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Profile: Greg Restall (University of Melbourne)
  1. Greg Restall, Models for Liars in Bradwardine's Theory of Truth.
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  2. Greg Restall, Models for Substructural Arithmetics.
    This paper explores models for arithmetics in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (linear logic plus distribution), R (relevant logic) and CK (C plus weakening). The eventual goal is to find negation complete models for arithmetic in R.
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  3. Greg Restall, Proof Theory and Meaning: On Second Order Logic.
  4. Greg Restall, Proof Theory and Meaning: The Context of Deducibility.
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  5. Greg Restall, Rebecca Kukla & Mark Lance, Appendix to Rebecca Kukla and Mark Lance 'Yo!' And 'Lo!': The Pragmatic Topography of the Space of Reasons.
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  6. JC Beall & Greg Restall, Logical Consequence.
    A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises. But in what sense do conclusions follow from premises? What is it for a conclusion to be a consequence of premises? Those questions, in many respects, are at the heart of logic (as a philosophical discipline). Consider the following argument: 1. If we charge high fees for university, only the rich will enroll. We charge high fees for university. Therefore, only the rich (...)
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  7. Greg Restall,      : Warren Goldfarb's Deductive Logic.
    Warren Goldfarb, Deductive Logic, Hackett Publishing Company, 2003.    : 0872206602. Deductive Logic is an introductory textbook in formal logic. The book is divided into four parts covering (i) truth-functional logic, (ii) monadic quantifi- cation, (iii) polyadic quantification and (iv) names and identity, and there are exercises for all these topics at the end of the book. In the truth-functional logic part, the reader learns to produce paraphrases of English statements and arguments in logical notation (this subsection (...)
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  8. Greg Restall, ,           .
          : In this paper I consider an interpretation of future contingents which motivates a unification of a Łukasiewicz-style logic with the more classical supervaluational semantics. This in turn motivates a new non-classical logic modelling what is “made true by history up until now.” I give a simple Hilbert-style proof theory, and a soundness and completeness argument for the proof theory with respect to the intended models.
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  9. Greg Restall, August 10, 1995.
    In this note I respond to Hartley Slater's argument 12 to the e ect that there is no such thing as paraconsistent logic. Slater's argument trades on the notion of contradictoriness in the attempt to show that the negation of paraconsistent logics is merely a subcontrary forming operator and not one which forms contradictories. I will show that Slater's argument fails, for two distinct reasons. Firstly, the argument does not consider the position of non-dialethic paraconsistency which rejects the possible truth (...)
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  10. Greg Restall, A Cut-Free Sequent System For.
    priori knowability. I show how a cut-free hypersequent calculus for 2d modal logic not only captures the logic precisely, but may be used to address issues in the episte-.
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  11. Greg Restall, Assertion, Denial.
    In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the (...)
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  12. Greg Restall, Assertion Denial, Acception, Rejecting, S y M M E T Ry, Paradox,.
    Proponents of “truth-value glut” responses to the paradoxes of self-reference, such as Priest [6, 7] argue that “truth-value gap” analyses of the paradoxes fall foul of the strengthened liar paradox: “this sentence is not true.” If we pay attention to the role of assertion and denial and the behaviour of negation in both “gap” and “glut” analyses, we see that the situation with these approaches has a pleasing symmetry: gap approaches take some denials to fail to be expressible by negation, (...)
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  13. Greg Restall, Barriers to Implication.
    Implication barrier theses deny that one can derive sentences of one type from sentences of another. Hume’s Law is an implication barrier thesis; it denies that one can derive an ‘ought’ (a normative sentence) from an ‘is’ (a descriptive sentence). Though Hume’s Law is controversial, some barrier theses are philosophical platitudes; in his Lectures on Logical Atomism, Bertrand Russell claims: You can never arrive at a general proposition by inference particular propositions alone. You will always have to have at least (...)
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  14. Greg Restall, Molinism and the Thin Red Line.
    Molinism is an attempt to do equal justice to divine foreknowledge and human freedom. For Molinists, human freedom fits in this universe for the future is open or unsettled. However, God’s middle knowledge — God’s contingent knowledge of what agents would freely do in this or that circumstance — underwrites God’s omniscience in the midst of this openness. In this paper I rehearse Nuel Belnap and Mitchell Green’s argument in “Indeterminism and the Thin Red Line” against the reality of (...)
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  15. Greg Restall, Truth Values.
    I present an account of truth values for classical logic, intuitionistic logic, and the modal logic s5, in which truth values are not a fundamental category from which the logic is defined, but rather, an idealisation of more fundamental logical features in the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of logical consequence.
