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  1. Contradictions and falling bridges: what was Wittgenstein’s reply to Turing?Ásgeir Berg Matthíasson - 2020 - British Journal for the History of Philosophy 29 (3).
    In this paper, I offer a close reading of Wittgenstein's remarks on inconsistency, mostly as they appear in the Lectures on the Foundations of Mathematics. I focus especially on an objection to Wittgenstein's view given by Alan Turing, who attended the lectures, the so-called ‘falling bridges’-objection. Wittgenstein's position is that if contradictions arise in some practice of language, they are not necessarily fatal to that practice nor necessitate a revision of that practice. If we then assume that we have adopted (...)
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  • Ultralogic as Universal?: The Sylvan Jungle - Volume 4.Richard Routley - 2019 - Cham, Switzerland: Springer Verlag.
    Ultralogic as Universal? is a seminal text in non-classcial logic. Richard Routley presents a hugely ambitious program: to use an 'ultramodal' logic as a universal key, which opens, if rightly operated, all locks. It provides a canon for reasoning in every situation, including illogical, inconsistent and paradoxical ones, realized or not, possible or not. A universal logic, Routley argues, enables us to go where no other logic—especially not classical logic—can. Routley provides an expansive and singular vision of how a universal (...)
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  • Resisting Non-causal Explanations.Otávio Bueno & Melisa Vivanco - 2019 - Analysis 79 (3):550-559.
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  • Inconsistent Models for Arithmetics with Constructible Falsity.Thomas Macaulay Ferguson - forthcoming - Logic and Logical Philosophy:1.
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  • Idealist Origins: 1920s and Before.Martin Davies & Stein Helgeby - 2014 - In Graham Oppy & Nick Trakakis (eds.), History of Philosophy in Australia and New Zealand. Dordrecht: Springer. pp. 15-54.
    This paper explores early Australasian philosophy in some detail. Two approaches have dominated Western philosophy in Australia: idealism and materialism. Idealism was prevalent between the 1880s and the 1930s, but dissipated thereafter. Idealism in Australia often reflected Kantian themes, but it also reflected the revival of interest in Hegel through the work of ‘absolute idealists’ such as T. H. Green, F. H. Bradley, and Henry Jones. A number of the early New Zealand philosophers were also educated in the idealist tradition (...)
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  • Axioms for finite collapse models of arithmetic.Andrew Tedder - 2015 - Review of Symbolic Logic 8 (3):529-539.
    The collapse models of arithmetic are inconsistent, nontrivial models obtained from ℕ and set out in the Logic of Paradox (LP). They are given a general treatment by Priest (Priest, 2000). Finite collapse models are decidable, and thus axiomatizable, because finite. LP, however, is ill-suited to normal axiomatic reasoning, as it invalidates Modus Ponens, and almost all other usual conditional inferences. I set out a logic, A3, first given by Avron (Avron, 1991), and give a first order axiom system for (...)
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  • Notes on the Model Theory of DeMorgan Logics.Thomas Macaulay Ferguson - 2012 - Notre Dame Journal of Formal Logic 53 (1):113-132.
    We here make preliminary investigations into the model theory of DeMorgan logics. We demonstrate that Łoś's Theorem holds with respect to these logics and make some remarks about standard model-theoretic properties in such contexts. More concretely, as a case study we examine the fate of Cantor's Theorem that the classical theory of dense linear orderings without endpoints is $\aleph_{0}$-categorical, and we show that the taking of ultraproducts commutes with respect to previously established methods of constructing nonclassical structures, namely, Priest's Collapsing (...)
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  • Paraconsistency: Logic and Applications.Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.) - 2012 - Dordrecht, Netherland: Springer.
    A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...)
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  • On a First-Order Bi-Sorted Semantically Closed Language.Fernanda Birolli Abrahão & Edelcio Gonçalves de Souza - forthcoming - Studia Logica:1-13.
    This paper is about the concept of semantically closed languages. Roughly speaking, those are languages which can name their own sentences and apply to them semantic predicates, such as the truth or satisfaction predicates. Hence, they are “self-referential languages,” in the sense that they are capable of producing sentences about themselves or other sentences in the same language. In section one, we introduce the concept informally; in section two, we provide the formal definition of first-order semantically closed languages, which is (...)
