148 found
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  1.  21
    Synonymous Logics.Francis Jeffry Pelletier & Alasdair Urquhart - 2003 - Journal of Philosophical Logic 32 (3):259-285.
    This paper discusses the general problem of translation functions between logics, given in axiomatic form, and in particular, the problem of determining when two such logics are "synonymous" or "translationally equivalent." We discuss a proposed formal definition of translational equivalence, show why it is reasonable, and also discuss its relation to earlier definitions in the literature. We also give a simple criterion for showing that two modal logics are not translationally equivalent, and apply this to well-known examples. Some philosophical morals (...)
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  2.  4
    Functional Interpretations of Feasibly Constructive Arithmetic.Stephen Cook & Alasdair Urquhart - 1993 - Annals of Pure and Applied Logic 63 (2):103-200.
    A notion of feasible function of finite type based on the typed lambda calculus is introduced which generalizes the familiar type 1 polynomial-time functions. An intuitionistic theory IPVω is presented for reasoning about these functions. Interpretations for IPVω are developed both in the style of Kreisel's modified realizability and Gödel's Dialectica interpretation. Applications include alternative proofs for Buss's results concerning the classical first-order system S12 and its intuitionistic counterpart IS12 as well as proofs of some of Buss's conjectures concerning IS12, (...)
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  3.  23
    Semantics for Relevant Logics.Alasdair Urquhart - 1972 - Journal of Symbolic Logic 37 (1):159-169.
  4.  23
    Relevance Logic: Problems Open and Closed.Alasdair Urquhart - 2016 - Australasian Journal of Logic 13 (1).
    I discuss a collection of problems in relevance logic. The main problems discussed are: the decidability of the positive semilattice system, decidability of the fragments of R in a restricted number of variables, and the complexity of the decision problem for the implicational fragment of R. Some related problems are discussed along the way.
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  5.  79
    Supervaluation Fixed-Point Logics of Truth.Philip Kremer & Alasdair Urquhart - 2008 - Journal of Philosophical Logic 37 (5):407-440.
    Michael Kremer defines fixed-point logics of truth based on Saul Kripke’s fixed point semantics for languages expressing their own truth concepts. Kremer axiomatizes the strong Kleene fixed-point logic of truth and the weak Kleene fixed-point logic of truth, but leaves the axiomatizability question open for the supervaluation fixed-point logic of truth and its variants. We show that the principal supervaluation fixed point logic of truth, when thought of as consequence relation, is highly complex: it is not even analytic. We also (...)
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  6.  12
    The Undecidability of Entailment and Relevant Implication.Alasdair Urquhart - 1984 - Journal of Symbolic Logic 49 (4):1059-1073.
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  7.  20
    Distributive Lattices with a Dual Homomorphic Operation.Alasdair Urquhart - 1979 - Studia Logica 38 (2):201 - 209.
    The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail.
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  8. Many-Valued Logic.Alasdair Urquhart - 1986 - In D. Gabbay & F. Guenther (eds.), Handbook of Philosophical Logic, Vol. Iii. D. Reidel Publishing Co..
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  9.  11
    Failure of Interpolation in Relevant Logics.Alasdair Urquhart - 1993 - Journal of Philosophical Logic 22 (5):449 - 479.
    Craig's interpolation theorem fails for the propositional logics E of entailment, R of relevant implication and T of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids.
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  10.  16
    Handbook of Philosophical Logic.Alasdair Urquhart - 1987 - Canadian Journal of Philosophy 17 (2):483-489.
  11. The Boundary Between Mathematics and Physics.Alasdair Urquhart - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 407--416.
     
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  12. Beth's Definability Theorem in Relevant Logics.Alasdair Urquhart - 1999 - In E. Orłowska (ed.), Logic at Work. Heidelberg. pp. 229--234.
  13.  19
    Duality for Algebras of Relevant Logics.Alasdair Urquhart - 1996 - Studia Logica 56 (1-2):263 - 276.