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  16. Greg Restall, What Are We to Accept, and What Are We to Reject.
    I’m delighted to have the opportunity to respond to Hartry Field’s Saving Truth From Paradox [3]. This is a wonderful book: it’s clear and precise, interesting and engaging, and deep and important all at once. Truth and the paradoxes..
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  17. Greg Restall & John Slaney, Technical Report TR-ARP-2-95.
    In this paper we consider the implications for belief revision of weakening the logic under which belief sets are taken to be closed. A widely held view is that the usual belief revision functions are highly classical, especially in being driven by consistency. We show that, on the contrary, the standard representation theorems still hold for paraconsistent belief revision. Then we give conditions under which consistency is preserved by revisions, and we show that this modelling allows for the gradual revision (...)
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  18. John K. Slaney, Robert K. Meyer & Greg Restall, Technical Report TR-ARP-2-96.
    In classical and intuitionistic arithmetics, any formula implies a true equation, and a false equation implies anything. In weaker logics fewer implications hold. In this paper we rehearse known results about the relevant arithmetic R, and we show that in linear arithmetic LL by contrast false equations never imply true ones. As a result, linear arithmetic is desecsed. A formula A which entails 0 = 0 is a secondary equation; one entailed by 0 6= 0 is a secondary unequation. A (...)
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  19. JC Beall & Greg Restall, Defending Logical Pluralism.
    We are pluralists about logical consequence [1]. We hold that there is more than one sense in which arguments may be deductively valid, that these senses are equally good, and equally deserving of the name deductive validity. Our pluralism starts with our analysis of consequence. This analysis of consequence is not idiosyncratic. We agree with Richard Jeffrey, and with many other philosophers of logic about how logical consequence is to be defined. To quote Jeffrey.
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  20. Robert K. Meyer & Greg Restall, “Strenge” Arithmetics.
    In Entailment, Anderson and Belnap motivated their modification E of Ackermann’s strenge Implikation Π Π’ as a logic of relevance and necessity. The kindred system R was seen as relevant but not as modal. Our systems of Peano arithmetic R# and omega arithmetic R## were based on R to avoid fallacies of relevance. But problems arose as to which arithmetic sentences were (relevantly) true. Here we base analogous systems on E to solve those problems. Central to motivating E is the (...)
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  21. Daniel Nolan, Greg Restall & Caroline West, Moral Fictionalism.
    What would morality have to be like in order to answer to our everyday moral concepts'? What are we committed to when we make moral claims such as "female infibulation is wrong"; or "we ought give money to famine relief"; or "we have a duty to not to harm others", and when we go on to argue about these sorts of claims'? It has seemed to many — and it seems plausible to us — that when we assert and argue (...)
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  22. Graham Priest & Greg Restall, Envelopes and Indifference.
    Consider this situation: Here are two envelopes. You have one of them. Each envelope contains some quantity of money, which can be of any positive real magnitude. One contains twice the amount of money that the other contains, but you do not know which one. You can keep the money in your envelope, whose numerical value you do not know at this stage, or you can exchange envelopes and have the money in the other. You wish to maximise your money. (...)
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  23. Greg Restall, Anti-Realist Classical Logic and Realist Mathematics.
    I sketch an application of a semantically anti-realist understanding of the classical sequent calculus to the topic of mathematics. The result is a semantically anti-realist defence of a kind of mathematical realism. In the paper, I begin the development of the view and compare it to orthodox positions in the philosophy of mathematics.
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  24. Greg Restall, Always More.
    Perhaps such a proposition is not expressible in any language that you or I speak, but – so a familiar story goes – it is decided by each world, so it plays just the role that other propositions do, so it counts as a proposition in the same way. In fact, we can see just how it counts as a proposition: given all the worlds in S, our proposition p says that the world is one of the worlds in S. (...)
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  25. Greg Restall, A S s E Rt I O N, Denial, Commitment, Entitlement, and Incompatibility (and Some Consequence).
    In this short paper, I compare and contrast the kind of symmetric treatment of negation favoured in different ways by Huw Price (in “Why ‘Not’?”) and by me (in “Multiple Conclusions”) with Robert Brandom’s analysis of scorekeeping in terms of commitment, entitlement and incompatibility. Both kinds of account are what Brandom calls a normative pragmatics. They are both semantic anti-realist accounts of meaning in the significance of vocabulary is explained in terms of our rule-governed (normative) practice (pragmatics). These accounts differ (...)
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  26. Greg Restall, Comparing Modal Sequent Systems.
