‘The impossibility of certain higher-order non-classical logics with extensionality’, in Austin [2], pp. 261–280.

  15. Dunn, J. Michael, ‘Relevant predication 1: The formal theory’, Journal of Philosophical Logic, 16:347–381, 1987.

    Article  Google Scholar 

  16. Dunn, J. Michael, and Greg Restall. ‘Relevance logic’, in Dov M. Gabbay and Franz Günthner, (eds.), Handbook of Philosophical Logic, 2nd Edition, volume 6, Kluwer, 2002, pp. 1–128.

  17. Forster, Thomas, ‘Axiomatising set theory with a universal set’, 1982. Typeset 1997.

  18. Gilmore, Paul C., ‘The consistency of partial set theory without extensionality’, in Jech [21], pp. 147–153.

  19. Gödel, Kurt, ‘What is Cantor’s continuum problem?’, in P. Benacerraf and H. Putnam, (eds.),Philosophy of Mathematics, Cambridge, 1964, pp. 258–273.

  20. Hallett, Michael, Cantorian Set Theory and Limitation of Size, Oxford Logic Guides, 1984.

  21. Jech, Thomas, (ed.), Axiomatic Set Theory, American Mathematical Society, 1974.

  22. Kremer, Philip, ‘Relevant identity, Journal of Philosophical Logic, 28:199–222, 1999.

    Article  Google Scholar 

  23. Libert, Thierry, ‘Models for paraconsistent set theory’, Journal of Applied Logic, 3, 2004.

  24. Mares, Edwin, ‘Semantics for relevant logic with identity’, Studia Logica, 51:1–20, 1992.

    Article  Google Scholar 

  25. Mares, Edwin, Relevant Logic, Cambridge, 2004.

  26. Meyer, Robert K., Richard Routley, and J. Michael Dunn, ‘Curry’s paradox’, Analysis, 39:124–128, 1978.

    Article  Google Scholar 

  27. Mortensen, Chris, Inconsistent Mathematics, Kluwer Academic Publishers, 1995.

  28. Petersen, Uwe, ‘Logic without contraction as based on inclusion and unrestriced abstraction’, Studia Logica, 64:365–403, 2000.

    Article  Google Scholar 

  29. Priest, Graham, ‘Sense, entailment and Modus Ponens’, Journal of Philosophical Logic, 9:415–435, 1980.

    Article  Google Scholar 

  30. Priest, Graham, In Contradiction: A Study of the Transconsistent, Martinus Nijhoff, The Hague, 1987.

  31. Priest, Graham, ‘Reductio ad absurdum et modus tollendo ponens’, in Priest et al. [38], pp. 613–626.

  32. Priest, Graham, ‘Is arithmetic consistent?’, Mind, 103, 1994,

  33. Priest, Graham, ‘Paraconsistent logic’, in Dov M. Gabbay and Franz Günthner, (eds.), Handbook of Philosophical Logic, 2nd Edition, volume 6, pp. 287–394, Kluwer, 2002.

  34. Priest, Graham, Doubt Truth Be A Liar, Oxford, 2006.

  35. Priest, Graham, In Contradiction: A Study of the Transconsistent, Oxford, 2006, Second expanded edition of [30].

  36. Priest, Graham, An Introduction to Non-Classical Logic, Cambridge, 2008. Second edition.

  37. Priest, Graham, and Richard Routley, ‘Systems of paraconsistent logic’, in Priest et al. [38], pp. 151–186.

  38. Priest, Graham, Richard Routley, and Jean Norman, (eds.), Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, 1989.

  39. Rogerson, Susan and Sam Butchart, ‘Naïve comprehension and contracting implications’, Studia Logica, 71(1):119–132, 2002.

    Article  Google Scholar 

  40. Routley, Richard, ‘Ultralogic as universal?’, Relevance Logic Newsletter, 2:51–89, 1977, Reprinted in [41].

  41. Routley, Richard, Exploring Meinong’s Jungle and Beyond, Philosophy Department, RSSS, Australian National University, 1980, Interim Edition, Departmental Monograph number 3.

  42. Routley, Richard, and Robert K. Meyer, ‘Dialectical logic, classical logic and the consistency of the world’, Studies in Soviet Thought, 16:1–25, 1976.

    Article  Google Scholar 

  43. Slaney, J. K., ‘RWX is not curry-paraconsistent’, in Priest et al. [38], pp. 472–480.

  44. van Heijenoort, Jean, (ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, Mass., 1967.

  45. Weber, Zach, ‘Transfinite Numbers in Paraconsistent Set Theory’, Review of Symbolic Logic, 3(1), 2010.

  46. Weyl, Herman, The Continuum, Dover, 1919.

  47. White, Richard, ‘The consistency of the axiom of comprehension in the infinite valued predicate logic of Łukasiewicz’, Journal of Philosophical Logic 8:503–534, 1979.

    Article  Google Scholar 

  48. Zalta, Ed, ‘Frege’s theorem’, Stanford Encyclopedia of Philosophy, 2007.

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  1. School of Philosophical and Historical Inquiry, University of Sydney, Sydney, Australia

    Zach Weber

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  1. Zach Weber
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Weber, Z. Extensionality and Restriction in Naive Set Theory. Stud Logica 94, 87–104 (2010). https://doi.org/10.1007/s11225-010-9225-y

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