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  • Was Wittgenstein a radical conventionalist?Ásgeir Berg - 2024 - Synthese 203 (2):1-31.
    This paper defends a reading of Wittgenstein’s philosophy of mathematics in the Lectures on the Foundation of Mathematics as a radical conventionalist one, whereby our agreement about the particular case is constitutive of our mathematical practice and ‘the logical necessity of any statement is a direct expression of a convention’ (Dummett 1959, p. 329). -/- On this view, mathematical truths are conceptual truths and our practices determine directly for each mathematical proposition individually whether it is true or false. Mathematical truths (...)
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  • (1 other version)Mathematical Pluralism and Indispensability.Silvia Jonas - 2023 - Erkenntnis 1:1-25.
    Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...)
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  • Is there an inconsistent primitive recursive relation?Seungrak Choi - 2022 - Synthese 200 (5):1-12.
    The present paper focuses on Graham Priest’s claim that even primitive recursive relations may be inconsistent. Although he carefully presented his claim using the expression “may be,” Priest made a definite claim that even numerical equations can be inconsistent. His argument relies heavily on the fact that there is an inconsistent model for arithmetic. After summarizing Priest’s argument for the inconsistent primitive recursive relation, I first discuss the fact that his argument has a weak foundation to explain that the existence (...)
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  • How Do You Apply Mathematics?Graham Priest - 2022 - Axiomathes 32 (3):1169-1184.
    As far as disputes in the philosophy of pure mathematics goes, these are usually between classical mathematics, intuitionist mathematics, paraconsistent mathematics, and so on. My own view is that of a mathematical pluralist: all these different kinds of mathematics are equally legitimate. Applied mathematics is a different matter. In this, a piece of pure mathematics is applied in an empirical area, such as physics, biology, or economics. There can then certainly be a disputes about what the correct pure mathematics to (...)
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  • Can Gödel's Incompleteness Theorem be a Ground for Dialetheism?Seungrak Choi - 2017 - Korean Journal of Logic 20 (2):241-271.
    Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest’s argument for Dialetheism from Gödel’s theorem is unconvincing as the lesson of Gödel’s proof (or Rosser’s proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest’s inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying Gödel sentence to the inconsistent and complete theory of arithmetic. We argue, (...)
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  • Feng Ye. Strict Finitism and the Logic of Mathematical Applications.Nigel Vinckier & Jean Paul Van Bendegem - 2016 - Philosophia Mathematica 24 (2):247-256.
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  • The Scope of Gödel’s First Incompleteness Theorem.Bernd Buldt - 2014 - Logica Universalis 8 (3-4):499-552.
    Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem.
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  • On LP -models of arithmetic.J. B. Paris & A. Sirokofskich - 2008 - Journal of Symbolic Logic 73 (1):212-226.
    We answer some problems set by Priest in [11] and [12], in particular refuting Priest's Conjecture that all LP-models of Th(N) essentially arise via congruence relations on classical models of Th(N). We also show that the analogue of Priest's Conjecture for I δ₀ + Exp implies the existence of truth definitions for intervals [0,a] ⊂ₑ M ⊨ I δ₀ + Exp in any cut [0,a] ⊂e K ⊆ M closed under successor and multiplication.
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  • What Would a Phenomenology of Logic Look Like?James Kinkaid - 2020 - Mind 129 (516):1009-1031.
    The phenomenological movement begins in the Prolegomena to Husserl’s Logical Investigations as a philosophy of logic. Despite this, remarkably little attention has been paid to Husserl’s arguments in the Prolegomena in the contemporary philosophy of logic. In particular, the literature spawned by Gilbert Harman’s work on the normative status of logic is almost silent on Husserl’s contribution to this topic. I begin by raising a worry for Husserl’s conception of ‘pure logic’ similar to Harman’s challenge to explain the connection between (...)
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  • Inconsistency, Paraconsistency and ω-Inconsistency.Bruno Da Ré - 2018 - Principia: An International Journal of Epistemology 22 (1):171-188.