    This is an exploratory and expository paper, comparing display logic formulations of normal modal logics with labelled sequent systems. We provide a translation from display sequents into labelled sequents. The comparison between different systems gives us a different way to understand the difference between display systems and other sequent calculi as a difference between local and global views of consequence. The mapping between display and labelled systems also gives us a way to understand labelled systems as properly structural and not (...)
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  27. Greg Restall, Carnap's Tolerance, Language Change and Logical Pluralism.
    In this paper, I distinguish different kinds of pluralism about logical consequence. In particular, I distinguish the pluralism about logic arising from Carnap’s Principle of Tolerance from a pluralism which maintains that there are different, equally “good” logical consequence relations on the one language. I will argue that this second form of pluralism does more justice to the contemporary state of logical theory and practice than does Carnap’s more moderate pluralism.
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  28. Greg Restall, Defining Double Negation Elimination.
    In his paper “Generalised Ortho Negation” [2] J. Michael Dunn mentions a claim of mine to the effect that there is no condition on ‘perp frames’ equivalent to the holding of double negation elimination ∼∼A A. That claim is wrong. In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of different theses.1..
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  29. Greg Restall, Logical Consequence.
    A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises. But in what sense do conclusions follow from premises? What is it for a conclusion to be a consequence of premises? Those questions, in many respects, are at the heart of logic (as a philosophical discipline). Consider the following argument: If we charge high fees for university, only the rich will enroll. We charge high fees for university. Therefore, only the rich will (...)
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  30. Greg Restall, Logics, Situations and Channels.
    The notion of that information is relative to a context is important in many different ways. The idea that the context is small — that is, not necessarily a consistent and complete possible world — plays a role not only in situation theory, but it is also an enlightening perspective from which to view other areas, such as modal logics, relevant logics, categorial grammar and much more. In this article we will consider these areas, and focus then on one further (...)
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  31. Greg Restall, Multiple Conclusions.
    Our topic is the notion of logical consequence: the link between premises and conclusions, the glue that holds together deductively valid argument. How can we understand this relation between premises and conclusions? It seems that any account begs questions. Painting with very broad brushtrokes, we can sketch the landscape of disagreement like this: “Realists” prefer an analysis of logical consequence in terms of the preservation of truth [29]. “Anti-realists” take this to be unhelpful and o:er alternative analyses. Some, like Dummett, (...)
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  32. Greg Restall, Models for Liars.
    As has been made clear in many of the papers in this volume, the crucial feature in Bradwardine’s theory of truth is the notion of signification. Expressed by a ‘connecticate’, which I shall write with the simple infix colon “:”, whenever t is a singular term and p is a sentence..
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  33. Greg Restall, Modal Models.
    There are many different approaches to the logic of truth. We could agree with Tarski, that the appropriate way to formalise a truth predicate is in a hierarchy, in which the truth predicate in one language can apply only to sentences from another language. Or, we could attempt to do without type restrictions on the truth predicate. Bradwardine’s theory of truth takes the second of these options: it is type-free, and admits sentences which say of themselves that they are not (...)
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  34. Greg Restall, Nietzsche, God and the Good Life.
    First, a few words of introduction, setting the scene. IÕm not a Nietzsche scholar. IÕm not even an historian of philosophy of any stripe. I am one of the fortunate few who are paid to Ôdo philosophyÕ, but the areas I tend to do most of my work in are logic, philosophy of language and some philosophy of religion. So why am I presenting a paper on Nietzsche? Well, there are at least two reasons. Firstly, I teach philosophy of religion, (...)
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  35. Greg Restall, Question.
    I suppose the natural way to interpret this question is something like “why do formal methods rather than anything else in philosophy” but in my case I’d rather answer the related question “why, given that you’re interested in formal methods, apply them in philosophy rather than elsewhere?” I started off my academic life as an undergraduate student in mathematics, because I was good at mathematics and studying it more seemed like a good idea at the time. I enjoyed mathematics a (...)
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  36. Greg Restall, Relevant and Substructural Logics.
    This essay is structured around the bifurcation between proofs and models: The first section discusses Proof Theory of relevant and substructural logics, and the second covers the Model Theory of these logics. This order is a natural one for a history of relevant and substructural logics, because much of the initial work — especially in the Anderson–Belnap tradition of relevant logics — started by developing proof theory. The model theory of relevant logic came some time later. As we will see, (...)
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  37. Greg Restall, Symbolic Logic.
    Symbolic logic is sited at intersection of philosophy, mathematics, linguistics and computer science. It deals with the structure of reasoning, and the formal features of information. Work in symbolic logic has almost exclusively treated the deductive validity of arguments: those arguments for which it is impossible for the premises to be true and the conclusion false. However, techniques from twentieth-century logic have found a place in the study of inductive or probabilistic reasoning, in which premises need not render their conclusions (...)