    This paper defines a category of bounded distributive lattice-ordered grupoids with a left-residual operation that corresponds to a weak system in the family of relevant logics. Algebras corresponding to stronger systems are obtained by adding further postulates. A duality theoey piggy-backed on the Priestley duality theory for distributive lattices is developed for these algebras. The duality theory is then applied in providing characterizations of the dual spaces corresponding to stronger relevant logics.
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  14.  64
    The Complexity of Propositional Proofs.Alasdair Urquhart - 1995 - Bulletin of Symbolic Logic 1 (4):425-467.
    Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.
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  15.  61
    Anderson and Belnap's Invitation to Sin.Alasdair Urquhart - 2010 - Journal of Philosophical Logic 39 (4):453 - 472.
    Quine has argued that modal logic began with the sin of confusing use and mention. Anderson and Belnap, on the other hand, have offered us a way out through a strategy of nominahzation. This paper reviews the history of Lewis's early work in modal logic, and then proves some results about the system in which "A is necessary" is intepreted as "A is a classical tautology.".
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  16. The Complexity of Propositional Proofs.Alasdair Urquhart - 1995 - Bulletin of Symbolic Logic 1 (4):425-467.
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  17.  15
    An Interpretation of Many-Valued Logic.Alasdair Urquhart - 1973 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 19 (7):111-114.
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  18.  18
    Proof Theories for Semilattice Logics.Steve Giambrone & Alasdaire Urquhart - 1987 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (5):433-439.
  19. Intensional Languages Via Nominalization.Alasdair Urquhart - 1982 - Pacific Philosophical Quarterly 63 (2):186.
     
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  20.  15
    The Complexity of Decision Procedures in Relevance Logic II.Alasdair Urquhart - 1999 - Journal of Symbolic Logic 64 (4):1774-1802.
  21. The Theory of Types.Alasdair Urquhart - 2003 - In Nicholas Griffin (ed.), The Cambridge Companion to Bertrand Russell. Cambridge University Press. pp. 286--309.
     
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  22. Relevant Implication and Projective Geometry.Alasdair Urquhart - 1983 - Logique Et Analyse 26 (3):345-357.
  23.  22
    Mathematical Depth.Alasdair Urquhart - 2015 - Philosophia Mathematica 23 (2):233-241.
    The first part of the paper is devoted to surveying the remarks that philosophers and mathematicians such as Maddy, Hardy, Gowers, and Zeilberger have made about mathematical depth. The second part is devoted to the question of whether we can make the notion precise by a more formal proof-theoretical approach. The idea of measuring depth by the depth and bushiness of the proof is considered, and compared to the related notion of the depth of a chess combination.
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  24.  3
    An Interpretation of Many‐Valued Logic.Alasdair Urquhart - 1973 - Mathematical Logic Quarterly 19 (7):111-114.
  25.  24
    Beyond Analytic Philosophy.Alasdair Urquhart - 1987 - Canadian Journal of Philosophy 17 (2):477-482.
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  26. Four Variables Suffice.Urquhart Alasdair - 2007 - Australasian Journal of Logic 5 (3):66-73.
    What I wish to propose in the present paper is a new form of “career induction” for ambitious young logicians. The basic problem is this: if we look at the n-variable fragments of relevant propositional logics, at what point does undecidability begin? Focus, to be definite, on the logic R. John Slaney showed that the 0-variable fragment of R contains exactly 3088 non-equivalent propositions, and so is clearly decidable. In the opposite direction, I claimed in my paper of 1984 that (...)
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  27. The Beat of a Different Drum: The Life and Science of Richard Feynman.Alasdair Urquhart & Jagdish Mehra - 1997 - International Studies in the Philosophy of Science 11 (3).
  28.  5
    Matrix Identities and the Pigeonhole Principle.Michael Soltys & Alasdair Urquhart - 2004 - Archive for Mathematical Logic 43 (3):351-357.
    We show that short bounded-depth Frege proofs of matrix identities, such as PQ=I⊃QP=I (over the field of two elements), imply short bounded-depth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential-size bounded-depth Frege proofs, it follows that the propositional version of the matrix principle also requires bounded-depth Frege proofs of exponential size.
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  29.  24
    Completeness of Weak Implication.Alasdair I. F. Urquhart - 1971 - Theoria 37 (3):274-282.
