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Logical Pluralism
 To appear, Special Logic issue of the Australasian Journal of Philosophy
, 2000
"... Abstract: A widespread assumption in contemporary philosophy of logic is that there is one true logic, that there is one and only one correct answer as to whether a given argument is deductively valid. In this paper we propose an alternative view, logical pluralism. According to logical pluralism th ..."
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Abstract: A widespread assumption in contemporary philosophy of logic is that there is one true logic, that there is one and only one correct answer as to whether a given argument is deductively valid. In this paper we propose an alternative view, logical pluralism. According to logical pluralism there is not one true logic; there are many. There is not always a single answer to the question “is this argument valid?” 1 Logic, Logics and Consequence Anyone acquainted with contemporary Logic knows that there are many socalled logics. 1 But are these logics rightly socalled? Are any of the menagerie of nonclassical logics, such as relevant logics, intuitionistic logic, paraconsistent logics or quantum logics, as deserving of the title ‘logic ’ as classical logic? On the other hand, is classical logic really as deserving of the title ‘logic ’ as relevant logic (or any of the other nonclassical logics)? If so, why so? If not, why not? Logic has a chief subject matter: Logical Consequence. The chief aim of
A BURGESSIAN CRITIQUE OF NOMINALISTIC TENDENCIES IN CONTEMPORARY MATHEMATICS AND ITS HISTORIOGRAPHY
, 2012
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MEANING IN CLASSICAL MATHEMATICS: IS IT AT ODDS WITH INTUITIONISM?
, 2011
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Constructive Mathematics, in Theory and Programming Practice
, 1997
"... The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). It gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part ..."
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Cited by 7 (3 self)
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The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). It gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on MartinLof's theory of types as a formal system for BISH.
The Constructive Implicit Function Theorem and Applications in Mechanics
 Chaos, Solitons and Fractals
, 1997
"... We examine some ways of proving the Implicit Function Theorem and the Inverse Function Theorem within Bishop's constructive mathematics. Section 2 contains a new, entirely constructive proof of the Implicit Function Theorem. The paper ends with some comments on the application of the Implicit F ..."
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We examine some ways of proving the Implicit Function Theorem and the Inverse Function Theorem within Bishop's constructive mathematics. Section 2 contains a new, entirely constructive proof of the Implicit Function Theorem. The paper ends with some comments on the application of the Implicit Function Theorem in classical mechanics. 1 Introduction In this paper, which is written entirely within the framework of constructive mathematics #BISH# erected by the late Errett Bishop #2#, we examine a standard proof of the Implicit Function Theorem and give a completely new proof. As far as understanding constructive mathematics goes, the reader need only be aware that when working constructively,weinterpret #existence" strictly as #computability". To do so, we need to be careful about our logic. For example, when we prove a disjunction P Q; we need to either produce a proof of P or produce a proof of Q; it is not enough, constructively, to show that : #:P :Q#:To understand this better, con...
Constructive Aspects of the Dirichlet Problem 1
"... Abstract: We examine, within the framework of Bishop's constructive mathematics, various classical methods for proving the existence of weak solutions of the Dirichlet Problem, with a view to showing why those methods do not immediately translate into viable constructive ones. In particular, we ..."
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Abstract: We examine, within the framework of Bishop's constructive mathematics, various classical methods for proving the existence of weak solutions of the Dirichlet Problem, with a view to showing why those methods do not immediately translate into viable constructive ones. In particular, we discuss the equivalence of the existence of weak solutions of the Dirichlet Problem and the existence of minimizers for certain associated integral functionals. Our analysis pinpoints exactly what is needed to nd weak solutions of the Dirichlet Problem: namely, the computation of either the norm of a linear functional on a certain Hilbert space or, equivalently, the in mum of an associated integral functional.