School of Historical and Philosophical Studies - Research Publications

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    Geometric Models for Relevant Logics
    Restall, G ; Duntsch, I ; Mares, E (Springer International Publishing, 2022-01-01)
    Alasdair Urquhart’s work on models for relevant logics is distinctive in a number of different ways. One key theme, present in both his undecidability proof for the relevant logic R (Urquhart 1984) and his proof of the failure of interpolation in R (Urquhart 1993), is the use of techniques from geometry (Urquhart 2019). In this paper, inspired by Urquhart’s work, I explore ways to generate natural models of R+ from geometries, and different constraints that an accessibility relation in such a model might satisfy. I end by showing that a set of natural conditions on an accessibility relation, motivated by geometric considerations, is jointly unsatisfiable.
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    Negation on the Australian Plan
    Berto, F ; Restall, G (SPRINGER, 2019-12)
    We present and defend the Australian Plan semantics for negation. This is a comprehensiveaccount,suitableforavarietyofdifferentlogics.Itisbasedontwoideas.The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilitiesbetweensuchpoints.WedefuseanumberofobjectionstothisPlan,raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan.
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    Normal Proofs, Cut Free Derivations and Structural Rules
    Restall, G (Springer Netherlands, 2014)
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    First Degree Entailment, Symmetry and Paradox
    Restall, G (Nicolaus Copernicus, 2017-03-01)
    Here is a puzzle, which I learned from Terence Parsons in his “True Contradictions” [8]. First Degree Entailment (fde) is a logic which allows for truth value gaps as well as truth value gluts. If you are agnostic between assigning paradoxical sentences gaps and gluts (and there seems to be no very good reason to prefer gaps over gluts or gluts over gaps if you’re happy with fde), then this looks no different, in effect, from assigning them a gap value? After all, on both views you end up with a theory that doesn’t commit you to the paradoxical sentence or its negation. How is the fde theory any different from the theory with gaps alone? In this paper, I will present a clear answer to this puzzle an answer that explains how being agnostic between gaps and gluts is a genuinely different position than admitting gaps alone, by using the formal notion of a bi-theory, and showing that while such positions might agree on what is to be accepted, they differ on what is to be rejected.
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    Truth-Tellers in Bradwardine’s Theory of Truth
    Restall, G ; Kann, C ; Loewe, B ; Rode, C ; Uckelman, SL (Peeters, 2018)
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    Review of Advances in Proof-Theoretic Semantics
    RESTALL, G (University of Notre Dame, 2016-05-15)
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    Fixed-Point Models for Theories of Properties and Classes
    RESTALL, G (Victoria University of Wellington, 2017)
    There is a vibrant (but minority) community among philosophical logicians seeking to resolve the paradoxes of classes, properties and truth by way of adopting some non-classical logic in which trivialising paradoxical arguments are not valid. There is also a long tradition in theoretical computer science–going back to Dana Scott’s fixed point model construction for the untyped lambda-calculus–of models allowing for fixed points. In this paper, I will bring these traditions closer together, to show how these model constructions can shed light on what we could hope for in a non-trivial model of a theory for classes, properties or truth featuring fixed points.
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    Assertion, Denial, Accepting, Rejecting, Symmetry and Paradox
    RESTALL, G ; Caret, C ; Hjortland, O (Oxford University Press, 2015)
    Proponents of a dialethic or “truth-value glut” response to the paradoxes of self-reference argue that “truth-value gap” analyses of the paradoxes fall foul of the extended 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 not be expressible by negation, and glut approaches take some negations to not express denials. But in the light of this symmetry, considerations against a gap view point to parallel considerations against a glut view. Those who find some reason to prefer one view over another (and this is almost everyone) must find some reason to break this symmetry.
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    Pluralism and Proofs
    Restall, G (Kluwer Academic Publishers, 2014)
    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 derivations to produce derivations for intuitionistic logic and for dual intuitionistic logic. The result is another way to understand the manner in which we can have different consequence relations in the same language. Furthermore, the proof-theoretic perspective gives us a different explanation of how the one concept of negation can have three different truth conditions, those in classical, intuitionistic and dual-intuitionistic models. © 2014 Springer Science+Business Media Dordrecht.