31 found
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  1.  7
    A Problem in Pythagorean Arithmetic.Victor Pambuccian - 2018 - Notre Dame Journal of Formal Logic 59 (2):197-204.
    Problem 2 at the 56th International Mathematical Olympiad asks for all triples of positive integers for which ab−c, bc−a, and ca−b are all powers of 2. We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.
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  2.  23
    The Arithmetic of the Even and the Odd.Victor Pambuccian - 2016 - Review of Symbolic Logic 9 (2):359-369.
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  3.  7
    Addenda Et Corrigenda to “the Arithmetic of the Even and the Odd”.Stephen Menn & Victor Pambuccian - 2016 - Review of Symbolic Logic 9 (3):638-640.
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  4.  14
    A Reverse Analysis of the Sylvester-Gallai Theorem.Victor Pambuccian - 2009 - Notre Dame Journal of Formal Logic 50 (3):245-260.
    Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.
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  5.  9
    The Complexity of Plane Hyperbolic Incidence Geometry Is∀∃∀∃.Victor Pambuccian - 2005 - Mathematical Logic Quarterly 51 (3):277-281.
    We show that plane hyperbolic geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ∀∃∀, but that there is an axiom system, all of whose axioms are ∀∃∀∃ sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields.
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  6.  31
    The Simplest Axiom System for Plane Hyperbolic Geometry.Victor Pambuccian - 2004 - Studia Logica 77 (3):385 - 411.
    We provide a quantifier-free axiom system for plane hyperbolic geometry in a language containing only absolute geometrically meaningful ternary operations (in the sense that they have the same interpretation in Euclidean geometry as well). Each axiom contains at most 4 variables. It is known that there is no axiom system for plane hyperbolic consisting of only prenex 3-variable axioms. Changing one of the axioms, one obtains an axiom system for plane Euclidean geometry, expressed in the same language, all of whose (...)
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  7.  36
    Axiomatizations of Hyperbolic Geometry: A Comparison Based on Language and Quantifier Type Complexity.Victor Pambuccian - 2002 - Synthese 133 (3):331 - 341.
    Hyperbolic geometry can be axiomatized using the notions of order andcongruence (as in Euclidean geometry) or using the notion of incidencealone (as in projective geometry). Although the incidence-based axiomatizationmay be considered simpler because it uses the single binary point-linerelation of incidence as a primitive notion, we show that it issyntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type forallexistsforall, while the axiom system based on congruence and order can beformulated using only forallexists-axioms.
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  8.  4
    Axiomatizing Geometric Constructions.Victor Pambuccian - 2008 - Journal of Applied Logic 6 (1):24-46.
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  9.  26
    Ternary Operations as Primitive Notions for Constructive Plane Geometry.Victor Pambuccian - 1989 - Mathematical Logic Quarterly 35 (6):531-535.
  10.  17
    Early Examples of Resource-Consciousness.Victor Pambuccian - 2004 - Studia Logica 77 (1):81 - 86.
    As with the development of several logical notions, it is shown that the concept of resource-consciousness, i. e. the concern over the number of times that a given sentence is used in the proof of another sentence, has its origin in the foundations of geometry, pre-dating its appearence in logical circles as BCK-logic or affine logic.
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  11.  26
    Forms of the Pasch Axiom in Ordered Geometry.Victor Pambuccian - 2010 - Mathematical Logic Quarterly 56 (1):29-34.
    We prove that, in the framework of ordered geometry, the inner form of the Pasch axiom does not imply its outer form . We also show that OP can be properly split into IP and the weak Pasch axiom.
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  12.  17
    The Simplest Axiom System for Plane Hyperbolic Geometry Revisited.Victor Pambuccian - 2011 - Studia Logica 97 (3):347 - 349.
    Using the axiom system provided by Carsten Augat in [1], it is shown that the only 6-variable statement among the axioms of the axiom system for plane hyperbolic geometry (in Tarski's language L B =), we had provided in [3], is superfluous. The resulting axiom system is the simplest possible one, in the sense that each axiom is a statement in prenex form about at most 5 points, and there is no axiom system consisting entirely of at most 4-variable statements.