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  30.  2
    Nuel Belnap: Doctoral Students.Carlos Giannoni, Robert Meyer, J. Michael Dunn, Peter Woodruff, James Garson, Kent Wilson, Dorothy Grover, Ruth Manor, Alasdair Urquhart & Garrel Pottinger - 1990 - In J. Dunn & A. Gupta (eds.), Truth or Consequences. Kluwer Academic Publishers.
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  31.  6
    Implicational Formulas in Intuitionistic Logic.Alasdair Urquhart - 1974 - Journal of Symbolic Logic 39 (4):661-664.
  32.  15
    Pavel Pudlák. Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction. Springer Monographs in Mathematics. Springer, 2013. ISBN: 978-3-319-00118-0 ; 978-3-319-00119-7 . Pp. Xiv + 695. [REVIEW]Alasdair Urquhart - 2015 - Philosophia Mathematica 23 (3):435-438.
  33.  32
    Henry M. Sheffer and Notational Relativity.Alasdair Urquhart - 2012 - History and Philosophy of Logic 33 (1):33 - 47.
    Henry M. Sheffer is well known to logicians for the discovery (or rather, the rediscovery) of the ?Sheffer stroke? of propositional logic. But what else did Sheffer contribute to logic? He published very little, though he is known to have been carrying on a rather mysterious research program in logic; the only substantial result of this research was the unpublished monograph The General Theory of Notational Relativity. The main aim of this paper is to explain, as far as possible (given (...)
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  34. The Collected Papers of Bertrand Russell, Volume 4: Foundations of Logic, 1903-05.Alasdair Urquhart & Albert C. Lewis (eds.) - 1994 - Routledge.
    First published in 1994. Routledge is an imprint of Taylor & Francis, an informa company.
     
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  35.  30
    The Unnameable.Alasdair Urquhart - 2008 - Canadian Journal of Philosophy 38 (S1):119-135.
    Hans Herzberger as a philosopher and logician has shown deep interest both in the philosophy of Gottlob Frege, and in the topic of the inexpressible and the ineffable. In the fall of 1982, he taught at the University of Toronto, together with André Gombay, a course on Frege's metaphysics, philosophy of language, and foundations of arithmetic. Again, in the fall of 1986, he taught a seminar on the philosophy of language that dealt with 'the limits of discursive symbolism in several (...)
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  36.  11
    Meaning and Modality. [REVIEW]Alasdair Urquhart - 1978 - Journal of Philosophy 75 (8):438-446.
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  37.  12
    Reputation Among Logicians as Being Essentially Trivial. I Hope to Convince the Reader That It Presents Some of the Most Challenging and Intriguing Problems in Modern Logic. Although the Problem of the Complexity of Propositional Proofs is Very Natural, It has Been Investigated Systematically Only Since the Late 1960s. [REVIEW]Alasdair Urquhart - 1995 - Bulletin of Symbolic Logic 1 (4).
  38.  20
    Decidability and the Finite Model Property.Alasdair Urquhart - 1981 - Journal of Philosophical Logic 10 (3):367 - 370.
  39.  21
    Distributive Lattices with a Dual Homomorphic Operation. II.Alasdair Urquhart - 1981 - Studia Logica 40 (4):391 - 404.
    An Ockham lattice is defined to be a distributive lattice with 0 and 1 which is equipped with a dual homomorphic operation. In this paper we prove: (1) The lattice of all equational classes of Ockham lattices is isomorphic to a lattice of easily described first-order theories and is uncountable, (2) every such equational class is generated by its finite members. In the proof of (2) a characterization of orderings of with respect to which the successor function is decreasing is (...)
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  40.  11
    Emil Post.Alasdair Urquhart - 2009 - In Dov Gabbay (ed.), The Handbook of the History of Logic. Elsevier. pp. 5--617.
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  41.  10
    X1. Introduction. The Classical Propositional Calculus has an Undeserved Reputation Among Logicians as Being Essentially Trivial. I Hope to Convince the Reader That It Presents Some of the Most Challenging and Intriguing Problems in Modern Logic. Although the Problem of the Complexity of Propositional Proofs is Very. [REVIEW]Alasdair Urquhart - 1995 - Bulletin of Symbolic Logic 1 (4).
