Skip to main content
SpringerLink
Log in

Notes on Inconsistent Set Theory

  • Chapter
  • First Online:
Paraconsistency: Logic and Applications

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 26))

Abstract

The purpose of this paper is to highlight and discuss two ideas that play in to the axiomatic development of a paraconsistent naive set theory. We will focus on aspects of the theory that can be read right off the axioms, concerning intensional identity and unrestricted set existence. Both relate to inconsistency. To begin I lay out a relevant background logic, placing a strong emphasis on the restrictions such a logic must have in order to support an inconsistent set theory. The sections that follow proceed on the understanding that, while highly inconsistent, a good deal of control is being exerted on the theory through the weakened logic. The two features of a fully naive theory, identity and self-reference, dovetail throughout.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Following a distinction I first saw in Libert (2005), Axiom 17.1 is called abstraction, while the formulation in Theorem 17.3 below is called comprehension. There is a syntactic difference between abstraction and comprehension, and in weak paraconsistent logics the principles are not equally user-friendly, because the quantifier ∃is sometimes tricky to eliminate. Nevertheless, both formulations capture a core intuition and in informal discussion the names are used interchangeably, without intending to mark an important difference.

  2. 2.

    The first step in securing a set theoretic account of functions is defined ordered pairs and show them to behave according to the law \(\langle a,b\rangle =\langle c,d\rangle \dashv \vdash a = c,b = d.\) We will be assuming throughout that some approximation of standard mathematical functions is available.

  3. 3.

    Without full comprehension, one can prove that the set of all sets is Dedekind infinite by producing an injection into itself, say by a map x↦{x}, but, again, functions and cardinality arguments are mostly beyond our scope here.

  4. 4.

    There is a debate about the right definition of subset—see (Mares, 2004, p. 198), and Beall et al. (2006), for instance using a more restricted implication.

  5. 5.

    Compare this to Petersen’s characterization of the natural numbers, (Petersen, 2000, p. 386).

  6. 6.

    Conrad Asmus pointed this out.

References

  • Arruda, A.I., and D. Batens. 1982. Russell’s set and the universal set in paraconsistent set theory. Logique et Analyse 25: 121–133.

    Google Scholar 

  • Austin, D.F. (ed.). 1988. Philosophical analysis. Dordrecht: Holland

    Google Scholar 

  • Barwise, J., and L. Moss. 1996. Vicious circles. Stanford: csli Publications.

    Google Scholar 

  • Batens, D., C. Mortensen, G. Priest, and J.-P. van Bendegem. eds. 2000. Frontiers of paraconsistent logic. Philadelphia: Research Studies Press.

    Google Scholar 

  • Beall, J.C., R.T. Brady, A.P. Hazen, G. Priest, and G. Restall. 2006. Relevant restricted quantification. Journal of Philosophical Logic 35: 587–598.

    Article  Google Scholar 

  • Brady, R.T. 1989. The non-triviality of dialectical set theory. In Paraconsistent logic: Essays on the inconsistent, ed. G. Priest, R. Routley, and J. Norman, 437–470. Munich: Philosophia Verlag.

    Google Scholar 

  • Brady, R.T. 2006. Universal logic. Stanford: CSLI.

    Google Scholar 

  • Brady, R.T., and R. Routley. 1989. The non-triviality of extensional dialectical set theory. In Paraconsistent logic: Essays on the inconsistent, ed. G. Priest, R. Routley, and J. Norman, 415–436. Munich: Philosophia Verlag.

    Google Scholar 

  • Curry, H.B. 1942. The inconsistency of certain formal logics. Journal of Symbolic Logic 7: 115–117.

    Article  Google Scholar 

  • da Costa, N.C.A. 1974. On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic 15: 497–510.

    Article  Google Scholar 

  • da Costa, N.C.A. 2000. Paraconsistent mathematics. In Frontiers of paraconsistent logic, ed. D. Batens, C. Mortensen, G. Priest, and J.-P. van Bendegem, 165–180. Philadelphia: Research Studies Press.

    Google Scholar 

  • Dunn, J.M. 1988. The impossibility of certain higher-order non-classical logics with extensionality. In Philosophical analysis, ed. D.F. Austin, 261–280. Dordrecht: Holland

    Chapter  Google Scholar 

  • Lewis, C.I., and C.H. Langford. 1959. Symbolic logic. New York: Dover.

    Google Scholar 

  • Libert, T. 2005. Models for paraconsistent set theory. Journal of Applied Logic 3: 15–41.

    Article  Google Scholar 

  • Mares, E. 2004. Relevant logic. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Meyer, R.K., R. Routley, and J.M. Dunn. 1978. Curry’s paradox. Analysis 39: 124–128.

    Article  Google Scholar 

  • Petersen, U. 2000. Logic without contraction as based on inclusion and unrestricted abstraction. Studia Logica 64: 365–403.

    Article  Google Scholar 

  • Priest, G. 2000. Inconsistent models of arithmetic, ii: The general case. Journal of Symbolic Logic 65: 1519–29.

    Article  Google Scholar 

  • Priest, G., R. Routley, and J. Norman (eds.). 1989. Paraconsistent logic: Essays on the inconsistent. Munich: Philosophia Verlag.

    Google Scholar 

  • Restall, G. 1994. On logics without contraction. Ph.D. thesis. The University of Queensland.

    Google Scholar 

  • Routley, R. 1977. Ultralogic as universal? Relevance Logic Newsletter2: 51–89. Reprinted in Routley (1980).

    Google Scholar 

  • Routley, R. 1980. Exploring Meinong’s jungle and beyond. Departmental Monograph, Philosophy Department, RSSS, Australian National University, vol. 3. Canberra: RSSS, Australian National University, Canberra.

    Google Scholar 

  • Routley, R., and R.K. Meyer. 1976. Dialectical logic, classical logic and the consistency of the world. Studies in Soviet Thought 16: 1–25.

    Article  Google Scholar 

  • Slaney, J.K. 1989. Rwx is not curry-paraconsistent. In Paraconsistent logic: Essays on the inconsistent, ed. G. Priest, R. Routley, and J. Norman, 472–480. Munich: Philosophia Verlag.

    Google Scholar 

  • Weber, Z. 2010a. Extensionality and restriction in naive set theory. Studia Logica 94(1): 87–104.

    Article  Google Scholar 

  • Weber, Z. 2010b. Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic 3(1): 71–92.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Otago, 56, Dunedin, 9054, New Zealand

    Zach Weber

Authors
  1. Zach Weber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zach Weber .

Editor information

Editors and Affiliations

  1. , Department of Philosophy, University of Auckland, Arts 2, 18 Symonds Street, Auckland, 1142, New Zealand

    Koji Tanaka

  2. , Department of Philosophy, University of Aberdeen, King's College, High Street, Aberdeen, AB24 3UB, United Kingdom

    Francesco Berto

  3. , Department of Philosophy, Victoria University of Wellington, Murphy 518, Kelburn Parade, Wellington, New Zealand

    Edwin Mares

  4. , Department of Education, University of Cagliari, Via Is Mirrionis 1, Cagliari, 09123, Italy

    Francesco Paoli

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht.

About this chapter

Cite this chapter

Weber, Z. (2013). Notes on Inconsistent Set Theory. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_17

Download citation

Publish with us

Policies and ethics

Search

Navigation