    In this paper I’ll explore the relation between ω-inconsistency and plain inconsistency, in the context of theories that intend to capture semantic concepts. In particular, I’ll focus on two very well known inconsistent but non-trivial theories of truth: LP and STTT. Both have the interesting feature of being able to handle semantic and arithmetic concepts, maintaining the standard model. However, it can be easily shown that both theories are ω- inconsistent. Although usually a theory of truth is generally expected to (...)
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  • On elimination of quantifiers in some non‐classical mathematical theories.Guillermo Badia & Andrew Tedder - 2018 - Mathematical Logic Quarterly 64 (3):140-154.
    Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.
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  • Peeking at the Impossible.Chris Mortensen - 1997 - Notre Dame Journal of Formal Logic 38 (4):527-534.
    The question of the interpretation of impossible pictures is taken up. Penrose's account is reviewed. It is argued that whereas this account makes substantial inroads into the problem, there needs to be a further ingredient. An inconsistent account using heap models is proposed.
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  • Is strict finitism arbitrary?Nuno Maia - forthcoming - Philosophical Quarterly.
    Strict finitism posits a largest natural number. The view is usually thought to be objectionably arbitrary. After all, there seems to be no apparent reason as to why the natural numbers should ‘stop’ at a specific point and not a bit later on the natural line. Drawing on how arguments from arbitrariness are employed in mereology, I propose several ways of understanding this objection against strict finitism. No matter how it is understood, I argue that it is always found wanting.
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  • (1 other version)Inconsistent models of artihmetic Part II : The general case.Graham Priest - 2000 - Journal of Symbolic Logic 65 (4):1519-1529.
    The paper establishes the general structure of the inconsistent models of arithmetic of [7]. It is shown that such models are constituted by a sequence of nuclei. The nuclei fall into three segments: the first contains improper nuclei: the second contains proper nuclei with linear chromosomes: the third contains proper nuclei with cyclical chromosomes. The nuclei have periods which are inherited up the ordering. It is also shown that the improper nuclei can have the order type of any ordinal. of (...)
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  • A Note on Priest's Finite Inconsistent Arithmetics.J. B. Paris & N. Pathmanathan - 2006 - Journal of Philosophical Logic 35 (5):529-537.
    We give a complete characterization of Priest's Finite Inconsistent Arithmetics observing that his original putative characterization included arithmetics which cannot in fact be realized.
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  • Dunn–Priest Quotients of Many-Valued Structures.Thomas Macaulay Ferguson - 2017 - Notre Dame Journal of Formal Logic 58 (2):221-239.
    J. Michael Dunn’s Theorem in 3-Valued Model Theory and Graham Priest’s Collapsing Lemma provide the means of constructing first-order, three-valued structures from classical models while preserving some control over the theories of the ensuing models. The present article introduces a general construction that we call a Dunn–Priest quotient, providing a more general means of constructing models for arbitrary many-valued, first-order logical systems from models of any second system. This technique not only counts Dunn’s and Priest’s techniques as special cases, but (...)
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  • The ontological commitments of inconsistent theories.Mark Colyvan - 2008 - Philosophical Studies 141 (1):115 - 123.
    In this paper I present an argument for belief in inconsistent objects. The argument relies on a particular, plausible version of scientific realism, and the fact that often our best scientific theories are inconsistent. It is not clear what to make of this argument. Is it a reductio of the version of scientific realism under consideration? If it is, what are the alternatives? Should we just accept the conclusion? I will argue (rather tentatively and suitably qualified) for a positive answer (...)
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  • Second-Order Logic of Paradox.Allen P. Hazen & Francis Jeffry Pelletier - 2018 - Notre Dame Journal of Formal Logic 59 (4):547-558.
    The logic of paradox, LP, is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order LP is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding that it will be extremely (...)
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  • Inconsistency in mathematics and the mathematics of inconsistency.Jean Paul van Bendegem - 2014 - Synthese 191 (13):3063-3078.
    No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what (...)
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  • Wittgenstein on Incompleteness Makes Paraconsistent Sense.Francesco Berto - 2012 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Dordrecht, Netherland: Springer. pp. 257--276.
    I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...)
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