
643A Topological SoritesJournal of Philosophy 107 (6): 311325. 2010.This paper considers a generalisation of the sorites paradox, in which only topological notions are employed. We argue that by increasing the level of abstraction in this way, we see the sorites paradox in a new, more revealing light—a light that forces attention on cutoff points of vague predicates. The generalised sorites paradox presented here also gives rise to a new, more tractable definition of vagueness.

239DialetheismStanford Encyclopedia of Philosophy 2018 (2018). 2008.A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truthbearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false

137Can u do that?Analysis 71 (2): 280285. 2011.In his ‘On t and u and what they can do’, Greg Restall presents an apparent problem for a handful of wellknown nonclassical solutions to paradoxes like the liar. In this article, we argue that there is a problem only if classical logic – or classicalenough logic – is presupposed. 1. Background Many have thought that invoking nonclassical logic – in particular, a paracomplete or paraconsistent logic – is the correct response to the liar and related paradoxes. At the most basic level, the targ…Read more

116Transfinite numbers in paraconsistent set theoryReview of Symbolic Logic 3 (1): 7192. 2010.This paper begins an axiomatic development of naive set theoryin a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the wellordering principle). At the end I indicate how later developments of cardinal …Read more

104A Paraconsistent Model of VaguenessMind 119 (476): 10251045. 2010.Vague predicates, on a paraconsistent account, admit overdetermined borderline cases. I take up a new line on the paraconsistent approach, to show that there is a close structural relationship between the breakdown of soritical progressions, and contradiction. Accordingly, a formal picture drawn from an appropriate logic shows that any cutoff point of a vague predicate is unidentifiable, in a precise sense. A paraconsistent approach predicts and explains many of the most counterintuitive aspect…Read more

89Transfinite cardinals in paraconsistent set theoryReview of Symbolic Logic 5 (2): 269293. 2012.This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem

84Extensionality and Restriction in Naive Set TheoryStudia Logica 94 (1): 87104. 2010.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 subproblem 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,…Read more

77Inconsistent boundariesSynthese 192 (5): 12671294. 2015.Mereotopology is a theory of connected parts. The existence of boundaries, as parts of everyday objects, is basic to any such theory; but in classical mereotopology, there is a problem: if boundaries exist, then either distinct entities cannot be in contact, or else space is not topologically connected . In this paper we urge that this problem can be met with a paraconsistent mereotopology, and sketch the details of one such approach. The resulting theory focuses attention on the role of empty p…Read more

72Real Analysis in Paraconsistent LogicJournal of Philosophical Logic 41 (5): 901922. 2012.This paper begins an analysis of the real line using an inconsistencytolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistencyreliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other nontrivializing contradictions are left open

66Bad WorldsThought: A Journal of Philosophy 4 (2): 93101. 2015.The idea of relevant logic—that irrelevant inferences are invalid—is appealing. But the standard semantics for relevant logics involve baroque metaphysics: a threeplace accessibility relation, a star operator, and ‘bad’ worlds. In this article we propose that these oddities express a mismatch between nonclassical object theory and classical metatheory. A uniformly relevant semantics for relevant logic is a better fit

66What Is an Inconsistent Truth Table?Australasian Journal of Philosophy 94 (3): 533548. 2016.ABSTRACTDo truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistencyindependent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears …Read more

55Wittgenstein's Notes on Logic. By Michael Potter (review)Metaphilosophy 42 (12): 166170. 2011.

