Abstract
John Bell played a pivotal role in my intellectual life. I was first shown the beauty of mathematics at school, by a particularly gifted teacher. Inspired by this, I went up to Cambridge to read the subject. The process was disillusioning. With only few exceptions, the lectures were dull and boring. It was not uncommon for a lecturer to spend an hour writing on the blackboard with his or her back to the audience. Supervisors obviously derived pleasure from solving the problems that I could not solve, but this was with all the emotional engagement of cross-word puzzle solving. The sense of intellectual excitement that I had experienced at school evaporated.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Though I remember discussing some of the material that would become “The Logic of Paradox” (Priest, 1979), with John and my other examiner, Michael Dummett, at the PhD viva in Oxford. Neither, I think, saw much in it.
- 2.
What follows is essentially Chapter 18 of the second edition of (Priest, 1987). I am grateful for the permission of Oxford University Press to republish the material.
- 3.
One might also want to add an appropriate version of the Axiom of Choice to these. There are, however, ways of obtaining the Axiom from unrestricted comprehension. One way is to use the machinery of Hilbert’s ε -calculus. (See, e.g., (Leisenring, 1969, pp. 105–107).) Another, much more radical, way is to take Abs in an absolutely unrestricted form which allows α to contain “x” free. This delivers the Axioms of Choice (see Routley, 1977, p. 924f of reprint) whilst, surprisingly enough, maintaining non-triviality (see Brady, 1989).
- 4.
See (Priest, 1987, 6.2).
- 5.
For a survey of paraconsistent logics, see (Priest, 2002).
- 6.
Adopting the material strategy in some form goes half way towards meeting (Goodship, 1996), who advocates taking the main conditional of both the Comprehension Principle and the T-schema to be material. Would treating the conditionals in the two schemas show the paradoxes of self-reference to be of different kinds? No. They still all fit the Inclosure Schema (Priest, 1995, part 3), and so have the same essential structure.
- 7.
The rule for \(b=b\) means that this can be introduced at any time.
- 8.
It might be thought that without detachment the axioms cannot be shown to be inconsistent. This is false, though. An instance of Abs is \(\forall x(x\in r\leftrightarrow \lnot x\in x)\). Whence we have \(r\in r\leftrightarrow \lnot r\in r\); and cashing out the conditional in terms of negation and disjunction gives \(r\in r\wedge \) \(\lnot r\in r\). More generally, whenever α is a classical consequence of Σ, there is a β such that \(\alpha \vee (\beta \wedge \lnot \beta )\) follows from Σ. (See Priest 1987, Ch. 6.) Hence, any classically inconsistent theory is inconsistent in this logic also.
- 9.
For details of all this, see (Restall, 1992). Note that he defines “\(\,x=y\,\)” as “\(\,\forall z(z\in x\leftrightarrow z\in y)\,\).”
- 10.
A semantics with respect to which it is sound can be found in (Priest, 1987, 2nd ed, 19.8).
- 11.
The rules for identity are an exception. The rules for this are:
$$\begin{array}{ccc} \begin{array}{c} \\ . \\ \downarrow \\ b=b,+0 \end{array} & & \begin{array}{c} b=c,+i \\ \alpha (x/b),\pm j \\ \downarrow \\ \alpha (x/c),\pm j \end{array} \end{array}$$ - 12.
Much of this is spelled out in (Routley, 1977, § 8).
- 13.
- 14.
This is not the only sort of problem. Various natural arguments require the use of principles that involve nested →s, such as Permutation, \(\{\alpha \rightarrow (\beta \rightarrow \gamma )\}\vdash \beta \rightarrow (\alpha \rightarrow \gamma )\). The logic just described does not contain this principle. Whether it can be added whilst maintaining non-triviality is not known. There is certainly triviality in the area. See (Slaney, 1989).
- 15.
For a more systematic discussion of the issue, see (Dunn, 1988).
- 16.
See, e.g., (Dunn and Restall, 2002, p. 10). Sometimes, depending on the context, t gets interpreted as the conjunction of all logical truths.
- 17.
For a general discussion of restricted quantification in relevant logic, see (Beall et al., 2006), which suggests the use of a different, but closely related, kind of enthymematic conditional.
- 18.
Note, in particular, that ⇀ does not contrapose. So from the fact that \(x=y\) we cannot infer that \(\overline{x}=\overline{y}\).
- 19.
- 20.
- 21.
The fact that \(\mathcal{M}^{\sim }\) is a model of Abs is a special case of a more general lemma, to be found in (Restall, 1992).
- 22.
In fact, \(V_{\vartheta _{2}}\,\)behaves just like the set of all non-well-founded sets, given Mirimanoff’s paradox. It is well-founded, but it is also a member of itself, so is not well-founded.
- 23.
Some of these can be obtained by other applications of the Collapsing Lemma. Different methods of constucting models of inconsistent set theory, some of which also model ZF, are discussed in (Libert, 2003).
- 24.