  42.  26
    The Complexity of Analytic Tableaux.Noriko H. Arai, Toniann Pitassi & Alasdair Urquhart - 2006 - Journal of Symbolic Logic 71 (3):777 - 790.
    The method of analytic tableaux is employed in many introductory texts and has also been used quite extensively as a basis for automated theorem proving. In this paper, we discuss the complexity of the system as a method for refuting contradictory sets of clauses, and resolve several open questions. We discuss the three forms of analytic tableaux: clausal tableaux, generalized clausal tableaux, and binary tableaux. We resolve the relative complexity of these three forms of tableaux proofs and also resolve the (...)
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  43.  26
    Von Neumann, Gödel and Complexity Theory.Alasdair Urquhart - 2010 - Bulletin of Symbolic Logic 16 (4):516-530.
    Around 1989, a striking letter written in March 1956 from Kurt Gödel to John von Neumann came to light. It poses some problems about the complexity of algorithms; in particular, it asks a question that can be seen as the first formulation of the P=?NP question. This paper discusses some of the background to this letter, including von Neumann's own ideas on complexity theory. Von Neumann had already raised explicit questions about the complexity of Tarski's decision procedure for elementary algebra (...)
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  44.  31
    Decorated Linear Order Types and the Theory of Concatenation.Alasdair Urquhart & Albert Visser - 2010 - In F. Delon (ed.), Logic Colloquium 2007. Cambridge University Press. pp. 1.
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  45.  23
    The Depth of Resolution Proofs.Alasdair Urquhart - 2011 - Studia Logica 99 (1-3):349-364.
    This paper investigates the depth of resolution proofs, that is to say, the length of the longest path in the proof from an input clause to the conclusion. An abstract characterization of the measure is given, as well as a discussion of its relation to other measures of space complexity for resolution proofs.
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  46.  17
    A Semantical Theory of Analytic Implication.Alasdair Urquhart - 1973 - Journal of Philosophical Logic 2 (2):212 - 219.
  47.  27
    Synonymous Logics: A Correction. [REVIEW]Francis Jeffry Pelletier & Alasdair Urquhart - 2008 - Journal of Philosophical Logic 37 (1):95 - 100.
    In an earlier paper entitled Synonymous Logics, the authors attempted to show that there are two modal logics so that each is exactly translatable into the other, but they are not translationally equivalent. Unfortunately, there is an error in the proof of this result. The present paper provides a new example of two such logics, and a proof of the result claimed in the earlier paper.
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  48.  24
    Mathematical Methods in Philosophy: Editors' Introduction.Aldo Antonelli, Alasdair Urquhart & Richard Zach - 2008 - Review of Symbolic Logic 1 (2):143-145.
    Mathematics and philosophy have historically enjoyed a mutually beneficial and productive relationship, as a brief review of the work of mathematician–philosophers such as Descartes, Leibniz, Bolzano, Dedekind, Frege, Brouwer, Hilbert, Gödel, and Weyl easily confirms. In the last century, it was especially mathematical logic and research in the foundations of mathematics which, to a significant extent, have been driven by philosophical motivations and carried out by technically minded philosophers. Mathematical logic continues to play an important role in contemporary philosophy, and (...)
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  49.  15
    From Berkeley to Bourbaki.Alasdair Urquhart - 1999 - Dialogue 38 (03):587-.
    This has been a great century for logic and the foundations of mathematics. Ewald's excellent sourcebook is a welcome addition to the literature on the exciting developments of this and the past two centuries. The richness of the material on which Ewald is drawing is shown by the fact that he has assembled a broad and representative selection without once duplicating anything to be found in the famous sourcebooks of van Heijenoort and Benacerraf/Putnam.
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  50.  17
    Books Received. [REVIEW]Janusz Czelakowski, Alasdair Urquhart, Ryszard Wójcicki, Jan Woleński, Andrzej Sendlewski & Marcin Mostowski - 1990 - Studia Logica 49 (1):151-161.
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