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  38. Greg Restall, Truth-Tellers in Bradwardine's Theory of Truth.
    Bradwardine’s account from Buridan’s [5, 6]. What are we to make of this? If the argument fails, what distinguishes problematic truth-tellers (such as a sentence that explicitly says of itself that it is true) from benign truth tellers? It is my task in this paper to explain this distinction, and to clarify the behaviour of truth-tellers, given my the contemporary formal treatment of Bradwardine’s account of signification.
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  39. Greg Restall, Technical Report Tr-Arp-11-94.
    Many logics in the relevant family can be given a proof theory in the style of Belnap's display logic Belnap 1982. However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modi ed proof theory which more closely models relevant logics. In addition, we use this proof theory to provide decidability proofs for a large range of substructural logics.
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  40. Greg Restall, Łukasiewicz, S U P E Rva L Uat I O N S,.
    A B S T R AC T: In this paper I consider an interpretation of future contingents which motivates a unification of a Łukasiewicz-style logic with the more classical supervaluational semantics. This in turn motivates a new non-classical logic modelling what is “made true by history up until now.” I give a simple Hilbert-style proof theory, and a soundness and completeness argument for the proof theory with respect to the intended models.
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  41. Greg Restall (forthcoming). Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star). Bulletin of the Section of Logic.
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  42. Greg Restall (forthcoming). Substructural Logics. Stanford Encyclopedia of Philosophy.
    summary of work in relevant in the Anderson– tradition.]; Mares Troestra, Anne, 1992, Lectures on , CSLI Publications [A quick, easy-to.
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  43. Gillian Russell & Greg Restall (forthcoming). Barriers to Implication. In Charles Pigden (ed.), Hume on Is and Ought. Palgrave MacMillan.
    The formulation and proof of Hume’s Law and several related inference barrier theses.
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  44. Greg Restall (2014). Pluralism and Proofs. Erkenntnis 79 (2):279-291.
    Beall and Restall’s Logical Pluralism (2006) characterises pluralism about logical consequence in terms of the different ways cases can be selected in the analysis of logical consequence as preservation of truth over a class of cases. This is not the only way to understand or to motivate pluralism about logical consequence. Here, I will examine pluralism about logical consequence in terms of different standards of proof. We will focus on sequent derivations for classical logic, imposing two different restrictions on classical (...)
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  45. Sergei S. Goncharov, Byunghan Kim & Greg Restall (2013). With Student Day on December 14, 2011. Bulletin of Symbolic Logic 19 (2).
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  46. Greg Restall (2013). Assertion, Denial and Non-Classical Theories. In. In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. 81--99.
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  47. Jc Beall, Ross Brady, J. Michael Dunn, A. P. Hazen, Edwin Mares, Robert K. Meyer, Graham Priest, Greg Restall, David Ripley, John Slaney & Richard Sylvan (2012). On the Ternary Relation and Conditionality. Journal of Philosophical Logic 41 (3):595 - 612.
    One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley-Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing (...)
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  48. Jc Beall, Ross Brady, Michael Dunn, Allen Hazen, Edwin Mares, John Slaney, Robert K. Meyer, Graham Priest, Greg Restall, David Ripley & Richard Sylvan (2012). On the Ternary Relation and Conditionality. Journal of Philosophical Logic 41 (3):595-612.
    One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley–Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing (...)
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  49. Greg Restall (2012). A Cut-Free Sequent System for Two-Dimensional Modal Logic, and Why It Matters. Annals of Pure and Applied Logic 163 (11):1611-1623.
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  50. Greg Restall & Gillian Kay Russell (eds.) (2012). New Waves in Philosophy. Palgrave Macmillan.
    Machine generated contents note: -- Series Editors' PrefaceAcknowledgementsNotes on ContributorsHow Things Are Elsewhere; W. Schwarz Information Change and First-Order Dynamic Logic; B.Kooi Interpreting and Applying Proof Theories for Modal Logic; F.Poggiolesi & G.Restall The Logic(s) of Modal Knowledge; D.Cohnitz On Probabilistically Closed Languages; H.Leitgeb Dogmatism, Probability and Logical Uncertainty; B.Weatherson & D.Jehle Skepticism about Reasoning; S.Roush, K.Allen & I.HerbertLessons in Philosophy of Logic from Medieval Obligations; C.D.Novaes How to Rule Out Things with Words: Strong Paraconsistency and the Algebra of Exclusion; (...)
     
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