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  13.  12
    Constructive Axiomatizations of Plane Absolute, Euclidean and Hyperbolic Geometry.Victor Pambuccian - 2001 - Mathematical Logic Quarterly 47 (1):129-136.
    In this paper we provide quantifier-free, constructive axiomatizations for 2-dimensional absolute, Euclidean, and hyperbolic geometry. The main novelty consists in the first-order languages in which the axiom systems are formulated.
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  14.  13
    Groups and Plane Geometry.Victor Pambuccian - 2005 - Studia Logica 81 (3):387-398.
    We show that the first-order theory of a large class of plane geometries and the first-order theory of their groups of motions, understood both as groups with a unary predicate singling out line-reflections, and as groups acting on sets, are mutually inter-pretable.
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  15.  34
    Correction to “Axiomatizations of Hyperbolic Geometry”.Victor Pambuccian - 2005 - Synthese 145 (3):497-497.
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  16.  13
    Axiomatizations of Hyperbolic Geometry: A Comparison Based on Language and Quantifier Type Complexity.Victor Pambuccian - 2002 - Synthese 133 (3):331-341.
    Hyperbolic geometry can be axiomatized using the notions of order and congruence or using the notion of incidence alone. Although the incidence-based axiomatization may be considered simpler because it uses the single binary point-line relation of incidence as a primitive notion, we show that it is syntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type ∀∃∀, while the axiom system based on congruence and order can be formulated using only ∀∃-axioms.
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  17.  20
    Ternary Operations as Primitive Notions for Plane Geometry II.Victor Pambuccian - 1992 - Mathematical Logic Quarterly 38 (1):345-348.
    We proved in the first part [1] that plane geometry over Pythagorean fields is axiomatizable by quantifier-free axioms in a language with three individual constants, one binary and three ternary operation symbols. In this paper we prove that two of these operation symbols are superfluous.
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  18.  16
    Ternary Operations as Primitive Notions for Constructive Plane Geometry IV.Victor Pambuccian - 1994 - Mathematical Logic Quarterly 40 (1):76-86.
    In this paper we provide a quantifier-free constructive axiomatization for Euclidean planes in a first-order language with only ternary operation symbols and three constant symbols . We also determine the algorithmic theories of some ‘naturally occurring’ plane geometries.
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  19.  25
    Ternary Operations as Primitive Notions for Plane Geometry II.Victor Pambuccian - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):345-348.
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  20.  7
    Simplicity.Victor Pambuccian - 1988 - Notre Dame Journal of Formal Logic 29 (3):396-411.
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  21.  7
    Ternary Operations as Primitive Notions for Constructive Plane Geometry V.Victor Pambuccian - 1994 - Mathematical Logic Quarterly 40 (4):455-477.
    In this paper we provide a quantifier-free, constructive axiomatization of metric-Euclidean and of rectangular planes . The languages in which the axiom systems are expressed contain three individual constants and two ternary operations. We also provide an axiom system in algorithmic logic for finite Euclidean planes, and for several minimal metric-Euclidean planes. The axiom systems proposed will be used in a sequel to this paper to provide ‘the simplest possible’ axiom systems for several fragments of plane Euclidean geometry.
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  22.  19
    Constructive Axiomatization of Plane Hyperbolic Geometry.Victor Pambuccian - 2001 - Mathematical Logic Quarterly 47 (4):475-488.
    We provide a universal axiom system for plane hyperbolic geometry in a firstorder language with two sorts of individual variables, ‘points’ and ‘lines’ , containing three individual constants, A0, A1, A2, standing for three non-collinear points, two binary operation symbols, φ and ι, with φ = l to be interpreted as ‘[MATHEMATICAL SCRIPT SMALL L] is the line joining A and B’ , and ι = P to be interpreted as [MATHEMATICAL SCRIPT SMALL L]P is the point of intersection of (...)