51Intrinsic Value and the Last Last ManRatio 29 (4). 2016.Even if you were the last person on Earth, you should not cut down all the trees—or so goes the Last Man thought experiment, which has been taken to show that nature has intrinsic value. But ‘Last Man’ is caught on a dilemma. If Last Man is too far inside the anthropocentric circle, so to speak, his actions cannot be indicative of intrinsic value. If Last Man is cast too far outside the anthropocentric circle, though, then value terms lose their cogency. The experiment must satisfy conditions in…Read more

49A Note on ContractionFree Logic for ValidityTopoi 34 (1): 6374. 2015.This note motivates a logic for a theory that can express its own notion of logical consequence—a ‘syntactically closed’ theory of naive validity. The main issue for such a logic is Curry’s paradox, which is averted by the failure of contraction. The logic features two related, but different, implication connectives. A Hilbert system is proposed that is complete and nontrivial

42Computation in NonClassical Foundations?Philosophers' Imprint 16. 2016.The ChurchTuring Thesis is widely regarded as true, because of evidence that there is only one genuine notion of computation. By contrast, there are nowadays many different formal logics, and different corresponding foundational frameworks. Which ones can deliver a theory of computability? This question sets up a difficult challenge: the meanings of basic mathematical terms are not stable across frameworks. While it is easy to compare what different frameworks say, it is not so easy to compare …Read more

41On closure and truth in substructural theories of truthSynthese 115. forthcoming.Closure is the idea that what is true about a theory of truth should be true in it. Commitment to closure under truth motivates nonclassical logic; commitment to closure under validity leads to substructural logic. These moves can be thought of as responses to revenge problems. With a focus on truth in mathematics, I will consider whether a noncontractive approach faces a similar revenge problem with respect to closure under provability, and argue that if a noncontractive theory is to be genuin…Read more

39Explanation And Solution In The Inclosure ArgumentAustralasian Journal of Philosophy 88 (2): 353357. 2010.In a recent article, Emil Badici contends that the inclosure schema substantially fails as an analysis of the paradoxes of selfreference because it is questionbegging. The main purpose of this note is to show that Badici's critique highlights a necessity condition for the success of dialectic about paradoxes. The inclosure argument respects this condition and remains solvent

26Figures, Formulae, and FunctorsIn SunJoo Shin & Amirouche Moktefi (eds.), Visual Reasoning with Diagrams, Springer. pp. 153170. 2013.This article suggests a novel way to advance a current debate in the philosophy of mathematics. The debate concerns the role of diagrams and visual reasoning in proofs—which I take to concern the criteria of legitimate representation of mathematical thought. Drawing on the socalled ‘maverick’ approach to philosophy of mathematics, I turn to mathematical practice itself to adjudicate in this debate, and in particular to category theory, because there (a) diagrams obviously play a major role, and…Read more

26On paraconsistent ethicsSouth African Journal of Philosophy 26 (2): 239244. 2007.No. South African Journal of Philosophy Vol.26 (2) 2007:239244

16Notes on inconsistent set theoryIn Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications, Springer. pp. 315328. 2013.

16Review of Peter Schotch, Bryson brown, Raymond Jennings (eds.), On Preserving: Essays on Preservationism and Paraconsistent Logic (review)Notre Dame Philosophical Reviews 2009 (9). 2009.

14Intrinsic Value and the Last Last ManRatio 30 (2): 165180. 2017.Even if you were the last person on Earth, you should not cut down all the trees—or so goes the Last Man thought experiment, which has been taken to show that nature has intrinsic value. But ‘Last Man’ is caught on a dilemma. If Last Man is too far inside the anthropocentric circle, so to speak, his actions cannot be indicative of intrinsic value. If Last Man is cast too far outside the anthropocentric circle, though, then value terms lose their cogency. The experiment must satisfy conditions in…Read more

12Paraconsistent Measurement of the CircleAustralasian Journal of Logic 14 (1). 2017.A theorem from Archimedes on the area of a circle is proved in a setting where some inconsistency is permissible, by using paraconsistent reasoning. The new proof emphasizes that the famous method of exhaustion gives approximations of areas closer than any consistent quantity. This is equivalent to the classical theorem in a classical context, but not in a context where it is possible that there are inconsistent innitesimals. The area of the circle is taken 'up to inconsistency'. The fact that t…Read more

9Piotr Łukowski , Paradoxes , tr. Marek Gensler. Reviewed byPhilosophy in Review 32 (4): 307309. 2012.
Dunedin, Otago, New Zealand
Areas of Specialization
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Areas of Interest
Logic and Philosophy of Logic 
Philosophy of Mathematics 