Criticising the strategy under discussion here, (Weir, 2005, p. 398), says: “It will not do to say … that the models which … [do not have the desired properties] are ‘pathological’ or ‘unintended.’ All the dialetheist’s ZFC models are unintended in the sense that they do not capture anything like the full structure of the naive universe of sets. This compares unfavourably with the unintended models of first-order number-theory: they at least contain the ‘real’ structure of numbers.” This is simply question-begging. The thesis is precisely that one of these models does capture the full structure of the universe of sets. (Or, if there are many equally good models, then each captures the structure of an equally good universe.) From the dialetheic perspective, it is precisely the cumulative hierarchy that is an incomplete fragment of the universe of sets. And the models in question do contain the cumulative hierarchy as a fragment (at least up to an inaccessible cardinal).
- 25.
In particular, the argument constructing the interpretation \(\mathcal{M} ^{\sim }\) above can be carried out in ZF, and so is perfectly acceptable.
- 26.
Rescher (1969, p. 229) documents this claim, though he does not endorse it.
- 27.
- 28.
Bibliography
Beall, J., Brady, R., Hazen, A., Priest, G., and Restall, G. (2006). Restricted quantification in relevant logics. Journal of Philosophical Logic, 35:587–598.
Brady, R. (1989). The non-triviality of dialectical set theory. In Priest, G., Routley, R., and Norman, J., editors, Paraconsistent Logic: Essays on the Inconsistent, chapter 14. Philosophia Verlag, Munich.
Carroll, L. (1895). What the tortoise said to Achilles. Mind, 4:278–280.
Dummett, M. (1977). Elements of Intuitonism. Oxford University Press, Oxford.
Dunn, J. (1988). The impossibility of certain second-order non-classical logics with extensionality. In Austin, D., editor, Philosophical Analysis. Kluwer Academic Publishers, Dordrecht.
Dunn, J. and Restall, G. (2002). Relevance logic. In Gabbay, D. and Guenther, F., editors, Handbook of Philosophical Logic, volume 6, pages 1–128. Kluwer Academic Publishers, Dordrecht, second edition.
Goodship, L. (1996). On dialethism. Australasian Journal of Philosophy, 74:153–161.
Leisenring, A. (1969). Mathematical Logic and Hilbert’s ε–Symbol. Gordon and Breach, New York, NY.
Libert, T. (2003). ZF and the axiom of choice in some paraconsistent set theories. Logic and Logical Philosophy, 11:91–114.
Priest, G. (1979). Logic of paradox. Journal of Philosophical Logic, 8:219–241.
Priest, G. (1987). In Contradiction. Martinus Nijhoff, Dordrecht. Second, extended edition, Oxford University Press, 2006.
Priest, G. (1991). Minimally inconsistent LP. Studia Logica, 50:321–331. Reprinted with minor changes as Ch. 16 of the second edition of Priest (2006).
Priest, G. (1995). Beyond the Limits of Thought. Cambridge University Press, Cambridge. Second, extended edition, Oxford University Press, 2002.
Priest, G. (2001). Introduction to Non-Classical Logic. Cambridge University Press, Cambridge. Second, extended edition, 2008.
Priest, G. (2002). Paraconsistent logic. In Gabbay, D. and Guenther, F., editors, Handbook of Philosophical Logic, volume 6, pages 287–293. Kluwer Academic Publishers, Dordrecht, second edition.
Rescher, N. (1969). Many-Valued Logic. McGraw-Hill, New York, NY.
Restall, G. (1992). A note on naive set theory in LP. Notre Dame Journal of Formal Logic, 33:422–433.
Routley, R. (1977). Ultralogic as universal. Relevant Logic Newsletter, pages 50–89 and 138–175. Reprinted as an appendix in Exploring Meinong’s Jungle and Beyond, Canberra: Research School of Social Sciences, 1980.
Slaney, J. (1989). RWX is not Curry paraconsistent. In Priest, G., Routley, R., and Norman, J., editors, Paraconsistent Logic: Essays on the Inconsistent, chapter 17. Philosophia Verlag, Munich.
Weber, Z. (2009). Paradox and Foundation. PhD thesis, University of Melbourne.
Weber, Z. (2010a). Extensionality and restriction in naive set theory. Studia Logica, 94:87–104.
Weber, Z. (2010b). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic, 3:71–92.
Weir, A. (2005). There are no true contradictions. In Priest, G., Beall, J., and Armour-Garb, B., editors, The Law of Non-Contradiction: New Philosophical Essays, chapter 22. Oxford University Press, Oxford.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Priest, G. (2011). Paraconsistent Set Theory. In: DeVidi, D., Hallett, M., Clarke, P. (eds) Logic, Mathematics, Philosophy, Vintage Enthusiasms. The Western Ontario Series in Philosophy of Science, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0214-1_8
Download citation
DOI: https://doi.org/10.1007/978-94-007-0214-1_8
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0213-4
Online ISBN: 978-94-007-0214-1
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)