•  240
    Stanford 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 truth-bearer: 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
  •  104
    A Paraconsistent Model of Vagueness
    Mind 119 (476): 1025-1045. 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 cut-off 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
  •  644
    A Topological Sorites
    Journal of Philosophy 107 (6): 311-325. 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 cut-off points of vague predicates. The generalised sorites paradox presented here also gives rise to a new, more tractable definition of vagueness.
  •  55
    Wittgenstein's Notes on Logic. By Michael Potter (review)
    Metaphilosophy 42 (1-2): 166-170. 2011.
  •  62
    Reply to Bjørdal
    Review of Symbolic Logic 4 (1): 109-113. 2011.
  •  17
    Notes on inconsistent set theory
    In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications, Springer. pp. 315--328. 2013.
  •  5
    Front Matter
    Australasian Journal of Logic 14 (1). 2017.
    Editors' Introduction and List of Contributors
  •  66
    Bad Worlds
    Thought: A Journal of Philosophy 4 (2): 93-101. 2015.
    The idea of relevant logic—that irrelevant inferences are invalid—is appealing. But the standard semantics for relevant logics involve baroque metaphysics: a three-place accessibility relation, a star operator, and ‘bad’ worlds. In this article we propose that these oddities express a mismatch between non-classical object theory and classical metatheory. A uniformly relevant semantics for relevant logic is a better fit
  •  136
    Tolerating Gluts
    with David Ripley, Graham Priest, Dominic Hyde, and Mark Colyvan
    Mind 123 (491): 813-828. 2014.
  •  12
    Paraconsistent Measurement of the Circle
    Australasian 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
  •  14
    Intrinsic Value and the Last Last Man
    Ratio 30 (2): 165-180. 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
  •  84
    Extensionality and Restriction in Naive Set Theory
    Studia Logica 94 (1): 87-104. 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 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,…Read more
  •  137
    Can u do that?
    with J. Beall and G. Priest
    Analysis 71 (2): 280-285. 2011.
    In his ‘On t and u and what they can do’, Greg Restall presents an apparent problem for a handful of well-known non-classical solutions to paradoxes like the liar. In this article, we argue that there is a problem only if classical logic – or classical-enough logic – is presupposed. 1. Background Many have thought that invoking non-classical 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
  •  6
    Reply to bj??Rdal: Reply to bj??Rdal
    Review of Symbolic Logic 4 (1): 109-113. 2011.
  •  57
    Naive Validity
    Philosophical Quarterly 64 (254): 99-114. 2014.
  •  77
    Inconsistent boundaries
    Synthese 192 (5): 1267-1294. 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
  •  72
    Real Analysis in Paraconsistent Logic
    Journal of Philosophical Logic 41 (5): 901-922. 2012.
    This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant 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 non-trivializing contradictions are left open
  •  116
    Transfinite numbers in paraconsistent set theory
    Review of Symbolic Logic 3 (1): 71-92. 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 well-ordering principle). At the end I indicate how later developments of cardinal …Read more
  •  39
    Explanation And Solution In The Inclosure Argument
    Australasian Journal of Philosophy 88 (2): 353-357. 2010.
    In a recent article, Emil Badici contends that the inclosure schema substantially fails as an analysis of the paradoxes of self-reference because it is question-begging. 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
  •  49
    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 non-trivial
  • Spandrels of truth (review)
    Bulletin of Symbolic Logic 16 (2): 284-285. 2010.
  •  41
    Closure is the idea that what is true about a theory of truth should be true in it. Commitment to closure under truth motivates non-classical 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
  •  16
    Issue Introduction
    Essays in Philosophy 12 (2): 1. 2011.
  •  42
    Computation in Non-Classical Foundations?
    Philosophers' Imprint 16. 2016.
    The Church-Turing 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
  •  66
    What Is an Inconsistent Truth Table?
    with Guillermo Badia and Patrick Girard
    Australasian Journal of Philosophy 94 (3): 533-548. 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 consistency-independent 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