•  1395
    Uniformly convex Banach spaces are reflexive—constructively
    with Douglas S. Bridges and Hajime Ishihara
    Mathematical Logic Quarterly 59 (4-5): 352-356. 2013.
    We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.
  •  96
    Real Analysis in Paraconsistent Logic
    with Zach Weber
    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
  •  25
    Paraconsistent Measurement of the Circle
    with Zach Weber
    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
  •  20
    The Difficulties in Using Weak Relevant Logics for Naive Set Theory
    with Erik Istre
    In Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency, Springer Verlag. pp. 365-381. 2019.
    We discuss logical difficulties with the naive set theory based on the weak relevant logic DKQ. These are induced by the restrictive nature of the relevant conditional and its interaction with set theory. The paper concludes with some possible ways to mitigate these difficulties.
  •  16
    Classifying material implications over minimal logic
    with Hannes Diener
    Archive for Mathematical Logic 59 (7-8): 905-924. 2020.
    The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. I…Read more
  •  11
  •  9
    Front Matter
    Australasian Journal of Logic 14 (1). 2017.
    Editors' Introduction and List of Contributors