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  23.  9
    Negation-Free and Contradiction-Free Proof of the Steiner–Lehmus Theorem.Victor Pambuccian - 2018 - Notre Dame Journal of Formal Logic 59 (1):75-90.
    By rephrasing quantifier-free axioms as rules of derivation in sequent calculus, we show that the generalized Steiner–Lehmus theorem admits a direct proof in classical logic. This provides a partial answer to a question raised by Sylvester in 1852. We also present some comments on possible intuitionistic approaches.
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  24.  21
    Sperner Spaces and First‐Order Logic.Andreas Blass & Victor Pambuccian - 2003 - Mathematical Logic Quarterly 49 (2):111-114.
    We study the class of Sperner spaces, a generalized version of affine spaces, as defined in the language of pointline incidence and line parallelity. We show that, although the class of Sperner spaces is a pseudo-elementary class, it is not elementary nor even ℒ∞ω-axiomatizable. We also axiomatize the first-order theory of this class.
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  25.  16
    Corrigendum to “The Complexity of Plane Hyperbolic Incidence Geometry is ∀∃∀∃”.Victor Pambuccian - 2008 - Mathematical Logic Quarterly 54 (6):668-668.
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  26.  18
    Ternary Operations as Primitive Notions for Constructive Plane Geometry VI.Victor Pambuccian - 1995 - Mathematical Logic Quarterly 41 (3):384-394.
    In this paper we provide quantifier-free, constructive axiomatizations for several fragments of plane Euclidean geometry over Euclidean fields, such that each axiom contains at most 4 variables. The languages in which they are expressed contain only at most ternary operations. In some precisely defined sense these axiomatizations are the simplest possible.
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  27.  11
    Ternary Operations as Primitive Notions for Constructive Plane Geometry III.Victor Pambuccian - 1993 - Mathematical Logic Quarterly 39 (1):393-402.
    This paper continues the investigations begun in [6] and continued in [7] about quantifier-free axiomatizations of plane Euclidean geometry using ternary operations. We show that plane Euclidean geometry over Archimedean ordered Euclidean fields can be axiomatized using only two ternary operations if one allows axioms that are not first-order but universal Lw1,w sentences. The operations are: the transport of a segment on a halfline that starts at one of the endpoints of the given segment, and the operation which produces one (...)
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  28.  14
    Another Constructive Axiomatization of Euclidean Planes.Victor Pambuccian - 2000 - Mathematical Logic Quarterly 46 (1):45-48.
    H. Tietze has proved algebraically that the geometry of uniquely determined ruler and compass constructions coincides with the geometry of ruler and set square constructions. We provide a new proof of this result via new universal axiom systems for Euclidean planes of characteristic ≠ 2 in languages containing only operation symbols.
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  29.  8
    Schatunowsky's Theorem, Bonse's Inequality, and Chebyshev's Theorem in Weak Fragments of Peano Arithmetic.Victor Pambuccian - 2015 - Mathematical Logic Quarterly 61 (3):230-235.
  30.  5
    On Definitions in an Infinitary Language.Victor Pambuccian - 2002 - Mathematical Logic Quarterly 48 (4):522-524.
    We provide the syntactic equivalent for the theorem stating that all epimorphisms of finite projective planes are isomorphisms. The definition of the inequality relation that we provide adds little to our understanding of the theorem, since its very validity can be discerned only from the validity of the model-theoretic theorem regarding epimorphisms.
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  31.  4
    The Sum of Irreducible Fractions with Consecutive Denominators Is Never an Integer in PA -.Victor Pambuccian - 2008 - Notre Dame Journal of Formal Logic 49 (4):425-429.
    Two results of elementary number theory, going back to Kürschák and Nagell, stating that the sums $\sum_{i=1}^k \frac{m_i}{n+i}$ (with $k\geq 1$, $(m_i, n+i)=1$, $m_i\lessthan n+i$) and $\sum_{i=0}^k \frac{1}{m+in}$ (with $n, m, k$ positive integers) are never integers, are shown to hold in $\mathrm{PA}^{-}$, a very weak arithmetic, whose axiom system has no induction axiom.
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